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3 There was a celebrated Fourier at the Academy of Science, whom posterity has forgotten; and in some garret an obscure Fourier, whom the future will recall. Victor Hugo Les Misèrables Fourier Analysis and Synthesis 3.1 Introduction 3.9 Applications 3.2 Fourier Series Spectrogram 3.3 Fourier Transform FFT Based Digital Filters 3.4 Discrete Fourier Transform Digital BASS for Audio 3.5 Short Time Fourier Transform Radar Signal Processing 3.6 Fast Fourier Transform Pitch Modification 3.7 2-D Discrete Fourier Transform Image Compression 3.8 Discrete Cosine Transfrom Watermarking Bibliography, Exercises ean Baptiste Joseph Fourier (1768-1830) studied the mathematical theory of heat conduction in his major work, The Analytic Theory of Heat, (Théorie analytique de la chaleur). He established the partial differential equation governing heat diffusion and solved it using an infinite series of trigonometric functions. The description of a signal in terms of elementary trigonometric functions had a profound effect on the way signals are analysed. The Fourier method is the most extensively applied signal-processing tool. This is because the transform output lends itself to easy interpretation and manipulation, and leads to the concept of frequency analysis. Furthermore even biological systems such as the human auditory system perform some form of frequency analysis of the input signals. This chapter begins with an introduction to the complex Fourier series and the Fourier transform, and then considers the discrete Fourier transform, the Fast Fourier transform, the 2-D Fourier transform and the discrete cosine transform. Important engineering issues such as the trade-off between the time and frequency resolutions, problems with finite data length, windowing and spectral leakage are considered. The applications of the Fourier transform include filtering. telecommunication, music processing, pitch modification, signal coding and signal synthesis feature extraction for pattern identification as in speech recognition, image processing, spectral analysis in astrophysics, radar signal processing. e jk ω 0t k ω 0 t R e I m J

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Page 1: Fourier Analysis and Synthesis - umu.se · PDF fileFourier Analysis and Synthesis ... studied the mathematical theory of ... signals are sinusoidal signals with different periods giving

3

There was a celebrated Fourier at the Academy of Science, whom posterity has forgotten; and in some garret an

obscure Fourier, whom the future will recall. Victor Hugo Les Misèrables

Fourier Analysis and Synthesis 3.1 Introduction 3.9 Applications 3.2 Fourier Series Spectrogram 3.3 Fourier Transform FFT Based Digital Filters 3.4 Discrete Fourier Transform Digital BASS for Audio 3.5 Short Time Fourier Transform Radar Signal Processing 3.6 Fast Fourier Transform Pitch Modification 3.7 2-D Discrete Fourier Transform Image Compression 3.8 Discrete Cosine Transfrom Watermarking Bibliography, Exercises

ean Baptiste Joseph Fourier (1768-1830) studied the mathematical theory of heat conduction in his major work, The Analytic Theory of Heat, (Théorie analytique de la chaleur). He established the partial differential equation governing heat diffusion and solved it using an infinite series of trigonometric

functions. The description of a signal in terms of elementary trigonometric functions had a profound effect on the way signals are analysed. The Fourier method is the most extensively applied signal-processing tool. This is because the transform output lends itself to easy interpretation and manipulation, and leads to the concept of frequency analysis. Furthermore even biological systems such as the human auditory system perform some form of frequency analysis of the input signals. This chapter begins with an introduction to the complex Fourier series and the Fourier transform, and then considers the discrete Fourier transform, the Fast Fourier transform, the 2-D Fourier transform and the discrete cosine transform. Important engineering issues such as the trade-off between the time and frequency resolutions, problems with finite data length, windowing and spectral leakage are considered. The applications of the Fourier transform include filtering. telecommunication, music processing, pitch modification, signal coding and signal synthesis feature extraction for pattern identification as in speech recognition, image processing, spectral analysis in astrophysics, radar signal processing.

ejkω0t

kω0t

Re

I m

J

Page 2: Fourier Analysis and Synthesis - umu.se · PDF fileFourier Analysis and Synthesis ... studied the mathematical theory of ... signals are sinusoidal signals with different periods giving

2 Chap.3 Fourier Analysis and Synthesis

3.1 Introduction The objective of signal transformation is to express a signal as a combination of a set of basic “building block” signals, known as the basis functions. The transform output should lend itself to convenient analysis, interpretation and manipulation. A useful consequence of transforms, such as the Fourier and the Laplace, is that differential analysis on the time domain signal become simple algebraic operations on the transformed signal. In the Fourier transform the basic building block signals are sinusoidal signals with different periods giving rise to the concept of frequency. In Fourier analysis a signal is decomposed into its constituent sinusoids, i.e. frequencies, the amplitudes of various frequencies form the so-called frequency spectrum of the signal. In an inverse Fourier transform operation the signal can be synthesised by adding up its constituent frequencies. It turns out that many signals that we encounter in daily life such as speech, car engine noise, bird songs, music etc. have a periodic or quasi-periodic structure, and that the cochlea in the human hearing system performs a kind of harmonic analysis of the input audio signals. Therefore the concept of frequency is not a purely mathematical abstraction in that biological and physical systems have also evolved to make use of the frequency analysis concept. The power of the Fourier transform in signal analysis and pattern recognition is its ability to reveals spectral structures that may be used to characterise a signal. This is illustrated in Fig. 3.1 for the two extreme cases of a sine wave and a purely random signal. For a periodic signal the power is concentrated in extremely narrow bands of frequencies indicating the existence of structure and the predictable character of the signal. In the case of a pure sine wave as shown in Fig. 3.1.a the signal power is concentrated in one frequency. For a purely random signal as shown in Fig 3.1.b the signal power is spread equally in the frequency domain indicating the lack of structure in the signal.

t

x(t)

PXX(f)

t

(a)

x(t)

PXX(f)

(b)

Figure 3.1 The concentration or spread of power in frequency indicates the correlated orrandom character of a signal: (a) a predictable signal, (b) a random signal.

Page 3: Fourier Analysis and Synthesis - umu.se · PDF fileFourier Analysis and Synthesis ... studied the mathematical theory of ... signals are sinusoidal signals with different periods giving

Sec. 3.2 Fourier Series 3

Notation: In this chapter the symbols t and m denote continuous and discrete time variables, and f and k denote continuous and discrete frequency variables respectively. The variable ω=2πf denotes the angular frequency in units of rad/s and is used interchangably (within a scaling of factor of 2π) with the frequency variable f in units of Hz. 3.2 Fourier Series: Representation of Periodic Signals The following three sinusoidal functions form the basis functions for the Fourier analysis

ttx 01 cos)( ω= (3.1) ttx 02 sin)( ω= (3.2)

tetttx 0j003 sinjcos)( ωωω =+=

(3.3)

Fig. 3.2.a shows the cosine and the sine components of the complex exponential (cisoidal) signal of Eq. (3.3), and Fig. 3.2.b shows a vector representation of the complex exponential in a complex plane with real (Re) and imaginary (Im) dimensions. The Fourier basis functions are periodic with an angular frequency of ω0 rad/s and a period of T0=2π/ω0=1/F0 seconds, where F0 is the frequency in Hz. The following properties make the sinusoids the ideal choice as the elementary building block basis functions for signal analysis and synthesis: (i) Orthogonality; two sinusoidal functions of different frequencies have the

following orthogonal property :

ejk ω0t

kω0t

Re

I m

t

sin(kωot) cos(kωot)

T0 (a) (b) Figure 3.2 - Fourier basis functions: (a) real and imaginary parts of a complex sinusoid,

(b) vector representation of a complex exponential.

Page 4: Fourier Analysis and Synthesis - umu.se · PDF fileFourier Analysis and Synthesis ... studied the mathematical theory of ... signals are sinusoidal signals with different periods giving

4 Chap.3 Fourier Analysis and Synthesis

0)cos(21)cos(

21)sin()sin( 212121 =−++−= ∫∫∫

∞−

∞−

∞−

dttdttdttt ωωωωωω

(3.4)

For harmonically related sinusoids the integration can be taken over one period. Similar equations can be derived for the product of cosines, or sine and cosine, of different frequencies. Orthogonality implies that the sinusoidal basis functions are independent and can be processed independently. For example in a graphic equaliser we can change the relative amplitudes of one set of frequencies, such as the bass, without affecting other frequencies, and in subband coding different frequency bands are coded independently and allocated different number of bits.

(ii) Sinusoidal functions are infinitely differentiable. This is important, as most signal analysis, synthesis and manipulation methods require the signals to be differentiable.

(iii) Sine and cosine signals of the same frequency have only a phase difference of π/2 or equivalently a relative time delay of a quarter of one period i.e. T0/4.

Associated with the complex exponential function te 0jω is a set of harmonically related complex exponential of the form

[ ]K,,,,1 000 3j2jj ttt eee ωωω ±±± (3.5) The set of exponential signals in Eq. (3.5) are periodic with a fundamental frequency ω0=2π/T0=2πF0 where T0 is the period and F0 is the fundamental frequency. These signals form the set of basis functions for the Fourier analysis. Any linear combination of these signals of the form

∑∞

−∞=k

tjkk ec 0ω (3.6)

is also periodic with a period of T0. Conversely any periodic signal x(t) can be synthesised from a linear combination of harmonically related exponentials. The Fourier series representation of a periodic signal are given by the following synthesis and analysis equations:

LL ,1,0,1)( 0j −== ∑∞

−∞=

kectxk

tkk

ω Synthesis equation (3.7)

Page 5: Fourier Analysis and Synthesis - umu.se · PDF fileFourier Analysis and Synthesis ... studied the mathematical theory of ... signals are sinusoidal signals with different periods giving

Sec. 3.2 Fourier Series 5

LL ,1,0,1)(12/

2/

j

0

0

0

0 −== ∫−

− kdtetxT

cT

T

tkk

ω

Analysis equation (3.8)

The complex-valued coefficient ck conveys the amplitude (a measure of the strength) and the phase of the frequency content of the signal at kω0 Hz. Note from the analysis Eq. (3.8), that the coefficient ck may be interpreted as a measure of the correlation of the signal x(t) and the complex exponential tke 0j ω− . The set of complex coefficients … c−1, c0, c1, … are known as the signal spectrum. Eq. (3.7) is referred to as the synthesis equation, and can be used as a frequency synthesizer (as in music synthesizers) to generate a signal as a weighted combination of its elementary frequencies. The representation of a signal in the form of Eq. (3.7) as the sum of its constituent harmonics is also referred to as the complex Fourier series representation. Note from Eqs. (3.7) and (3.8) that the complex exponentials that form a periodic signal occur only at discrete frequencies which are integer multiples, i.e. harmonics, of the fundamental frequency ω0. Therefore the spectrum of a periodic signal, with a period of T0, is discrete in frequency with discrete spectral lines spaced at integer multiples of ω0=2π/T0. Example 3.1 Given the Fourier synthesis Eq. (3.7), obtain the frequency analysis Eq. (3.8). Solution: Multiply both sides of Eq. (3.7) by tjme 0ω− and integrate over one period to obtain

∑ ∫∑ ∫∫∞

−∞= −

−∞

−∞= −

− ==k

T

T

tmkk

k

T

T

tjmtkk

T

T

tm dtecdteecdtetx2/

2/

)(j2/

2/

j2/

2/

j0

0

0

0

0

00

0

0

0)( ωωωω

(3.9)

From the orthogonality principle the integral in the r.h.s of Eq. (3.9) is zero unless k=m in which case the integral is equal to T0. Hence

∫−

−=2/

2/0

0

0

0)(1T

T

tjmm dtetx

Tc ω

(3.10)

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6 Chap.3 Fourier Analysis and Synthesis

Example 3.2 Find the frequency spectrum of a 1 kHz sinewave shown in Fig 3.3.a

∞<<∞−= tttx )2000sin()( π (3.11)

Solution A: The Fourier synthesis Eq. (3.7) can be written as

LL ++++== −−

−∞=∑ tt

k

tkk eccecectx πππ 2000j

102000j

12000j)(

(3.12)

now the sine wave can be expressed as

tt eettx πππ 2000j2000j

j21

j21)2000sin()( −−== (3.13)

Equating the coefficients of Eqs. (3.12) and (3.13) yields

j21

1=c , j2

11 −=−c and 01=±≠kc ( 3.14)

Fig 3.3.b shows the magnitude spectrum of the sinewave, where the spectral lines c1 and c−1 correspond to the 1 kHz and −1 kHz frequencies respectively.

Solution B: Substituting tt eet πππ 2000j2000j

2j1

2j1)2000sin( −−= in the Fourier

analysis Eq. (3.8) yields

∫∫

+−

−−

−=

−=

2/

2/

2000)1(j

0

2/

2/

2000)1(

0

2/

2/

2000j2000j2000

0

0

0

0

0

0

0

j21

j21

j21

j211

T

T

tkT

T

tkj

T

T

tkttjk

dteT

dteT

dteeeT

c

ππ

πππ

(3.15)

x(t)

t

| ck |

f HT0=1 ms

-103 103

k-1 1

(a) (b)

Figure 3.3 - A sinewave and its magnitude spectrum.

Page 7: Fourier Analysis and Synthesis - umu.se · PDF fileFourier Analysis and Synthesis ... studied the mathematical theory of ... signals are sinusoidal signals with different periods giving

Sec. 3.2 Fourier Series 7

Since sine and cosine functions are positive-valued over one half a period and negative-valued over the other half, it follows that Eq. (3.15) is zero unless k=1 or k=−1.

j21

1=c and j2

11 −=−c and 01 =±≠kc (3.16)

Example 3.3 Find the frequency spectrum of a periodic train of pulses with amplitude of 1.0, a period of 1.o kHz and a pulse 'on' duration of 0.3 milliseconds. Solution: The pulse period T0=1/F0=0.001 s, and the angular frequency ω0=2πF0=2000π rad/s. Substituting the pulse signal in the Fourier analysis Eq. (3.8) gives

kk

kee

ke

dtedtetxT

c

kkt

t

tk

tkT

T

tkk

ππ

ππ

πππ

πω

)3.0sin(2j2j

001.01)(1

3.0j3.0j00015.0

00015.0

2000j

00015.0

00015.0

2000j2/

2/

j

0

0

0

0

=−

=−

=

==

−=

−=

− ∫∫ (3.17)

For k=0 as c0=sin(0)/0 is undefined, differentiate the numerator and denominator of Eq. (3.17) w.r.t. to the variable k (strictly this can only be done for a continuous variable k i.e. when the period T0 tends to infinity) to obtain

3.0)03.0cos(3.00 ==

πππc

(3.18

x(t)

t

c(k)

k2

-0 .1

-0 .05

0

0.05

0.1

0.15

0.2

0.25

0.3

1

Figure 3.4 - A rectangular pulse train and its discrete frequency ‘line’ spectrum.

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8 Chap.3 Fourier Analysis and Synthesis

Example 3.4 For the example 3.3 write the formula for synthesising the signal up to the Nth harmonic, and plot a few examples for the increasing number of harmonics. Solution: The equation for the synthesis of a signal upto the Nth harmonic content is given by

∑∑∑

=

=

=

=

−−

=−=

−+=

−−+

+++=

++==

N

kkk

N

kkk

N

kkk

N

k

tkk

N

k

tkk

N

Nk

tkk

tkctkcc

tktkcc

tktkccc

ececcectx

1000

01

0

01

00

1

j

1

j0

j

)]sin()Im(2)cos()Re(2[

)]sin(j))][cos(Im(j)[Re(

)]sin(j))][cos(Im(j)[Re(

)( 000

ωω

ωω

ωω

ωωω

(3.19)

The following MatLab code generates a synthesised pulse train composed of N harmonics. In this example there are 5 cycles in an array of 1000 samples. Fig. 3.5 shows the waveform for the number of harmonics equal to; 1,3,6, and 100. NHarmonics=100; Ncycles=5; Nsamples=1000; y(1:Nsamples)=0.3; j=1:Nsamples; for k=1:NHarmonics x(j)=(2*sin(0.3*pi*k)/(pi*k))*cos(k*2*pi*Ncycles*j/Nsamples); y=y+x; plot(y); pause; end

(a) (b)

(c) (d)

x(t) x(t)

x(t) x(t)t t

t t Figure 3.5 Illustration of the Fourier synthesis of a periodic pulse train, and the Gibbs

phenomenon, with the increasing number of harmonics in the Fourier synthesis: (a) N=1, (b) N=3, (c) N=6, and (d) N=100.

Page 9: Fourier Analysis and Synthesis - umu.se · PDF fileFourier Analysis and Synthesis ... studied the mathematical theory of ... signals are sinusoidal signals with different periods giving

Sec. 3.3 Fourier Transform 9

3.2.1 Fourier Synthesis of Discontinuous Signals: Gibbs Phenomenon The sinusoidal basis functions of the Fourier transform are smooth and infinitely differentiable. In the vicinity of a discontinuity the Fourier synthesis of a signal exhibits ripples as shown in the Fig 3.5. The peak amplitude of the ripples does not decrease as the number of harmonics used in the signal synthesis increases. This behaviour is known as the Gibbs phenomenon. For a discontinuity of unity height, the partial sum of the harmonics exhibits a maximum value of 1.09 (that is an overshoot of 9%) irrespective of the number of harmonics used in the Fourier series. As the number of harmonics used in the signal synthesis increases, the ripples become compressed toward the discontinuity but the peak amplitude of the ripples remains constant. 3.3 Fourier Transform: Representation of Aperiodic Signals The Fourier series representation of periodic signals consist of harmonically related spectral lines spaced at the integer multiples of the fundamental frequency. The Fourier representation of aperiodic signals can be developed by regarding an aperiodic signal as a special case of a periodic signal with an infinite period. If the period of a signal is infinite, then the signal does not repeat itself and is aperiodic. Now consider the discrete spectra of a periodic signal with a period of T0, as

x(t)

t k

X(f)

c(k)

x(t)

t f

(a)

(b)

Ton

Toff

∞=offT

T0=Ton+Toff

1T0

Figure 3.6 – (a) A periodic pulse train and its line spectrum, (b) a single pulse from the periodic train in (a) with an imagined ‘off’ duration of infinity; its spectrum is the envelope

of the spectrum of the periodic signal in (a).

Page 10: Fourier Analysis and Synthesis - umu.se · PDF fileFourier Analysis and Synthesis ... studied the mathematical theory of ... signals are sinusoidal signals with different periods giving

10 Chap.3 Fourier Analysis and Synthesis

shown in Fig. 3.6.a. As the period T0 is increased, the fundamental frequency F0=1/T0 decreases, and successive spectral lines become more closely spaced. In the limit as the period tends to infinity (i.e. as the signal becomes aperiodic) the discrete spectral lines merge and form a continuous spectrum. Therefore the Fourier equations for an aperiodic signal, (known as the Fourier transform), must reflect the fact that the frequency spectrum of an aperiodic signal is continuous. Hence to obtain the Fourier transform relation the discrete-frequency variables and operations in the Fourier series Eqs. (3.7) and (3.8) should be replaced by their continuous-frequency counterparts. That is the discrete summation sign Σ should be replaced by the continuous summation integral ∫ ,

the discrete harmonics of the fundamental frequency kF0 should be replaced by the continuous frequency variable f, and the discrete frequency spectrum ck must be replaced by a continuous frequency spectrum say )( fX . The Fourier synthesis and analysis equations for aperiodic signals, the so-called Fourier transform pair, are given by

∫∞

∞−

= dfefXtx ftπ2j)()( (3.20)

∫∞

∞−

−= dtetxfX ftπ2j)()( (3.21)

Note from Eq. (3.21), that )( fX may be interpreted as a measure of the

correlation of the signal x(t) and the complex sinusoid fte π2j− . The condition for existence and computability of the Fourier transform integral of a signal x(t) is that the signal must have finite energy

∞<∫∞

∞−

dttx 2)( (3.22)

Example 3.5 Derivation of inverse Fourier transform Given the Fourier transform Eq. (3.21) derive the Fourier synthesis Eq. (3.20). Solution: Consider the Fourier analysis Eq. (3.8) for a periodic signal and Eq. (3.21) for its non-periodic version (consisting of one period only). Comparing these equations reproduced below

∫−

−=2/

2/

2j

0

0

0

0)(1T

T

tkFk dtetx

Tc π

∫∞

∞−

−= dtetxfX ftπ2j)()(

we have

Page 11: Fourier Analysis and Synthesis - umu.se · PDF fileFourier Analysis and Synthesis ... studied the mathematical theory of ... signals are sinusoidal signals with different periods giving

Sec. 3.3 Fourier Transform 11

∞→= 000

as)(1 TkFXT

ck

(3.23)

where F0=1/T0. Using Eq. (3.23) the Fourier synthesis Eq. (3.7) for a periodic signal can be rewritten as

FeTkXtxk

tkF ∆= ∑∞

−∞=

02j0 )/()( π

(3.24)

where ∆F=1/T0=F0 is the frequency spacing between successive spectral lines of the spectrum of a periodic signal as shown in Fig. 3.6. Now as the period T0 tends to infinity, ∆F=1/T0 tends to zero, then the discrete frequency variables kF0 and ∆F should be replaced by a continuous frequency variable f, and the discrete summation sign by the continuous integral sign. Thus Eq. (3.24) becomes

∫∞

∞−∞→⇒ dfefXtx ft

T

π2j)()(0

(3.25)

Example 3.6 The spectrum of an Impulse Function Consider the unit-area pulse p(t) shown in Fig 3.7.a. As the pulse width ∆ tends to zero the pulse tends to an impulse. The impulse function shown in Fig 3.7.b is defined as a pulse with an infinitesimal time width as

∆>

∆≤∆==

→∆ 2/0

2//1)(limit)(

0 t

ttptδ (3.26)

p(t)

t∆

δ(t)

t

∆(f)

f

1/∆

(a) (b) (c)

As ∆ 0

Figure 3.7 (a) A unit-area pulse, (b) the pulse becomes an impulse as 0→∆ , (c) the

spectrum of the impulse function.

Page 12: Fourier Analysis and Synthesis - umu.se · PDF fileFourier Analysis and Synthesis ... studied the mathematical theory of ... signals are sinusoidal signals with different periods giving

12 Chap.3 Fourier Analysis and Synthesis

It is easy to see that the integral of the impulse function is given by

11)( =∆

×∆=∫∞

∞−

dttδ

(3.27)

The important sampling property of the impulse function is defined as

)()()( TxdtTttx =−∫∞

∞−

δ (3.28)

The Fourier transform of the impulse function is obtained as

1)()( 02 ===∆ ∫∞

∞−

− edtetf ftj πδ

(3.29)

The impulse function is used as a test function to obtain the impulse response of a system. This is because as shown in Fig 3.7.c an impulse is a spectrally rich signal containing all frequencies in equal amounts. Example 3.7 Find the spectrum of the 10 kHz periodic pulse train shown in Fig. 3.8.a with an amplitude of 1 mV and expressed as

∑∑∞

−∞=

−∞=→∆

−=∆×−×=mm

mTtAmTtpAtx )()(limit)( 000δ

(3.30)

where p(t) is a unit-area pulse with width ∆ as shown in Fig. 3.7.a, and the

function δ(t-mT0) is now assumed to have a unit amplitude representing the area

under the impulse. Solution: T0=1/F0=0.1 ms and A=1 mV. For the time interval −T0/2<t<T0/2 x(t)=Aδ(t), hence we have

T0 1/T0

... ...t

)(tx kc

k (a) (b)

Figure 3.8 A periodic impulse train x(t) and its Fourier series spectrum ck.

Page 13: Fourier Analysis and Synthesis - umu.se · PDF fileFourier Analysis and Synthesis ... studied the mathematical theory of ... signals are sinusoidal signals with different periods giving

Sec. 3.3 Fourier Transform 13

V10)(1)(1

0

2/0

2/0

0j

0

2/0

2/0

0j

0

==== ∫∫−

TAdtetA

Tdtetx

Tc

T

T

tkT

T

tkk

ωω δ

(3.31)

As shown in Fig. 3.8.b the spectrum of a periodic impulse train in time is a periodic impulse train in frequency Example 3.8 The Spectrum of a Rectangular Function: Sinc Function The rectangular pulse is particularly important in digital signal analysis and digital communication. For example, the spectrum of a rectangular pulse can be used to calculate the bandwidth required by pulse radar systems or communication systems that transmit pulses. The Fourier transform of a rectangular pulse Fig. 3.9.a of duration T seconds is obtained as

)(sinc)sin(j2

.1)()(

2/j22/j2

2/

2/

j2j2

fTTfT

fTTfee

dtedtetrfR

fTfT

T

T

ftft

ππ

ππ

ππ

ππ

==−

=

==

−∞

∞−

− ∫∫ (3.32)

Fig. 3.9.b shows the spectrum of the rectangular pulse. Note that most of the pulse energy is concentrated in the main lobe within a bandwidth of 2/T. However there are pulse energy in the side lobes that may interfere with other electronic devices operating at the side lobe frequencies. Matlab code for drawing the spectrum of a rectangular pulse. % Pulsewidth T, Number of frequency samples N, Frequency resolution for plot df. T=.001;df=10; N=1000; for i=1:N if (i~=N/2)x(i)=T*sin(2*pi*df*T*(i-N/2))/(2*pi*df*T*(i-N/2));end end x(N/2)=T; plot(x);

1T

2T

3T

-1T

-2T f

T

T

r(t)R(f)

t

1

(a) (b)

Figure 3.9 A rectangular pulse and its spectrum.

Page 14: Fourier Analysis and Synthesis - umu.se · PDF fileFourier Analysis and Synthesis ... studied the mathematical theory of ... signals are sinusoidal signals with different periods giving

14 Chap.3 Fourier Analysis and Synthesis

m

e–σtIm[e–j2πft]

m m

e–σtIm[e–j2πmf] e–σtIm[e–j2πmf]

σ > 0 σ =0 σ < 0 Figure 3.10 – The Laplace basis functions.

3.3.1 The Relation Between the Laplace and the Fourier Transforms The Laplace transform of x(t) is given by the integral

∫∞

−=0

)()( dtetxsX st

(3.33)

where the complex variable s=σ+jω, and the lower limit of t=0− allows the

possibility that the signal x(t) may include an impulse. The inverse Laplace

transform is defined by

∫∞+

∞−

=j

j

st dsesXtx1

1

)()(σ

σ (3.34)

where σ1 is selected so that X(s) is analytic (no singularities) for s>σ1. The basis functions for the Laplace transform are damped or growing sinusoids of the form

ttst eee ωσ j−−− = as shown in Fig. 3.10. These are particularly suitable for transient signal analysis. The Fourier basis functions are steady complex exponential, tje ω− , of time-invariant amplitudes and phase, suitable for steady state or time-invariant signal analysis. The Laplace transform is a one-sided transform with the lower limit of integration at −= 0t , whereas the Fourier transform Eq. (3.21) is a two-sided transform with the lower limit of integration at −∞=t . However for a one-sided signal, which is zero-valued for −< 0t , the limits of integration for the Laplace and the Fourier transforms are identical. In that case if the variable s in the Laplace transform is replaced with the frequency variable jω then the Laplace integral becomes the Fourier integral. Hence for a one-sided signal, the Fourier transform is a special case of the Laplace transform corresponding to s=jω and σ=0. The relation between the Fourier and the Laplace transforms are discussed further in Chapter 4. 3.3.2 Properties of the Fourier Transform

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Sec. 3.3 Fourier Transform 15

There are a number of Fourier Transform properties that provide further insight into the transform and are useful in reducing the complexity of the solutions of Fourier transforms and inverse transforms. These are:

Linearity The Fourier transform is a linear operation, this mean the principle of superposition applies. Hence if:

)()()( tbytaxtz += (3.35) then

)()()( fbYfaXfZ += (3.36) Symmetry This property states that if the time domain signal x(t) is real (as is often the case in practice) then

)()( * fXfX −= (3.37) Where the superscript asterisk * denotes the complex conjugate operation. From Eq. (3.37) it follows that Re{ )( fX } is an even function of f and Im{ )( fX } is an odd function of f. Similarly the magnitude of )( fX is an even function and the phase angle is an odd function. Time Shifting and Frequency Modulation (FM) Let ])([)( txfX F= be the Fourier transform of x(t). If the time domain signal x(t) is delayed by an amount T0, the effect on its spectrum )( fX is a phase shift of

fTe 02j π− as [ ] )()( 02j

0 fXeTtx fTπ−=−F (3.38) Conversely if X(f) is shifted by an amount F0, the effect is

[ ] )()( 02j0

1 txeFfX tFπ=−−F (3.39) Note that the modulation Eq. (3.39) states that multiplying a signal x(t) by tFje 02π translates the spectrum of x(t) onto the frequency F0, this is the frequency modulation (FM) principle. Differentiation and Integration Let x(t) be a continuous time signal with Fourier transform )( fX .

∫∞

∞−

= dfefXtx ftπ2j)()( (3.40)

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16 Chap.3 Fourier Analysis and Synthesis

Then by differentiating both sides of the Fourier transform Eq. (3.40) we obtain

∫∞

∞−

= dfefXftdtxd ft

dttxd

ππ 2j

/)(ofnsformFourierTra

)(2j)(43421

(3.41)

That is multiplication of )( fX by the factor j2πf in the frequency domain is equivalent to differentiation of x(t) in time. Similarly division of )( fX by j2πf is equivalent to integration of the function of time x(t)

)()0()(2j1)( fXfX

fdx

t

δππ

ττ +→←∫∞−

F (3.42)

Where the impulse term on the right-hand side reflects the dc or average value that can result from the integration. Time and Frequency Scaling If x(t) and X(f) are Fourier transform pairs then

→←

ααα fXtx 1)( F

(3.43)

For example try to say something very slowly, then α >1, your voice spectrum will be compressed and you may sound like a slowed down tape or disc, you can do the reverse and the spectrum would be expanded and your voice shifts to higher frequencies, This property is further illustrated in section 3.9.4. Convolution The convolution integral of two signals x(t) and h(t) is defined as

∫∞

∞−

−= τττ dthxty )()()( (3.44)

The convolution integral is also written as

)(*)()( thtxty = (3.45)

where asterisk * denotes the convolution operation. The convolution integral is used to obtain the time-domain response of linear systems to arbitrary inputs as will be discussed in later sections.

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Sec. 3.3 Fourier Transform 17

The Convolution Property of the Fourier Transform. It can be shown that convolution of two signals in the time domain corresponds to multiplication of the signals in the frequency domain, and conversely multiplication in the time domain corresponds to convolution in the frequency domain. To derive the convolutional property of the Fourier transform take the Fourier transform of the convolution of the signals x(t) and h(t) as

)()(

)()( )()( )(2jf2jf2j

fHfX

dtethdexdtedthx tft

=

−=

− ∫ ∫∫ ∫

∞−

∞−

−−−−∞

∞−

∞−

τπτππ ττττττ

(3.46)

Duality Comparing the Fourier transform and the inverse Fourier transform relations we observe a symmetric relation between them. In fact the main difference between Eqs. (3.20) and (3.21) is a negative sign in the exponent of fte π2j−

in Eq (3.21).

This symmetry leads to a property of Fourier transform known as the duality principle and stated as

)( )(

)( )(

fxtX

fXtx

→←

→←F

F

(3.47)

As illustrated in Fig 3.11. the Fourier transform of a rectangular function of time r(t) has the form of a sinc pulse function of frequency )(sinc f . From the duality

principle the Fourier transform of a sinc function of time sinc(t) is a rectangular function of frequency R(f).

1T

2T

3T

-1T

-2T f

T

T

x(t)X (f)

t

1T1

-1T1

f

T

T1

X (t) x(f)

t

Figure 3.11 Illustration of the principle of duality.

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18 Chap.3 Fourier Analysis and Synthesis

Parseval's Theorem: Energy Relationship in Time and Frequency Parseval’s relation states that the energy of a signal can be computed by integrating the squared magnitude of the signal either over the time domain or over the frequency domain. If x(t) and X(f) are a Fourier transform pair, then

∫∫∞

∞−

∞−

== dffXdttx 22 |)(||)(|Energy (3.48)

This expression referred to as Parseval's relation follows from a direct application of the Fourier transform. Example 3.9 The Spectrum of a Finite Duration Signal Find and sketch the frequency spectrum of the following finite duration signal

2/2/)2sin()( 000 NTtNTtFtx ≤≤−= π (3.49)

where T0=1/F0 is the period and ω0= 2π/T0. Solution: Substitute for x(t) and its non-zero valued limits in the Fourier transform Eq. (3.21)

∫−

−=2/

2/

20

0

0

)2sin()(NT

NT

ftj dtetFfX ππ

(3.50)

substituting j2/)()2sin( 00 2j2j

0tFtF eetF πππ −−= in (3.50) gives

∫∫

+−

−−

−−

−=

−=

2/

2/

)(2j2/

2/

)(2j

2/

2/

2j2j2j

0

0

00

0

0

0

0

00

j2j2

j2)(

NT

NT

tFfNT

NT

tFf

NT

NT

fttFtF

dtedte

dteeefX

ππ

πππ

(3.51)

Evaluating the integrals yields

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Sec. 3.3 Fourier Transform 19

( ) ( )000

000

0

)(j)(j

0

)(j)(j

)(sin)(2π

j)(sin)(2π

j)(4)(4

)(00000000

NTFfFf

NTFfFf

Ffee

FfeefX

NTFfNTFfNTFfNTFf

++

+−−

−=

+−

−−−

=++−−−−

ππ

ππ

ππππ

(3.52) Matlab code for drawing the spectrum of a finite duration sinwave. % Sinewave Period T0, Number of Cycles in the window N T0=.001; F0=1/T0; N=2000; for Nc=1:4; for f=1:999 if ((f-f0)~=0) x(Nc,f)=Nc*T0*sin(pi*(f-F0)*T0*Nc)/(pi*(f-F0)*T0*Nc); end end x(Nc,1000)=Nc*T0; end plot(x);

Nc=1

Nc=4

Nc=2

Nc=3

fF0

X(f)

Figure 3.12 The spectrum of a finite duration sine wave with the increasing length of

observation. Nc is the number of cycles in the observation window. Note that the width of the main lobe depends inversely on the duration of the signal.

The signal spectrum can be expressed as the sum of two shifted sinc functions.

( ) ( )000000 )(sincj)(sincj)( NTFfNTNTFfNTfX ++−−= ππ (3.53)

Note that energy of a finite duration sinewave is spread in frequency between the main lobe and the side lobes of the sinc function. Fig. 3.12 demonstrates that as the window length increases the energy becomes more concentrated in the main lobe and in the limit for an infinite duration window the spectrum tends to an impulse positioned at the frequency F0.

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20 Chap.3 Fourier Analysis and Synthesis

Example 3.10 Calculation of the bandwidth for transmission of data at a rate of rb bits per second. In its simplest form binary data can be represented as a sequence of amplitude modulated pulses as

∑−

=

−=1

0)()()(

N

mbmTtpmAtx ( )

where A(m) may be +1 or a –1 and

>

≤=

00

2/1)(

t

Tttp b ( )

The Fourier transform of x(t) is given by

∑∑−

=

−−

=

− ==1

0

21

0

2 )()()()()(N

m

fmTjN

m

fmTj bb emAfPefPmAfX ππ ( )

The power spectrum of this signal is obtained as

[ ]

∑ ∑

∑∑−

=

=

−−

−∞=

−∞=

=

=

1

0

1

0

)(22

2*2

)]()([)(

)()()()()(*)(

N

m

N

n

fTnmj

n

fmTj

m

fmTj

b

bb

enAmAEfP

efPnAefPmAEfXfXE

π

ππ

( )

Now assuming that the data is uncorrelated we have

≠=

=nmnm

nAmAE01

)]()([ ( )

Substituting Equations () in () we have

[ ] 2)()(*)( fPNfXfXE = ( ) From Equation () the bandwidth required from a sequence of pulses is basically the same as the bandwidth of a single pulse. From Figure (3.11) the main lobe width is 2rb.

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Sec. 3.3 Fourier Transform 21

3.3.3 Fourier Transform of a Sampled Signal v A sampled signal x(m) can be modelled as

∑∞

−∞=

−=m

smTttxmx )()()( δ (3.54)

where m is the discrete time variable and Ts is the sampling period. The Fourier transform of x(m), a sampled version of a continuous signal x(t), can be obtained from Eq. (3.21) as

∑ ∫∫ ∑∞

−∞=

−∞=

∞−

−∞

∞−

−∞

−∞=

=

−=−=

m

Fmfjs

m

ftjs

ftj

m

semTx

dtemTttxdtemttxfX

/2

22

)(

)()()()()(

π

ππ δδ

(3.55)

For convenience of notation and without loss of generality it is often assumed that sampling frequency Fs=1/Ts=1, hence

∑∞

−∞=

−=m

mfs emxfX π2j)()(

(3.56)

The inverse Fourier transform of a sampled signal is defined as

∫−

=2/1

2/1

2j)()( dfefXmx fms

π (3.57)

Note that x(m) and )( fX s are equivalent in that they contain the same information in different domains. In particular, as expressed by the Parseval's theorem, the energy of the signal may be computed either in the time or in the frequency domain a

∫∑−

−∞=

==2/1

2/1

22 |)(|)(EnergySignal dffXmx sm

(3.58)

Example 3.10 Show that the spectrum of a sampled signal is periodic with a period equal to the sampling frequency Fs.

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22 Chap.3 Fourier Analysis and Synthesis

Solution: substitute f+kFs for the frequency variable f in Eq. (3.56)

)()()()(1

2j2j)(

2jfXeemTxemTxkFfX s

s

ss

s

FkF

m

m

Ffm

sm

FkFf

m

ss ===+=

−∞

−∞=

−∞

−∞=

+−

∑∑ 43421

πππ

(3.59) Fig. 3.13.a shows the spectrum of a band-limited continuous-time signal. As shown in Fig. 3.13.b after the signal is sampled its spectrum becomes periodic.

-Fs Fs 2Fs

......

X(f) Xs(f)

f f-2Fs00

(a) (b)

Figure 3.13 The spectrum of : (a) a continuous signal, and (b) its sampled version.

x(0)

x(2)

x(N–2)

x(1)

x(N – 1)

X(0)

X(2)

X(N – 2)

X(1)

X(N– 1)

X(k) = ∑ m=0

N–1

x(m) e

–jN

2πkn

Discrete Fourier Transform

.

.

.

.

.

.

Figure 3.14 Illustration of the DFT as a parallel-input parallel-output signal processor.

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Sec. 3.5 Short-Time Fourier Transform 23

3.4 Discrete Fourier Transform (DFT) When a non-periodic signal is sampled, its Fourier transform becomes a periodic but continuous function of frequency, as shown in Eq. (3.59). The discrete Fourier transform (DFT) is derived from sampling the Fourier transform of a discrete-time signal. For a finite duration discrete-time signal x(m) of length N samples, the discrete Fourier transform (DFT) is defined as N uniformly spaced spectral samples

mkN

N

memxkX

π2j1

0)()(

−−

=∑= k = 0, . . ., N−1 (3.60)

Comparing Eqs. (3.60) and (3.56) we see that the DFT consists of N equi-spaced samples taken from one period (2π) of the continuous spectrum of the discrete time signal x(m). The inverse discrete Fourier transform (IDFT) is given by

mkN

N

kekX

Nmx

π2j1

0)(1)( ∑

=

= m= 0, . . ., N−1 (3.61)

A periodic signal has a discrete spectrum. Conversely any discrete frequency spectrum belongs to a periodic signal. Hence the implicit assumption in the DFT theory, is that the input signal x(m) is periodic with a period equal to the observation window length of N samples. Example: Derivation of inverse discrete Fourier transform Obtain the inverse DFT from the DFT equation. The discrete Fourier transform (DFT) is given by

kmN

jN

memxkX

π21

0)()(

−−

=∑= k = 0, . . ., N−1

Multiply both sides of the DFT equation by e-j2πkn/N and take the summation as

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24 Chap.3 Fourier Analysis and Synthesis

2 2 21 1 1

0 0 0

21 1 ( )

0 0

0

( ) ( )

( )

N N Nj kn j km j knN N N

k k m

N N j k m nN

k m

N if m notherwise

X k e x m e e

x m e

π π π

π

− − −+ − +

= = =

− − − −

= =

=

=

=

∑ ∑∑

∑ ∑1442443

Using the orthogonality principle,the inverse DFT equation can be derived as

kmN

jN

kekX

Nmx

π21

0)(1)(

−−

=∑=

3.4.1 Time and Frequency Resolutions: The Uncertainty Principle Signals such as speech, music or image are composed of nonstationary i.e. time-varying and/or space varying events. For example speech is composed of a string of short-duration sounds called phonemes, and an image is composed of various objects. When using the DFT it is desirable to have a high enough time and space resolution in order to obtain the spectral characteristics of each individual elementary event or object in the input signal. However there is a fundamental trade-off between the length, i.e. the time or space resolution, of the input signal and the frequency resolution of the output spectrum. The DFT takes as the input a window of N uniformly spaced time domain samples [x(0), x(1), …, x(N−1)] of duration ∆T=N.Ts, and outputs N spectral samples [X(0), X(1), …, X(N−1)] spaced uniformly between zero Hz and the sampling frequency Fs=1/Ts Hz. Hence the frequency resolution of the DFT spectrum ∆f, i.e. the space between successive frequency samples, is given by

NF

NTTf s

s

==∆

=∆11

(3.62)

Note that the frequency resolution ∆f and the time resolution ∆T are inversely proportional in that they can not both be simultanously increased, in fact ∆T∆f=1. This is known as the uncertainty principle. Example 3.11 A DFT is used in a DSP system for the analysis of an analog signal with a frequency content of up to 10 kHz. Calculate: (i) the minimum sampling rate Fs required, and (ii) the number of samples required for the DFT to achieve a frequency resolution of 10 Hz at the minimum sampling rate.

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Sec. 3.5 Short-Time Fourier Transform 25

Solution: (i) Sampling rate > 2×10 kHz, say 22 kHz, and

(ii) NF

f s=∆ N

2200010 = 2200≥N .

Example 3.12 Write a MATLAB program to explore the spectral resolution of a signal consisting of two sinewaves, closely spaced in frequency, with the varying length of the observation window. Solution: In the following program the two sinwaves have frequencies of 100 Hz and 110 Hz, and the sampling rate is 1 kHz. We experiment with two time windows of length N1=1024 with a theoretical frequency resolution of ∆f=1000/1024=0.98 Hz, and N2=64 with a theoretical frequency resolution ∆f=1000/64=15.7 Hz.

Fs=1000; F1=100; F2=110; N=1:1024; N1=1024; N2=64; x1=sin(2*pi*F1*N/Fs); x2=sin(2*pi*F2*N/Fs); y=x1+x2;

0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0- 2

- 1

0

1

2

T i m e

Am

plitu

de

(a)

0 10 20 30 40 50 60 70

20

40

60

80

Feq k

Am

plitu

de

100 200 300 400 500 6000

150

300

450

Feq k

Am

plitu

de

(b) (c)

Figure 3.15 Illustration of time and frequency resolutions: (a) sum of two sinewaves with 10 Hz difference in their frequencies, (b) the spectrum of a segment of 64 samples from

demonstrating insufficient frequency resolution to separate the sinewaves, (c) the spectrum of a segment of 1024 samples has sufficient resolution to show the two sinewaves.

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26 Chap.3 Fourier Analysis and Synthesis

Y1=abs(fft(y(1:N1)));Y2=abs(fft(y(1:N2))); figure(1); plot(y); figure(2); plot(Y1(1:N1/2)); figure(3); plot(Y2(1:N2/2)); 3.4.2 The Effect of Finite Length Data on DFT (Windowing) In a practical situation we have either with a short length signal, or with a long signal of which the DFT can only handle one segment at a time. Having a short segment of N samples of a signal or taking a slice of N samples from a signal is equivalent to multiplying the signal by a unit-amplitude rectangular pulse window of N samples. Therefore an N-sample segment of a signal x(m) is equivalent to

)()()( mxmwmxw = (3.63)

where w(m) is a rectangular pulse of N samples duration given is

−≤≤

=otherwise

Nmmw

0101

)( (3.64)

Multiplying two signals in time is equivalent to the convolution of their frequency spectra. Thus the spectrum of a short segment of a signal is convolved with the spectrum of a rectangular pulse as

X k W k X kw ( ) ( ) * ( )= (3.65) The result of this convolution is some spreading of the signal energy in the frequency domain as illustrated in the next example. Example 3.13 Find the DFT of a rectangular window given by

−≤≤

=otherwise

Nmmw

0101

)( (3.66)

Solution: Taking the DFT of w(m), and using the convergence formula for the partial sum of a geomertic series, described in the appendix A, we have

)/sin()sin(

1

1)()()1(j

2j

2j1

0

2j

Nkke

e

eemwkWk

NN

kN

kN

m

mkN

πππ

π

ππ −−

−−

=

−=

−== ∑ (3.67)

Note that for the integer values of k, w(k) is zero except for k=0. Example 3.14 Find the spectrum of an N-sample segment of a complex sinewave with a fundamental frequency F0=1/T0.

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Sec. 3.5 Short-Time Fourier Transform 27

Solution: Taking the DFT of mFjemx 02)( π−= we have

)/)(sin())(sin(

11

)(

0

0)()1(j

/)(2j

)(2j

1

0

)(2j1

0

22j

0

0

0

00

NkNFkNF

eee

eeekX

kNFN

N

NkNF

kNF

N

m

mNkFN

m

mkN

jmF

−−

=−−

=

==

−−

−−

−−

=

−−−

=

−− ∑∑

πππ

π

π

πππ

(3.68)

Note that for integer values of k, X(k) is zero at all samples but one k=0. 3.4.3 End-Point Effects in DFT; Spectral Energy Leakage and Windowing In DFT the input signal is assumed to be periodic, with a period equal to the length of the observation window of N samples. Now, for a sinusoidal input signal if there is an integer number of cycles within the observation window, as in Fig. 3.16.a, then the assumed periodic waveform is the same as an infinite length pure sinusoid. But if the observation window contains a non-integer number of cycles of a sinusoid then the assumed periodic waveform will not be a pure sine wave and will have end-point discontinuities. The spectrum of the signal then differs from the spectrum of sinewave as illustrated in Fig. 3.16.b. The overall effects of finite length window and end-point discontinuities are:

1. The spectral energy which could have been concentrated at a single point, or in a narrow band of frequencies is spread over a larger band of frequencies.

2 cycles 2.25 cycles

time time

FrequencyFrequencyF0 F0

(a) (b) Figure 3.16 The DFT spectrum of exp(j2πfm) : (a) an integer number of cycles with in the N-sample analysis window, (b) a non-integer number of cycles in the window.

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28 Chap.3 Fourier Analysis and Synthesis

2. A smaller amplitude signal, located in frequency near a larger amplitude

signal, may be obscured by one of the larger signal’s side-lobes. That is side-lobes of a large amplitude signal may interfere with the main-lobe of a nearby small amplitude signal.

The end-point problems may be alleviated using a window that gently drops to zero. One such window is a raised cosine window of the form

( )

−≤≤−−

=otherwise

NmNm

mw0

102cos1)(

παα (3.69)

For α=0.5 we have the Hanning window also known as the raised cosine window

102cos5.05.0)( −≤≤−= NmNmmwHan

π (3.70)

For α=0.54 we have the Hamming window

102cos46.054.0)( −≤≤−= NmNmmwHam

π (3.71)

0- 8 0- 7 0- 6 0- 5 0- 4 0- 3 0- 2 0

- 1 0

0

0- 6 0

- 4 0- 3 0

- 2 0

- 1 0

0

0- 3 0

- 2 5

- 2 0

- 1 5

- 1 0

- 5

0

T i m e F r e q u e n c y

- 5 0

F s / 2

F s / 2

F s / 2

6 40

6 40

0 6 4 Figure 3.17 (a) Rectangular window frequency response, (b) Hamming window frequency response, (c) Hanning window frequency response. 3.4.3 Spectral Smoothing

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Sec. 3.5 Short-Time Fourier Transform 29

The spectrum of a short length signal can be interpolated to obtain a smoother spectrum. Interpolation of the frequency spectrum X(k) is achieved by zero-padding of the time domain signal x(m). Consider a signal of length N samples [x(0), . . ., x(N−1)]. Increase the signal length from N to 2N samples by padding N zeros to obtain the padded sequence [x(0), . . ., x(N−1), 0, . . ., 0]. The DFT of the padded signal is given by

mk

NN

m

mkN

N

m

emx

emxkX

π

π

j1

0

22

j12

0

)(

)()(

−−

=

−−

=

=

= k = 0, . . ., 2N−1 (3.72)

The spectrum of the zero-padded signal, Eq. (3.72), is composed of 2N spectral samples; N of which, [X(0), X(2), X(4), X(6), . . . X(2N−2)] are the same as those that would be obtained from a DFT of the original N samples, and the other N samples [X(1), X(3), X(5), X(6), . . . X(2N−1)] are interpolated spectral lines that result from zero-padding. Note that zero padding does not increase the spectral resolution, it merely has an interpolating or smoothing effect in the frequency domain, as illustrated in Fig 3.18.

Padded zero’s

Time Frequency

Frequency Frequency

r(m) |R(f)|

|R(f)| |R(f)|

without zero padding

with 10 padded zeros with 20 padded zeros

Figure 3.18 Illustration of the interpolating effect, in the frequency domain, of zero

padding a signal in the time domain.

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30 Chap.3 Fourier Analysis and Synthesis

3.5 Short-Time Fourier Transform In Fourier transform it is assumed that the signal is stationary, meaning the signal statistics, such as the mean, the power, and the power spectrum, are time-invariant. Most real life signals such as speech, music and image signals are nonstationary in that their amplitude, power, spectral composition and other features changes continuously with time. To apply Fourier transform to nonstationary signals the signal is divided into appropriately short-time windows, such that within each window the signal can be assumed to be time-invariant. The Fourier transform applied to the short signal segment within each window is known as the short-time Fourier Transform. Fig. 3.19 illustrates the segmentation of a speech signal into a sequence of overlapping, hamming windowed, short segments. The choice of window length is a compromise between the time resolution and the frequency resolution. For Audio signals a time window of about 25 ms, corresponding to a frequency resolution of 40 Hz is normally adopted.

time

Figure 3.19 - Segmentation of speech using Hamming window for STFT.

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Fourier Analysisand WaveletAnalysisJames S. Walker

658 NOTICES OF THE AMS VOLUME 44, NUMBER 6

In this article we will compare the classicalmethods of Fourier analysis with the newermethods of wavelet analysis. Given a sig-nal, say a sound or an image, Fourier analy-sis easily calculates the frequencies and

the amplitudes of those frequencies which makeup the signal. This provides a broad overview ofthe characteristics of the signal, which is im-portant for theoretical considerations. However,although Fourier inversion is possible undercertain circumstances, Fourier methods are notalways a good tool to recapture the signal, par-ticularly if it is highly nonsmooth: too muchFourier information is needed to reconstructthe signal locally. In these cases, wavelet analy-sis is often very effective because it provides asimple approach for dealing with local aspectsof a signal. Wavelet analysis also provides us withnew methods for removing noise from signalsthat complement the classical methods ofFourier analysis. These two methodologies aremajor elements in a powerful set of tools for the-oretical and applied analysis.

This article contains many graphs of discretesignals. These graphs were created by the com-puter program FAWAV, A Fourier–Wavelet An-alyzer, being developed by the author.

Frequency Information, DenoisingAs an example of the importance of frequency in-formation, we will examine how Fourier analysis canbe used for removing noise from signals. Considera signal f (x) defined over the unit interval (where herex stands for time). The period 1 Fourier series ex-pansion of f is defined by

∑n∈Z cnei2πnx, with

cn =∫ 10 f (x)e−i2πnx dx. Each Fourier coefficient, cn , is

an amplitude associated with the frequency n of theexponential ei2πnx. Although each of these expo-nentials has a precise frequency, they all suffer froma complete absence of time localization in that theirmagnitudes, |ei2πnx| , equal 1 for all time x.

To see the importance of frequency information,let us examine a problem in noise removal. In Figure 1(a)[top] we show the graph of the signal

(1) f (x) =

(5 cos 2πνx) [ e−640π (x−1/8)2

+ e−640π (x−3/8)2 + e−640π (x−4/8)2

+ e−640π (x−6/8)2 + e−640π (x−7/8)2 ]

where the frequency, ν, of the cosine factor is 280.Such a signal might be used by a modem for trans-mitting the bit sequence 1 0 1 1 0 1 1. The Fourier co-efficients for this signal are shown in Figure 1(b)[top].The highest magnitude coefficients are concentratedaround the frequencies ±280. Suppose that when thissignal is received, it is severely distorted by addednoise; see Figure 1(a)[middle]. Using Fourier analy-sis, we can remove most of this noise. Computing thenoisy signal’s Fourier coefficients, we obtain thegraph shown in Figure 1(b)[middle]. The original sig-nal’s largest magnitude Fourier coefficients are clus-

James S. Walker is professor of mathematics at theUniversity of Wisconsin-Eau Claire. His e-mail addressis [email protected].

The author would like to thank Hugo Rossi, StevenKrantz, his colleague Marc Goulet, and two anony-mous reviewers for their helpful comments during thewriting of this article.

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tered around the frequency positions ±280. TheFourier coefficients of the added noise are lo-calized around the origin, and they decrease inmagnitude until they are essentially zero nearthe frequencies ±280. Thus, the original sig-nal’s coefficients and the noise’s coefficientsare well separated. To remove the noise from thesignal, we multiply the noisy signal’s coefficientsby a filter function, which is 1 where the signal’scoefficients are concentrated and 0 where thenoise’s coefficients are concentrated. We then re-cover essentially all of the signal’s coefficients;see Figure 1(b)[bottom]. Performing a Fourierseries partial sum with these recovered coeffi-cients, we obtain the denoised signal, which isshown in Figure 1(a)[bottom]. Clearly, the bit se-quence 1 0 1 1 0 1 1 can now be determined fromthe denoised signal, and the denoised signal isa close match of the original signal. In the sec-tion “Signal Denoising” we shall look at anotherexample of this method and also discuss howwavelets can be used for noise removal.

Signal CompressionAs the example above shows, Fourier analysis isvery effective in problems dealing with frequencylocation. However, it is often very ineffective atrepresenting functions. In particular, there aresevere problems with trying to analyze tran-sient signals using classical Fourier methods.For example, in Figure 2(a)[top] we show a dis-crete signal obtained from M = 1024 values{fj = F (j/M)}M−1

j=0 of the function F (x) =

e−105π (x−.6)2. For this example, we compute thediscrete Fourier series coefficients fn defined by

fn =M−1∑M−1j=0 fje−i2πnj/M

for n = −12M + 1, . . . ,0, . . . , 1

2M . The discreteFourier coefficients fn can be calculated by a fastFourier transform (FFT) algorithm and are thediscrete analog of the Fourier coefficients cnfor F , when fj = F (j/M) . Moreover, fn is just aRiemann sum approximation of the integral thatdefines cn .1 The magnitudes of the discreteFourier coefficients for this transient damp downto zero very slowly (their graph is a very widebell-shaped curve with maximum at the origin).Consequently, to represent the transient well, onemust retain most if not all of these Fourier co-efficients. In Figure 2(a)[bottom] we show the re-sults obtained from trying to compress the tran-sient by computing a discrete partial sum∑104n=−104 fnei2πnj using only one-fifth of the

Fourier coefficients.2 Clearly, even a moderatecompression ratio of 5:1 is not effective.

Wavelets, however, are often very effective atrepresenting transients. This is because they aredesigned to capture information over a largerange of scales. A wavelet series expansion of afunction f is defined by∑

n,k∈Z

βnk 2n/2ψ(2nx− k)

withβnk =

∫∞−∞f (x) 2n/2ψ(2nx− k) dx .

The function ψ(x) is called the wavelet, and thecoefficients βnk are called the wavelet coeffi-cients. The function 2n/2ψ(2nx− k) is the

Figure 1. (a)[top] Signal. (b)[top] Fourier coefficients of signal. (a)[middle] Signal after adding noise.(b)[middle] Fourier coefficients of noisy signal and filter function. (b)[bottom] Fourier coefficients

after multiplication by filter function. (a)[bottom] Recovered signal.

1For further discussion of discrete Fourier coefficients,see [2] or [11].2The sum

∑512n=−511 fnei2πnj, which uses all of the dis-

crete Fourier coefficients, equals fj.

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660 NOTICES OF THE AMS VOLUME 44, NUMBER 6

wavelet shrunk by a factor of 2n if n is positive(magnified by a factor of 2−n if n is negative) andshifted by k2−n units. The factor 2n/2 in the ex-pression 2n/2ψ(2nx− k) preserves the L2-norm.

Since the wavelet series depends on two pa-rameters of scale and translation, it can often bevery effective in analyzing signals. These para-meters make it possible to analyze a signal’sbehavior at a dense set of time locations and withrespect to a vast range of scales, thus providingthe ability to zoom in on the transient behaviorof the signal. For example, let us examine the ear-lier transient using a discretized version of awavelet series. We shall use a Daubechies order4 wavelet (Daub4 for short; see the section“Daubechies Wavelets”). In Figure 2(b)[top] weshow all of the 1024 wavelet coefficients of thistransient and observe that most of these coef-ficients are close to 0 in magnitude. Conse-quently, by retaining only the largest magnitudecoefficients for use in a wavelet series, we ob-tain significant compression. In Figure 2(b)[bot-tom] we show the reconstruction of the transientusing only the top 4% in magnitude of the waveletcoefficients, a 25:1 compression ratio. Noticehow accurately the transient is represented. Infact, the maximum error at all computed pointsis less than 9.95× 10−14. There is an importantapplication here to the field of signal transmis-sion. By transmitting only these 4% of the waveletcoefficients, the information in the signal can betransmitted 25 times faster than if we transmit-ted all of the original signal. This provides a con-siderable boost in efficiency of transmission.

We shall look at more examples of com-pression in the section “Compression of Sig-nals”, but first we shall describe how waveletanalysis works.

The Haar WaveletIn order to understand how wavelet analysisworks, it is best to begin with the simplestwavelet, the Haar wavelet. Let 1A(x) denote theindicator function of the set A , defined by1A(x) = 1 if x ∈ A and 1A(x) = 0 if x /∈ A. TheHaar wavelet ψ is defined byψ(x) = 1[0, 12 )(x)− 1[ 1

2 ,1)(x) . It is 0 outside of

[0,1), so it is well localized in time, and it sat-isfies∫∞

−∞ψ(x)dx = 0,

∫∞−∞|ψ(x)|2 dx = 1.

The Haar wavelet ψ(x) is closely related to thefunction φ(x) defined by φ(x) = 1[0,1)(x). Thisfunction φ(x) is called the Haar scaling function.Clearly, the Haar wavelet and scaling functionsatisfy the identities

(2)ψ(x) = φ(2x)−φ(2x− 1),φ(x) = φ(2x) +φ(2x− 1),

and the scaling function satisfies∫∞−∞φ(x)dx = 1,

∫∞−∞|φ(x)|2 dx = 1.

The Haar wavelet ψ(x) generates the systemof functions {2n/2ψ(2nx− k)}. It is possible toshow directly that {2n/2ψ(2nx− k)} is an or-thonormal basis for L2(R), but it is more illu-minating to put the discussion on an axiomaticlevel. This axiomatic approach leads to theDaubechies wavelets and many other waveletsas well. We begin by defining the subspaces{Vn}n∈Z of L2(R) in the following way:

Figure 2. (a)[top] Signal. (a)[bottom] Fourier series using 205 coefficients, 5:1 compression. (b)[top]Wavelet coefficients for signal. (b)[bottom] Wavelet series using only largest 4% in magnitude ofwavelet coefficients, 25:1 compression.

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Vn ={step functions in L2(R), constant

on the intervals [k2n,k + 12n

), k ∈ Z}.

This set of subspaces {Vn}n∈Z satisfies the fol-lowing five axioms [6]:

Axioms for a Multi-Resolution Analysis

(MRA)

Scaling: f (x) ∈ Vn if and only if f (2x) ∈ Vn+1.

Inclusion: Vn ⊂ Vn+1, for each n.

Density: closure { ⋃n∈Z

Vn}

= L2(R).

Maximality:⋂n∈Z

Vn = {0}.Basis: ∃φ(x) such that {φ(x− k)}k∈Z is an

orthonormal basis for V0.To satisfy the basis axiom, we shall use the

Haar scaling function φ defined above. Then, bycombining the scaling axiom with the basisaxiom, we find that {2n/2φ(2nx− k)}k∈Z is anorthonormal basis for Vn. But the totality of allthese orthonormal bases, consisting of the set{2n/2φ(2nx− k)}k,n∈Z, is not an orthonormalbasis for L2(R) because the spaces Vn are notmutually orthogonal. To remedy this difficulty,we need what are called wavelet subspaces. De-fine the wavelet subspace Wn to be the orthog-onal complement of Vn in Vn+1. That is, Wn sat-isfies the equation Vn+1 = Vn ⊕Wn where ⊕denotes the sum of mutually orthogonal sub-spaces. From the density axiom and repeated ap-plication of the last equation, we obtainL2(R) = V0 ⊕

⊕∞n=0 Wn. Decomposing V0 in a

similar way, we obtain L2(R) =⊕n∈Z Wn. Thus,

L2(R) is an orthogonal sum of the wavelet sub-spaces Wn.

Using (2) and the MRA axioms, it is easy toprove the following lemma.

Lemma 1. The functions {ψ(x− k)}k∈Z are anorthonormal basis for the subspace W0.

It follows from the scaling axiom that{2n/2ψ(2nx− k)}k∈Z is an orthonormal basisfor Wn. Therefore, since L2(R) is the orthogonalsum of all the wavelet subspaces Wn, we haveobtained the following result.

Theorem 1. The functions

{2n/2ψ(2nx− k)}k,n∈Z

are an orthonormal basis for L2(R).This orthonormal basis is the Haar basis for

L2(R). There is also a Haar basis for L2[0,1). Toobtain it, we first define periodic wavelets ψn,kby

(3) ψn,k(x) =∑j∈Z

2n/2ψ(2n(x + j)− k) .

Note that these wavelets have period 1. Fur-thermore, ψn,k ≡ 0 for n < 0, andψn,k+2n = ψn,k for all k ∈ Z and n ≥ 0. On the in-terval [0,1), the periodic Haar wavelets ψn,k sat-isfy ψn,k(x) = 2n/2ψ(2nx− k) for n ≥ 0 andk = 0,1, . . . ,2n − 1. So we have the following the-orem as a consequence of Theorem 1.

Theorem 2. The functions 1 and ψn,k for n ≥ 0and k = 0,1, . . . ,2n − 1 are an orthonormal basisfor L2[0,1).

Remark. In the section “Daubechies Wavelets”we will make use of periodized scaling func-tions, φn,k, defined by

(4) φn,k(x) =∑j∈Z

2n/2φ(2n(x + j)− k)

for n ≥ 0 and k = 0,1, . . . ,2n − 1.

Fast Haar Transform

The relation between the Haar scaling functionφ and wavelet ψ leads to a beautiful set of re-lations between their coefficients as bases. Let{αnk} and {βnk} be defined by

(5)αnk =

∫∞−∞f (x) 2n/2φ(2nx− k)dx,

βnk =∫∞−∞f (x) 2n/2ψ(2nx− k)dx.

Substituting 2nx in place of x in the identitiesin (2), we obtain

2n/2φ(2nx) =1√2

[2n+1

2 φ(2n+1x)]

+1√2

[2n+1

2 φ(2n+1x− 1)]

2n/2ψ(2nx) =1√2

[2n+1

2 φ(2n+1x)]

− 1√2

[2n+1

2 φ(2n+1x− 1)].

It then follows that

(6)αnk =

1√2αn+1

2k +1√2αn+1

2k+1,

βnk =1√2αn+1

2k − 1√2αn+1

2k+1.

This result shows that the nth level coefficientsαnk and βnk are obtained from the (n + 1)st levelcoefficients αn+1

k through multiplication by thefollowing orthogonal matrix:

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662 NOTICES OF THE AMS VOLUME 44, NUMBER 6

(7) O =

1√2

1√2

1√2

−1√2

.

Successively applying this orthogonal matrix O,we obtain {αnk} and {βnk} starting from somehighest level coefficients {αNk } for some largeN. Because of the density axiom, N can be cho-sen large enough to approximate f by∑k∈ZαNk 2N/2φ(2Nx− k) in L2-norm as closely

as desired.Let us now discretize these results. Suppose

that we are working with data {fj}M−1j=0 associ-

ated with the time values {xj = j/M}M−1j=0 on the

unit interval. If we shrink the Haar scaling func- tion φ(x) = 1[0,1)(x) enough, it covers only thefirst point, x0 = 0. Consequently, by choosing alarge enough N, we may assume that our scal-ing coefficients {αNk } satisfy αNk = fk, fork = 0,1, . . . ,M − 1. Assuming that M is a powerof 2, say M = 2R , it follows that N = R .

The next step involves expressing the coeffi-cient relations in (6) in a matrix form. Let A⊕ Bstand for the orthogonal sum of the matrices Aand B, that is, A⊕ B =

(A 00 B

). Now let HM denote

the M ×M orthogonal matrix defined byHM = O⊕O⊕ · · · ⊕O, where the orthogonalmatrix sums are applied M/2 times and O is thematrix defined in (7). Then, by applying the co-efficient relations in (6) and using the fact thatfk = αRk , we obtain

HM [f0, f1, . . . , fM−1]T

=[αR−1

0 , βR−10 , αR−1

1 , βR−11 ,

. . . , αR−112M−1

, βR−112M−1

]T.

To sort the coefficients properly into two groups,we apply an M ×M permutation matrix PM asfollows:

PM[αR−1

0 , βR−10 , . . . , αR−1

12M−1

, βR−112M−1

]T

=[αR−1

0 , . . . , αR−112M−1

, βR−10 , . . . , βR−1

12M−1

]T.

If we go to the next lower level, the transfor-mations just described are repeated, only nowthe matrices used are HM/2 and PM/2, and theyoperate only on the M/2 -length vector

{αR−10 , . . . , αR−1

12M−1

} to obtain the next level

wavelet coefficients {βR−20 , . . . , βR−2

14M−1

} and

scaling coefficients {αR−20 , . . . , αR−2

14M−1

} . These

operations continue until we can no longer divide the number of components by 2. At theR = log2M step, we obtain a single scale coeffi-

cient α00 and a single wavelet coefficient β0

0,and at this last step the permutation P2 is un-necessary. The complete transformation, de-noted by H , satisfies

H = (H2 ⊕ IM−2)

· · · (PM/2 ⊕ IM/2) (HM/2 ⊕ IM/2)PM HM

where IN is the N ×N identity matrix.These matrix multiplications can be per-

formed rapidly on a computer. Multiplying byHM requires only O(M) operations, since HMconsists mostly of zeroes. Similarly, the per-mutation PM requires O(M) operations. There-fore, the whole transformation requiresO(M) + O(M/2) + · · · + O(2) = O(M) operations.The transformation H is called a fast Haartransform. It should be noted that FFTs, whichhave revolutionized scientific practice duringthe last thirty years, are O(M logM) algorithms.

Since each Hk is an orthogonal matrix, andso is every permutation Pk, it follows that H isinvertible. Its inverse is

H−1 = HTM P

TM (HT

M/2 ⊕ IM/2) (PTM/2 ⊕ IM/2)

· · · (HT2 ⊕ IM−2).

Therefore, the inverse operation is also an O(M)operation.

Discrete Haar SeriesThe fast Haar transform can be used for com-puting partial sums of the discretized versionof the following Haar wavelet series in L2[0,1):

(8) α00 +

∞∑n=0

2n−1∑k=0

βnk 2n/2ψ(2nx− k).

Let us assume, as in the previous section, thatwe have a discrete signal {fj}M−1

j=0 associated

with the time values {xj = j/M}M−1j=0 on the unit

interval. Substituting these time values into (8)and restricting the upper limit of n, we obtain

fj = α00 CM +

R−1∑n=0

2n−1∑k=0

βnk CM 2n/2ψ(2nxj − k).

The right side of this equation is just the trans-formation H−1H applied to {fj} . The firstpart of this transformation, H{fj}, produces the

coefficients α00, β

00, β

10, β

11, . . . , β

R−112M−1, and the

second part, the application of H−1, repro-duces the original data {fj}. The constant CMis a scale factor which ensures that the constantvector CM and the vectors {CM2n/2ψ(2nxj − k)}are unit vectors in RM , using the standard innerproduct. Consequently, CM =

√1/M .

There are many ways of forming partial sumsof discretized Haar series. The simplest ones con-

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sist of multiplying the data by H , then settingsome of the resulting coefficients equal to 0, andthen multiplying by H−1. A widely used methodinvolves specifying a threshold. All coefficientswhose magnitudes lie below this threshold areset equal to 0. This method is frequently usedfor noise removal, where coefficients whosemagnitudes are significant only because of theadded noise will often lie below a well-chosenthreshold. We shall give an example of this in thesection “Signal Denoising”. A second methodkeeps only the largest magnitude coefficients,while setting the rest equal to 0. This methodis convenient for making comparisons when itis known in advance how many terms are needed.We used it in the compression example in the sec-tion “Signal Compression”. A third method,which we shall call the energy method, involvesspecifying a fraction of the signal’s energy, wherethe energy is the square root of the sum of thesquares of the coefficients.3 We then retain theleast number of the largest magnitude coeffi-cients whose energy exceeds this fraction of thesignal’s energy and set all other coefficientsequal to 0. The energy method is useful for the-oretical purposes: it is clearly helpful to be ableto specify in advance what fraction of the sig-nal’s energy is contained in a partial sum. Weshall use the energy method frequently below.

Let us look at an example. Suppose our sig-nal is

{fj = F (j/8192)}8191j=0

where F (x) = x1[0,.5)(x) + (x− 1)1[.5,1)(x). In Fig-ure 3(a)[top] we show a Haar series partial sum,

created by the energy method, which contains99.5% of the energy of this signal. This partialsum, which used 229 coefficients out of a pos-sible 8192, provides an acceptable visual repre-sentation of the signal. In fact, the sum of thesquares of the errors is 2.0× 10−3. By compar-ison a 229 coefficient Fourier series partial sumsuffers from serious drawbacks (see Figure3(b)[top]). The sum of the squares of the errorsis 4.5× 10−1, and there is severe oscillation anda Gibbs’ effect near x = 0.5. Although these lat-ter two defects could be ameliorated using othersummation methods ([11], Ch. 4), there wouldstill be a significant deviation from the originalsignal (especially near x = 0.5).

This example illustrates how wavelet analy-sis homes right in on regions of high variabilityof signals and that Fourier methods try tosmooth them out. The size of a function’s Fouriercoefficients is related to the frequency contentof the function, which is measured by integra-tion of the function against completely unlo-calized basis functions. For a function having adiscontinuity, or some type of transient behav-ior, this produces Fourier coefficients that de-crease in magnitude at a very slow rate. Conse-quently, a large number of Fourier coefficientsare needed to accurately represent such sig-nals.4 Wavelet series, however, use compactlysupported basis functions which, at increasinglevels of resolution, have rapidly decreasing sup-ports and can zoom in on transient behavior. Thetransient behaviors contribute to the magnitudeof only a small portion of the wavelet coefficients.

Figure 3. (a)[top] Haar series partial sum, 229 terms. (b)[top] Fourier series partial sum, 229 terms.(a)[bottom] Haar series partial sum, 92 terms. (b)[bottom] Daub4 series partial sum, 22 terms.

3The Haar transform is orthogonal, so it makes senseto specify energy in this way.

4In recent years, though, significant improvementshave been achieved using local cosine bases [3, 7,9, 5, 1].

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664 NOTICES OF THE AMS VOLUME 44, NUMBER 6

Consequently, a small number of wavelet coeffi-cients are needed to accurately represent such sig-nals.

The Haar system performs well when the signalis constant over long stretches. This is because theHaar wavelet is supported on [0,1) and satisfies a0th order moment condition,

∫∞−∞ψ(x)dx = 0. There-

fore, if the signal {fj} is constant over an intervala ≤ xj < b such that [k2−n, (k + 1)2−n) ⊂ [a, b), then

the wavelet coefficient βnk equals 0. For example, sup-

pose {fj = F (j/8192)}8191j=0 where

F (x) = (8x− 1)1[.125,.25)(x)

+ 1[.25,.75)(x) + (7− 8x)1[.75,.875)(x).

In Figure 3(a)[bottom] we show a Haar series partialsum which contains 99.5% of the energy of this sig-nal and uses only 92 coefficients out of a possible8192. The fact that the signal is constant over threelarge subintervals of [0,1) accounts for the excellentcompression in this example. In order to obtainwavelet bases that provide considerably more com-pression, we need a compactly supported waveletψ(x) which has more moments equal to zero. Thatis, we want

(9)∫∞−∞xj ψ(x)dx = 0, for j = 0,1, . . . , L− 1

for an integer L ≥ 2. We say that such a wavelet hasits first L moments equal to zero. For example, aDaub4 wavelet has its first 2 moments equal to zero.Using a Daub4 wavelet series for the signal above, itis possible to capture 99.5% of the energy using only22 coefficients! See Figure 3(b)[bottom]. This im-provement in compression is due to the fact that the

Daub4 wavelet is supported on [0,3) and satis-fies

∫∞−∞ψ(x)dx = 0 and

∫∞−∞ xψ(x)dx = 0. Con-

sequently, if the signal is constant or linear overan interval [a, b) which contains[k2−n,3(k + 1)2−n) , then the wavelet coefficientβnk will equal 0. In Figure 4(a) we show graphsof the magnitudes of the highest level Haar co-efficients and Daub4 coefficients. Each magni-tude |β12

k | is plotted at the x-coordinate k2−12

for k = 0,1, . . . ,212 − 1. These graphs show thatthe highest level Haar coefficients are near 0 overthe constant parts of F , while the highest levelDaub4 coefficients are near 0 over the constantand linear parts of F . In Figure 4(b) we showgraphs of the sums of the squares of all the co-efficients, which show that almost all the Daub4coefficients are near 0 over the constant and lin-ear parts of F , while the Haar coefficients arenear 0 only over the constant parts of F . Fur-thermore, the largest magnitude Daub4 coeffi-cients are concentrated around the locations ofthe points of nondifferentiability of F. This kindof local analysis illustrates one of the powerfulfeatures of wavelet analysis.

Looking again at Figure 3(b)[bottom], we seethat the most serious defects of the Daub4 com-pressed signal are near the points where F is non-differentiable. If, however, we consider the in-terval [0.4,0.6] where F is constant, the Daub4compressed signal values differ from the valuesof F by no more than 1.2× 10−15 at all of the1641 discrete values of x in this interval. Incontrast, a Fourier series partial sum using 23coefficients differs by more than 10−3 at 1441of these 1641 values of x. The Fourier series par-tial sum exhibits oscillations of amplitude6.5× 10−3 around the value 1 over this subin-

Figure 4. (a) Magnitudes of highest level coefficients for a function F: [top] Haar coefficients, [middle]Daub4 coefficients; [bottom] Graph of F. (b) Sums of squares of all coefficients: [top] Haar coefficients,[middle] Daub4 coefficients; [bottom] Graph of F. Vertical scales for highest level coefficients and sumsof squares are logarithmic.

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JUNE/JULY 1997 NOTICES OF THE AMS 665

terval. Because the Fourier coefficients for F areonly O (n−2), using just the first 23 coefficientsproduces an oscillatory approximation to F overall of [0,1), including the subinterval [0.4,0.6].The highest magnitude wavelet coefficients,however, are concentrated at the corner pointsfor F , and their terms affect only a small por-tion of the partial sum (since their basis func-tions are compactly supported). Consequently,the wavelet series provides an extremely closeapproximation of F over the subinterval[0.4,0.6].

A major defect of the Haar wavelet is its dis-continuity. For one thing, it is unsatisfying to usediscontinuous functions to approximate con-tinuous ones. Even with discrete signals there canbe undesirable jumps in Haar series partial sumvalues (see Figure 3(a)[bottom]). Therefore, wewant to have a wavelet that is continuous. In thenext section we will describe the Daubechieswavelets, which have their first L ≥ 2 momentsequal to zero and are continuous.

Daubechies WaveletsIt is possible to generalize the construction ofthe Haar wavelet so as to obtain a continuousscaling function φ(x) and a continuous waveletψ(x). Moreover, Daubechies has shown how tomake them compactly supported. We will brieflysketch the main ideas; more details can be foundin [4, 5, 8, 10].

Generalizing from the case of the Haarwavelets, we require that φ(x) and ψ(x) satisfy

(10)

∫∞−∞φ(x)dx = 1,

∫∞−∞|φ(x)|2 dx = 1,∫∞

−∞|ψ(x)|2 dx = 1.

The MRA axioms tell us that φ(x) must gener-ate a subspace V0 and that V0 ⊂ V1. Therefore,

(11) φ(x) =∑k∈Z

ck√

2φ(2x− k)

for some constants {ck}. A wavelet ψ(x), forwhich {ψ(x− k)} spans the wavelet subspaceW0, can be defined by5

(12) ψ(x) =∑k∈Z

(−1)kc1−k√

2φ(2x− k).

Equations (11) and (12) generalize the equationsin (2) for the Haar case. The orthogonality of φand ψ leads to the following equation

(13)∑k∈Z

(−1)kc1−kck = 0.

This equation, and the second equation in (14)below, imply the orthogonality of the matrices,WN, used in the fast wavelet transform which weshall discuss later in this section.

Combining (11) with the first two integrals in(10), it follows that

(14)∑k∈Z

ck =√

2,∑k∈Z

|ck|2 = 1.

Similarly, assuming that L = 2, the equations in(9) combined with (12) imply

(15)∑k∈Z

(−1)kck = 0,∑k∈Z

k(−1)kck = 0.

5A simple proof, based on the MRA axioms, that{ψ(x− k)} spans W0 can be found in [10].

Figure 5. (a)[top] Signal. (b)[top] 37-term Daub4 approximation. (a)[bottom] 257-term Fourier cosineseries approximation. (b)[bottom] Highest level Daub4 wavelet coefficients of signal.

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666 NOTICES OF THE AMS VOLUME 44, NUMBER 6

And, for L > 2, equations (9) and (12) yield ad-ditional equations similar to the ones in (15).There is a finite set of coefficients that solvesthe equations in (14) and (15), namely,

(16)c0 =

1 +√

34√

2, c1 =

3 +√

34√

2,

c2 =3−√3

4√

2, c3 =

1−√34√

2

with all other ck = 0. Using these values of ck,the following iterative solution of (11)

(17)

φ0(x) = 1[0,1)(x),

φn(x) =∑k∈Z

ck√

2φn−1(2x− k),

for n ≥ 1,

converges to a continuous function φ(x) sup-ported on [0,3]. It then follows from (12) thatthe wavelet ψ(x) is also continuous and com-pactly supported on [0,3]. This wavelet ψ wehave been referring to as the Daub4 wavelet.The set of coefficients {ck} in (16) is the small-est set of coefficients that produce a continuouscompactly supported scaling function. Othersets of coefficients, related to higher values ofL, are given in [4] and [12].

Once the scaling function φ(x) and thewavelet function ψ(x) have been found, thenwe proceed as we did above in the Haar case. Wedefine the coefficients {αnk} and {βnk} by theequations in (5), where now φ and ψ are theDaubechies scaling function and wavelet, re-spectively. The scaling identity (11) and thewavelet definition (12) yield the following coef-ficient relations:

(18)

αnk =∑m∈Z

cm αn+1m+2k,

βnk =∑m∈Z

(−1)m c1−mαn+1m+2k.

In order to perform calculations in L2[0,1), wedefine the periodized wavelet ψn,k and the pe-riodized scaling function φn,k by equations (3)and (4), only now using the Daubechies waveletψ and scaling function φ in place of the Haarwavelet and scaling function. Theorem 2 re-mains valid using these periodic wavelets, butthe proof is more involved (see section 4.5 of [5]or section 3.11 of [8]). Therefore, for eachf ∈ L2[0,1) we can write

(19) f (x) = α00 +

∞∑n=0

2n−1∑k=0

βnkψn,k(x),

where α00 =

∫ 10 f (x)dx and βnk =∫ 1

0 f (x)ψn,k(x)dx . We also define the coefficients

αnk by αnk =∫ 10 f (x)φn,k(x)dx. And, we periodi-

cally extend f with period 1, also denoting thisperiodic extension by f. Then, for n ≥ 0 andk = 0,1, . . . ,2n − 1, we have

(20)

αnk =∫ 1

0f (x)2n/2

∑j∈Z

φ(2n(x + j)− k) dx

=∑j∈Z

∫ j+1

jf (x− j)2n/2φ(2nx− k)dx

= αnk.

Similar arguments show that βnk = βnk andαnk+2n = αnk and βnk+2n = βnk for n ≥ 0 andk = 0,1, . . . ,2n − 1. After periodizing (11) and(12), it follows that

(21)

αnk =∑m∈Z

cm αn+1m+2k,

βnk =∑m∈Z

(−1)m c1−m αn+1m+2k.

Remark. In the section “The Haar Wavelet” wesaw, for the Haar wavelet ψ , thatψn,k(x) = 2n/2ψ(2nx− k) for n ≥ 0 andk = 0,1, . . . ,2n − 1. Almost exactly the same re-sult holds for the Daubechies wavelets. For in-stance, if ψ is the Daub4 wavelet, then ψ is sup-ported on [0,3]. It follows, for n ≥ 2, that on theunit interval ψn,0 is supported on [0,3 · 2−n]. Onthe unit interval, we then haveψn,k(x) = 2n/2ψ(2nx− k) for k = 0,1, . . . ,2n − 3.Hence, for n ≥ 2, the periodized Daub4 waveletsψn,k are identical in L2[0,1) with the waveletfunctions 2n/2ψ(2nx− k) , except whenk = 2n − 2 and 2n − 1. Similar results hold for allthe Daubechies wavelets.

We can discretize the series in (19) by analogywith the Haar series. The coefficient relations in(21) yield a fast wavelet transform, W, an or-thogonal matrix defined by

W = (W2 ⊕ IM−2)

· · · (PM/2 ⊕ IM/2) (WM/2 ⊕ IM/2) PM WM

where each matrix WN is an N ×N orthogonalmatrix (as follows from (13) and the second equa-tion in (14)). The matrix WN is used to producethe (N − 1)st level coefficients {αN−1

k } and{βN−1k } from the Nth level scaling coefficients

{αNk } as follows:

WN[αN0 , . . . , α

N2N−1

]T

=[αN−1

0 , βN−10 , . . . , αN−1

2N−1−1, βN−12N−1−1

]T.

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If we use the coefficients c0, c1, c2, c3 definedin (16), then for N > 2, WN has the followingstructure :

For N = 2,W2 =( c2+c0 c3+c1

c1+c3 −c0−c2

). For other

Daubechies wavelets, there are other finite co-efficient sequences {ck}, and the matrices WNare defined similarly. The permutation matrix PNis the same one that we defined for the Haartransform and is used to sort the (N − 1)st levelcoefficients so that WN−1 can be applied to thescaling coefficients {αN−1

k }.As initial data for the wavelet transform we

can, as we did for the Haar transform, use dis-crete data of the form {fj}M−1

j=0 . The equationsin (15) then provide a discrete analog of the zeromoment conditions in (9) [for L = 2]; hence thewavelet coefficients will be 0 where the data islinear. In the last section, we saw how this canproduce effective compression of signals whenjust the 0th and 1st moments of ψ are 0.

It is often the case that the initial data are val-ues of a measured signal, i.e. {fk} = {F (xk)} , forxk = k2−n, where F is a signal obtained from ameasurement process. As shown in the previoussection, we can interpret the behavior of the dis-crete case based on properties of the functionF . A measured signal F is often described by aconvolution: F (x) =

∫∞−∞ g(t)µ(x− t)dt, where g

is the signal being measured and µ is called theinstrument function. Such convolutions generallyhave greater regularity than a typical functionin L2(R). For instance, if g ∈ L1(R) and is sup-ported on a finite interval and µ = 1[−r ,r ] forsome positive r , then F is continuous and sup-ported on a finite interval. By a linear change ofvariables, we may then assume that F is sup-ported on [0,1]. The data {F (xk)} then provideapproximations for the highest level scale coef-ficients {αRk } . If we assume that F has period 1,then

αRk = 2−R/2∫∞−∞F (x) 2R φ

(2R(x− xk)

)dx

≈ CMF (xk) .

Here we have replaced 1/√

2R by the scale fac-tor CM and used

∫∞−∞F (x) 2R φ(2R(x− xk))dx ≈ F (xk) .

This approximation will hold for all period 1 con-tinuous functions F and will be more accurate thelarger the value of 2R =M .

The higher the order of a Daubechies wavelet,the more of its moments are zero. A Daubechieswavelet of order 2L is defined by 2L nonzero coef-ficients {ck}, has its first L moments equal to zero,and is supported on the interval [0,2L− 1]. Gener-ally speaking, the more moments that are zero, themore wavelet coefficients that are nearly vanishingfor smooth functions F . This follows from consid-ering Taylor expansions. Suppose F (x) has an L-term Taylor expansion about the point xk = k2−n.That is,

F (x) =L−1∑j=0

1j !F (j)(xk) (x− xk)j

+1L!F (L)(tx) (x− xk)L

where tx lies between x and xk. Suppose also thatψ is supported on [−a,a] and that ψ has its first Lmoments equal to 0 and that |F (L)(x)| is boundedby a constant B on [(k− a)2−n, (k + a)2−n]. It thenfollows that

(22) |βnk| ≤B√

L + 1/2L!

( a2n

)L+1/2.

This inequality shows why ψ(x) having zero mo-ments produces a large number of small wavelet co-efficients. If F has some smoothness on an interval(c, d), then wavelet coefficients βnk corresponding tothe basis functions ψ (2n(x− xk)) whose supportsare contained in (c, d) will approach 0 rapidly as nincreases to ∞.

In addition to the Daubechies wavelets, there isanother class of compactly supported wavelets calledcoiflets. These wavelets are also constructed usingthe method outlined above. A coiflet of order 3L isdefined by 3L nonzero coefficients {ck} and has itsfirst L moments equal to zero and is supported onthe interval [−L,2L− 1]. A coiflet of order 3L is dis-tinguished from a Daubechies wavelet of order 2Lin that, in addition to ψ having its first L momentsequal to zero, the scaling function φ for the coifletalso has L− 1 moments vanishing. In particular,∫∞−∞ xjφ(x)dx = 0, for j = 1, . . . , L− 1. For a coiflet of

order 3L , supported on [−a,a], an argument simi-lar to the one that proves (22) shows that

|αRk − CMF (xk)| ≤ B√L + 1/2L!

( a2R

)L+1/2.

This inequality provides a stronger theoretical jus-tification for using the data {CMF (xk)} in place of

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668 NOTICES OF THE AMS VOLUME 44, NUMBER 6

the highest level scaling coefficients, beyondthe argument we gave above for Daubechieswavelets. The construction of coiflets was firstcarried out by Daubechies and named after Coif-man (who first suggested them).

Compression of SignalsOne of the most important applications ofwavelet analysis is to the compression of signals.As an example, let us use a Daub4 series to com-

press the signal {fj = F (j/1024)}1023j=0 where

F (x) = − log |x− 0.2|. See Figure 5(a)[top]. Forthis signal, a partial sum containing 99% of theenergy required only 37 coefficients (see Figure5(b)[top]). It certainly provides a visually ac-ceptable approximation of {fj}. In particular, thesharp maximum in the signal near x = 0.2 seemsto be reproduced quite well. The compressionratio is 1024: 37 ≈ 27: 1, which is an excellent re-sult considering that we also have 99% accu-racy. In addition, wavelet analysis has identi-fied the singularities of F . Notice in Figure5(b)[bottom] the peak in the wavelet coefficientsis near x = 0.2, where F has a singularity, andthe largest wavelet coefficient is near x = 1,where the periodic extension of F has a jump dis-continuity.

Turning to Fourier series, since the even pe-riodic extension is continuous, we used a discreteFourier cosine series to compress this signal. InFigure 5(a)[bottom] we show a 257-term discreteFourier cosine series partial sum for {fj}. Evenusing seven times as many coefficients as thewavelet series, the cosine series cannot repro-duce the sharp peak in the signal. Better resultscould be obtained in this case by either seg-

menting the interval and performing a cosine ex-pansion on each segment, or by using a smootherversion of the same idea involving local cosinebases [3, 9, 12, 1, 5].

One way to quantify the accuracy of theseapproximations is to use relative R.M.S. differ-ences. Given two sets of data {fj}M−1

j=0 and{gj}M−1

j=0 , their relative R.M.S. difference,

relative to {fj}, is defined by

D(f , g) =

√√√√√M−1∑j=0

|fj − gj |2/√√√√√M−1∑

j=0

|fj |2 .

For the example above, if we denote the waveletapproximation by fw , then D(f , fw ) =9.8× 10−3. For the Fourier cosine series ap-proximation, call it f c , we haveD(f , f c ) = 2.7× 10−2. A rule of thumb for a vi-sually acceptable approximation is to have a rel-ative R.M.S. difference of less than 10−2. The ap-proximations in this example are consistent withthis rule of thumb.

We can also do more localized analysis withR.M.S. differences. For example, over the subin-terval [.075, .325] centered on the singularityof F , we find that D(f , fw ) = 9.7× 10−3 andD(f , f c ) = 3.2× 10−2. These numbers confirmour visual impression that the wavelet seriesdoes a better job reproducing the sharp peak inthe signal. Or, using the subinterval [.25, .75], weget D(f , fw ) = 1.0× 10−2 and D(f , f c ) =3.3× 10−3, confirming our impression that bothseries do an adequate job approximating {fj}over this subinterval.

Figure 6. (a)[top] Signal. (b)[top] Signal’s coiflet30 transform, 7th level coefficients lie above the dottedline. (b)[middle] 7th level coefficients. (a)[bottom] Signal with added noise. (b)[bottom] Noisy signal’scoiflet30 transform. The horizontal lines are thresholds equal to ±0.15.

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Although in the examples we have discussedso far Fourier analysis did not compress the sig-nals very well, we do not wish to create the im-pression that this will always be true. In fact, ifa signal is composed of relatively few sinusoids,then Fourier analysis will provide very goodcompression. For example, consider the signal{fj = f (j/1024)}1023

j=0 where f (x) is defined in (1)with ν = 280. The Fourier coefficients for f aregraphed in Figure 1(b)[top]. They tend rapidly to0 away from the frequencies ±280; hence thesignal is composed of relatively few sinusoids.By computing a Fourier series partial sum thatuses only the 122 Fourier coefficients whosefrequencies are within ±30 of ±280, we ob-tained a signal g that was visually indistin-guishable from the original signal. In fact,D(f , g) = 5.1× 10−3. However, by compressing{fj} with the largest 122 Daub4 wavelet coef-ficients, we obtained D(f , fw ) = 2.7× 10−1 andthe compressed signal fw was only a crude ap-proximation of the original signal. The reasonthat compactly supported wavelets performpoorly in this case is that the large number ofrapid oscillations in the signal produce a corre-spondingly large number of high magnitudewavelet coefficients at the highest levels. Con-sequently, a significant fraction of all the waveletcoefficients are of high magnitude, so it is notpossible to significantly compress the signalusing compactly supported wavelets. This ex-ample illustrates that wavelet analysis is not apanacea for the problem of signal compression.In fact, much work has been done in creatinglarge collections of wavelet bases and Fourierbases and choosing for each signal a basis whichbest compresses it [12, 9, 3, 5].

Signal DenoisingWavelet analysis can also be used for removingnoise from signals. As an example, we show inFigure 6(a)[top] a discrete signal {f (j/1024)}1023

j=0where f (x) is defined by formula (1) with ν = 80.Each term of the form

(5 cos 2πνx)e−640π (x−k/8)2

we shall refer to as a blip. Notice that each blipis concentrated around x = k/8 , sincee−640π (x−k/8)2 rapidly decreases to 0 away fromx = k/8. This signal can be interpreted as rep-resenting the bit sequence 1 0 1 1 0 1 1. In Figure6(a)[bottom] we show this signal after it hasbeen corrupted by adding noise. In Figure6(b)[top] we show the coiflet30 wavelet coeffi-cients for the original signal. The rationale forusing wavelets to remove the noise is that theoriginal signal’s wavelet coefficients are closelycorrelated with the points near x = k/8 where theblips are concentrated. To demonstrate this, weshow in Figure 6(b)[middle] a graph of the 7th

level wavelet coefficients {β7k}

corresponding to the points {k2−7}27−1k=0 on the

unit interval. Comparing this to Figure 6(a)[top],we can see that the positions of this level’slargest magnitude wavelet coefficients are closelycorrelated with the positions of the blips. Simi-lar graphs could also be drawn for other levels,but the 7th level coefficients have the largestmagnitude. In Figure 6(b)[bottom] we show thecoiflet30 transform of the noisy signal. In spiteof the noise, the 7th level coefficients clearlystand out, although in a distorted form. By in-troducing a threshold, in this case 0.15, we canretain these 7th level coefficients and remove

Figure 7. (a)[top] Denoised signal using wavelet analysis. (a)[bottom] Denoised signal using Fourieranalysis. (b)[top] Fourier coefficients of noisy signal and filter function. (b)[middle] Moduli-squared of

Fourier coefficients of original signal. (b)[bottom] Moduli-squared of Fourier coefficients of waveletdenoised signal.

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670 NOTICES OF THE AMS VOLUME 44, NUMBER 6

most of the noise. In Figure 7(a)[top] we showthe reconstructed signal obtained by computinga partial sum using only those coefficients whosemagnitudes do not fall below 0.15. This recon-struction is not a flawless reproduction of theoriginal signal, but nevertheless the amount ofnoise has been greatly reduced, and the bit se-quence 1 0 1 1 0 1 1 can be determined.

In Figure 7(a)[bottom] we show the denoisedsignal obtained by filtering the Fourier coeffi-cients of the noisy signal (see Figure 7(b)[top])using the method of denoising described in thesection “Frequency Information, Denoising”. Incontrast to the wavelet denoising, the Fourier de-noising has retained a significant amount ofnoise in the spaces between the blips. The sourceof this retained noise is that most of the origi-nal noise’s Fourier coefficients are of uniformmagnitude distributed across all frequencies.Consequently, the filter preserves noise coeffi-cients corresponding to frequencies that werenot present in the original signal. These coeffi-cients generate sinusoids that oscillate across theentire interval [0,1]. The noise’s wavelet coeffi-cients also have almost uniform magnitude, butthe thresholding process eliminates them all,except the ones modifying the 7th level coeffi-cients of the original signal. Since these coeffi-cients’ wavelet basis functions are compactlysupported, this causes distortions in the recov-ered signal that are limited to neighborhoods ofthe positions of the 7th level coefficients. Con-sequently, there is still noise distorting the blips,but very little noise in between them.

It is also interesting to observe that thewavelet reconstructed signal and the originalsignal have similar frequency content. In Figure7(b)[middle] and Figure 7(b)[bottom], we havegraphed the moduli-squared of the Fourier co-efficients of the original signal and of the waveletdenoised signal, respectively. These graphs showthat the frequencies of the wavelet reconstruc-tion are, like the frequencies of the original sig-nal, concentrated around ±80 with the highestmagnitude frequencies located precisely at ±80.This shows that the coiflet30 wavelet has the abil-ity to extract frequency information. Much workhas been done in refining this ability, includingthe development of another class of bases calledwavelet packets [12, 9, 3, 5].

ConclusionIn this paper we have tried to show how the twomethodologies of Fourier analysis and waveletanalysis are used for various kinds of work. Ofcourse, we have only scratched the surface ofboth fields. Much more information can be foundin the references and their bibliographies.

References[1] P. Auscher, G. Weiss, and M. V. Wickerhauser,

Local sine and cosine basis of Coifman and Meyerand the construction of smooth wavelets, Wavelets:A Tutorial in Theory and Applications (C. K. Chui,ed.), Academic Press, 1992.

[2] W. Briggs and V. E. Henson, The DFT. Anowner’s manual for the Discrete Fourier Trans-form, SIAM, 1995.

[3] R. Coifman and M. V. Wickerhauser, Wavelets andadapted waveform analysis, Wavelets: Math-ematics and Applications (J. Benedetto and M. Fra-zier, eds.), CRC Press, 1994.

[4] I. Daubechies, Ten lectures on wavelets, SIAM,1992.

[5] E. Hernandez and G. Weiss, A first course onwavelets, CRC Press, 1996.

[6] S. Mallat, Multiresolution approximation andwavelet orthonormal bases of L2(R), Trans. Amer.Math. Soc. 315 (1989), 69–87.

[7] H. Malvar, Signal processing with lapped trans-forms, Artech House, 1992.

[8] Y. Meyer, Wavelets and operators, CambridgeUniv. Press, 1992.

[9] ———, Wavelets: Algorithms and applications, SIAM,1993.

[10] R. Strichartz, How to make wavelets, MAAMonthly 100, no. 6 (June–July 1993).

[11] J. Walker, Fast Fourier transforms, second edi-tion, CRC Press, 1996.

[12] M. V. Wickerhauser, Adapted wavelet analysisfrom theory to software, IEEE and A. K. Peters,1994.

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1. Wavelet-based Image Coding: An Overview 23

This spectral density is shown in Figure 10. From the shape of this densitywe see that in order to obtain segments in which the spectrum is flat, weneed to partition the spectrum finely at low frequencies, but only coarselyat high frequencies. The subbands we obtain by this procedure will be ap-proximately vectors of white noise with variances proportional to the powerspectrum over their frequency range. We can use an procedure similar tothat described for the KLT for coding the output. As we will see below,this particular partition of the spectrum is closely related to the wavelettransform.

4 Wavelets: A Different Perspective

4.1 Multiresolution Analyses

The discussion so far has been motivated by probabilistic considerations.We have been assuming our images can be reasonably well-approximatedby Gaussian random vectors with a particular covariance structure. Theuse of the wavelet transform in image coding is motivated by a ratherdifferent perspective, that of approximation theory. We assume that ourimages are locally smooth functions and can be well-modeled as piecewisepolynomials. Wavelets provide an efficient means for approximating suchfunctions with a small number of basis elements. This new perspectiveprovides some valuable insights into the coding process and has motivatedsome significant advances.We motivate the use of the wavelet transform in image coding using the

notion of a multiresolution analysis. Suppose we want to approximate acontinuous-valued square-integrable function f(x) using a discrete set ofvalues. For example, f(x) might be the brightness of a one-dimensional im-age. A natural set of values to use to approximate f(x) is a set of regularly-spaced, weighted local averages of f(x) such as might be obtained from thesensors in a digital camera.A simple approximation of f(x) based on local averages is a step function

approximation. Let φ(x) be the box function given by φ(x) = 1 for x ∈ [0, 1)and 0 elsewhere. A step function approximation to f(x) has the form

Af(x) =∑n

fnφ(x− n), (1.22)

where fn is the height of the step in [n, n + 1). A natural value for theheights fn is simply the average value of f(x) in the interval [n, n + 1).This gives fn =

∫ n+1

n f(x)dx.We can generalize this approximation procedure to building blocks other

than the box function. Our more generalized approximation will have the

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24 Geoffrey M. Davis, Aria Nosratinia

formAf(x) =

∑n

〈φ(x − n), f(x)〉φ(x− n). (1.23)

Here φ(x) is a weight function and φ(x) is an interpolating function cho-sen so that 〈φ(x), φ(x − n)〉 = δ[n]. The restriction on φ(x) ensures thatour approximation will be exact when f(x) is a linear combination of thefunctions φ(x − n). The functions φ(x) and φ(x) are normalized so that∫ |φ(x)|2dx = ∫ |φ(x)|2dx = 1. We will further assume that f(x) is periodicwith an integer period so that we only need a finite number of coefficientsto specify the approximation Af(x).We can vary the resolution of our approximations by dilating and con-

tracting the functions φ(x) and φ(x). Let φj(x) = 2j2φ(2jx) and φj(x) =

2j2 φ(2jx). We form the approximation Ajf(x) by projecting f(x) onto thespan of the functions {φj(x− 2−jk)}k∈Z, computing

Ajf(x) =∑

k

〈f(x), φj(x− 2−jk)〉φj(x − 2−jk). (1.24)

Let Vj be the space spanned by the functions {φj(x−2−jk)}. Our resolutionj approximation Ajf is simply a projection (not necessarily an orthogonalone) of f(x) onto the span of the functions φj(x − 2−jk).For our box function example, the approximation Ajf(x) corresponds to

an orthogonal projection of f(x) onto the space of step functions with stepwidth 2−j. Figure 11 shows the difference between the coarse approxima-tion A0f(x) on the left and the higher resolution approximation A1f(x) onthe right. Dilating scaling functions give us a way to construct approxima-tions to a given function at various resolutions. An important observationis that if a given function is sufficiently smooth, the differences betweenapproximations at successive resolutions will be small.Constructing our function φ(x) so that approximations at scale j are

special cases of approximations at scale j + 1 will make the analysis ofdifferences of functions at successive resolutions much easier. The functionφ(x) from our box function example has this property, since step functionswith width 2−j are special cases of step functions with step width 2−j−1.For such φ(x)’s the spaces of approximations at successive scales will benested, i.e. we have Vj ⊂ Vj+1.The observation that the differences Aj+1f−Ajf will be small for smooth

functions is the motivation for the Laplacian pyramid [26], a way of trans-forming an image into a set of small coefficients. The 1-D analog of theprocedure is as follows: we start with an initial discrete representation ofa function, the N coefficients of Ajf . We first split this function into thesum

Ajf(x) = Aj−1f(x) + [Ajf(x) − Aj−1f(x)]. (1.25)

Because of the nesting property of the spaces Vj , the difference Ajf(x) −Aj−1f(x) can be represented exactly as a sum of N translates of the func-

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1. Wavelet-based Image Coding: An Overview 25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

FIGURE 11. A continuous function f(x) (plotted as a dotted line) and box func-tion approximations (solid lines) at two resolutions. On the left is the coarseapproximation A0f(x) and on the right is the higher resolution approximationA1f(x).

tion φj(x). The key point is that the coefficients of these φj translates willbe small provided that f(x) is sufficiently smooth, and hence easy to code.Moreover, the dimension of the space Vj−1 is only half that of the spaceVj , so we need only N

2coefficients to represent Aj−1f . (In our box-function

example, the function Aj−1f is a step function with steps twice as wide asAjf , so we need only half as many coefficients to specify Aj−1f .) We havepartitioned Ajf into N difference coefficients that are easy to code andN2 coarse-scale coefficients. We can repeat this process on the coarse-scalecoefficients, obtaining N

2easy-to-code difference coefficients and N

4coarser

scale coefficients, and so on. The end result is 2N −1 difference coefficientsand a single coarse-scale coefficient.Burt and Adelson [26] have employed a two-dimensional version of the

above procedure with some success for an image coding scheme. The mainproblem with this procedure is that the Laplacian pyramid representationhas more coefficients to code than the original image. In 1-D we have twiceas many coefficients to code, and in 2-D we have 4

3 as many.

4.2 Wavelets

We can improve on the Laplacian pyramid idea by finding a more efficientrepresentation of the difference Dj−1f = Ajf − Aj−1f . The idea is thatto decompose a space of fine-scale approximations Vj into a direct sumof two subspaces, a space Vj−1 of coarser-scale approximations and itscomplement,Wj−1. This spaceWj−1 is a space of differences between coarseand fine-scale approximations. In particular, Ajf −Aj−1f ∈ W j−1 for anyf . Elements of the space can be thought of as the additional details thatmust be supplied to generate a finer-scale approximation from a coarse one.Consider our box-function example. If we limit our attention to functions

on the unit interval, then the space Vj is a space of dimension 2j. We can

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26 Geoffrey M. Davis, Aria Nosratinia

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

FIGURE 12. D0f(x), the difference between the coarse approximation A0f(x)and the finer scale approximation A1f(x) from figure 11.

1 1

1

1

-1 -1

φ (x) (x)ψ

FIGURE 13. The Haar scaling function and wavelet.

decompose Vj into the space Vj−1, the space of resolution 2−j+1 approxi-mations, andWj−1, the space of details. Because Vj−1 is of dimension 2j−1,Wj−1 must also have dimension 2j−1 for the combined space Vj to havedimension 2j. This observation about the dimension ofWj provides us witha means to circumvent the Laplacian pyramid’s problems with expansion.Recall that in the Laplacian pyramid we represent the difference Dj−1f

as a sum ofN fine-scale basis functions φj(x). This is more information thanwe need, however, because the space of functions Dj−1f is spanned by justN2 basis functions. Let c

jk be the expansion coefficient 〈φj(x− 2−jk), f(x)〉

in the resolution j approximation to f(x). For our step functions, eachcoefficient cj−1

k is the average of the coefficients cj2k and cj

2k+1 from theresolution j approximation. In order to reconstruct Ajf from Aj−1f , weonly need the N

2 differences cj2k+1 − cj

2k. Unlike the Laplacian pyramid,there is no expansion in the number of coefficients needed if we store thesedifferences together with the coefficients for Aj−1f .The differences cj

2k+1 − cj2k in our box function example correspond

(up to a normalizing constant) to coefficients of a basis expansion of thespace of details Wj−1. Mallat has shown that in general the basis forWj consists of translates and dilates of a single prototype function ψ(x),called a wavelet [27]. The basis for Wj is of the form ψj(x − 2−jk) whereψj(x) = 2

j2 ψ(x).

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1. Wavelet-based Image Coding: An Overview 27

Figure 13 shows the scaling function (a box function) for our box func-tion example together with the corresponding wavelet, the Haar wavelet.Figure 12 shows the function D0f(x), the difference between the approxi-mations A1f(x) and A1f(x) from Figure 11. Note that each of the intervalsseparated by the dotted lines contains a translated multiple of ψ(x).The dynamic range of the differences D0f(x) in Figure 12 is much smaller

than that of A1f(x). As a result, it is easier to code the expansion coeffi-cients of D0f(x) than to code those of the higher resolution approximationA1f(x). The splitting A1f(x) into the sum A0f(x) + D0f(x) performs apacking much like that done by the Karhunen-Loeve transform. For smoothfunctions f(x) the result of the splitting of A1f(x) into a sum of a coarserapproximation and details is that most of the variation is contained in A0f ,and D0f is near zero. By repeating this splitting procedure, partitioningA0f(x) into A−1f(x) + D−1f(x), we obtain the wavelet transform. Theresult is that an initial function approximation Ajf(x) is decomposed intothe telescoping sum

Ajf(x) = Dj−1f(x) +Dj−2f(x) + . . .+Dj−nf(x) + Aj−nf(x). (1.26)

The coefficients of the differences Dj−kf(x) are easier to code than theexpansion coefficients of the original approximation Ajf(x), and there isno expansion of coefficients as in the Laplacian pyramid.

4.3 Recurrence Relations

For the repeated splitting procedure above to be practical, we will need anefficient algorithm for obtaining the coefficients of the expansions Dj−kffrom the original expansion coefficients for Ajf . A key property of ourscaling functions makes this possible.One consequence of our partitioning of the space of resolution j approx-

imations, Vj , into a space of resolution j − 1 approximations Vj−1 andresolution j − 1 details Wj−1 is that the scaling functions φ(x) possessself-similarity properties. Because Vj−1 ⊂ Vj , we can express the functionφj−1(x) as a linear combination of the functions φj(x − n). In particularwe have

φ(x) =∑

k

hkφ(2x− k). (1.27)

Similarly, we have

φ(x) =∑

k

hkφ(2x− k)

ψ(x) =∑

k

gkφ(2x− k)

ψ(x) =∑

k

gkφ(2x− k). (1.28)

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28 Geoffrey M. Davis, Aria Nosratinia

These recurrence relations provide the link between wavelet transformsand subband transforms. Combining ( 4.3) and ( 4.3) with ( 4.1), we ob-tain a simple means for splitting the N expansion coefficients for Ajf intothe N

2 expansion coefficients for the coarser-scale approximationAj−1f andthe N

2 coefficients for the details Dj−1f . Both the coarser-scale approxima-

tion coefficients and the detail coefficients are obtained by convolving thecoefficients of Ajf with a filter and downsampling by a factor of 2. For thecoarser-scale approximation, the filter is a low-pass filter with taps given byh−k. For the details, the filter is a high-pass filter with taps g−k. A relatedderivation shows that we can invert the split by upsampling the coarser-scale approximation coefficients and the detail coefficients by a factor of2, convolving them with synthesis filters with taps hk and gk, respectively,and adding them together.We begin the forward transform with a signal representation in which

we have very fine temporal localization of information but no frequency lo-calization of information. Our filtering procedure splits our signal into low-pass and high-pass components and downsamples each. We obtain twicethe frequency resolution at the expense of half of our temporal resolution.On each successive step we split the lowest frequency signal componentin to a low pass and high pass component, each time gaining better fre-quency resolution at the expense of temporal resolution. Figure 14 showsthe partition of the time-frequency plane that results from this iterativesplitting procedure. As we discussed in Section 3.5, such a decomposition,with its wide subbands in the high frequencies and narrow subbands at lowfrequencies leads to effective data compression for a common image model,a Gaussian random process with an exponentially decaying autocorrelationfunction.The recurrence relations give rise to a fast algorithm for splitting a fine-

scale function approximation into a coarser approximation and a detailfunction. If we start with an N coefficient expansion Ajf , the first splitrequires kN operations, where k depends on the lengths of the filters we use.The approximation AJ−1 has N

2 coefficients, so the second split requires

Initial time/frequencylocalization

Time/frequencylocalization afterfirst split

Time/frequencylocalization aftersecond split

first high pass

first low pass

first high pass

second high pass

second low pass

split

split

time

frequency

FIGURE 14. Partition of the time-frequency plane created by the wavelet trans-form.

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1. Wavelet-based Image Coding: An Overview 29

kN2 operations. Each successive split requires half as much work, so the

overall transform requires O(N) work.

4.4 Wavelet Transforms vs. Subband Decompositions

The wavelet transform is a special case of a subband transform, as thederivation of the fast wavelet transform reveals. What, then, does thewavelet transform contribute to image coding? As we discuss below, thechief contribution of the wavelet transform is one of perspective. The math-ematical machinery used to develop the wavelet transform is quite differentthan that used for developing subband coders. Wavelets involve the analy-sis of continuous functions whereas analysis of subband decompositions ismore focused on discrete time signals. The theory of wavelets has a strongspatial component whereas subbands are more focused in the frequencydomain.The subband and wavelet perspectives represent two extreme points in

the analysis of this iterated filtering and downsampling process. The filtersused in subband decompositions are typically designed to optimize thefrequency domain behavior of a single filtering and subsampling. Becausewavelet transforms involve iterated filtering and downsampling, the analysisof a single iteration is not quite what we want. The wavelet basis functionscan be obtained by iterating the filtering and downsampling procedure aninfinite number of times. Although in applications we iterate the filteringand downsampling procedure only a small number of times, examinationof the properties of the basis functions provides considerable insight intothe effects of iterated filtering.A subtle but important point is that when we use the wavelet machin-

ery, we are implicitly assuming that the values we transform are actuallyfine-scale scaling function coefficients rather than samples of some function.Unlike the subband framework, the wavelet framework explicitly specifiesan underlying continuous-valued function from which our initial coefficientsare derived. The use of continuous-valued functions allows the use of pow-erful analytical tools, and it leads to a number of insights that can be usedto guide the filter design process. Within the continuous-valued frameworkwe can characterize the types of functions that can be represented exactlywith a limited number of wavelet coefficients. We can also address issuessuch as the smoothness of the basis functions. Examination of these is-sues has led to important new design criteria for both wavelet filters andsubband decompositions.A second important feature of the wavelet machinery is that it involves

both spatial as well as frequency considerations. The analysis of subbanddecompositions is typically more focused on the frequency domain. Coef-ficients in the wavelet transform correspond to features in the underlyingfunction in specific, well-defined locations. As we will see below, this ex-plicit use of spatial information has proven quite valuable in motivating

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30 Geoffrey M. Davis, Aria Nosratinia

some of the most effective wavelet coders.

4.5 Wavelet Properties

There is an extensive literature on wavelets and their properties. See [28], [23],or [29] for an introduction. Properties of particular interest for image com-pression are the the accuracy of approximation , the smoothness, and thesupport of these bases.The functions φ(x) and ψ(x) are the building blocks from which we con-

struct our compressed images. When compressing natural images, whichtend contain locally smooth regions, it is important that these buildingblocks be reasonably smooth. If the wavelets possess discontinuities orstrong singularities, coefficient quantization errors will cause these dis-continuities and singularities to appear in decoded images. Such artifactsare highly visually objectionable, particularly in smooth regions of images.Procedures for estimating the smoothness of wavelet bases can be foundin [30] and [31]. Rioul [32] has found that under certain conditions thatthe smoothness of scaling functions is a more important criterion than astandard frequency selectivity criterion used in subband coding.Accuracy of approximation is a second important design criterion that

has arisen from wavelet framework. A remarkable fact about wavelets isthat it is possible to construct smooth, compactly supported bases that canexactly reproduce any polynomial up to a given degree. If a continuous-valued function f(x) is locally equal to a polynomial, we can reproduce thatportion of f(x) exactly with just a few wavelet coefficients. The degreeof the polynomials that can be reproduced exactly is determined by thenumber of vanishing moments of the dual wavelet ψ(x). The dual waveletψ(x) has N vanishing moments provided that

∫xkψ(x)dx = 0 for k =

0, . . . , N . Compactly supported bases for L2 for which ψ(x) hasN vanishingmoments can locally reproduce polynomials of degree N − 1.The number of vanishing moments also determines the rate of conver-

gence of the approximationsAjf to the original function f as the resolutiongoes to infinity. It has been shown that ‖f−Ajf‖ ≤ C2−jN‖f(N)‖ where Nis the number of vanishing moments of ψ(x) and f(N) is the Nth derivativeof f [33, 34, 35].The size of the support of the wavelet basis is another important de-

sign criterion. Suppose that the function f(x) we are transforming is equalto polynomial of degree N − 1 in some region. If ψ has has N vanishingmoments, then any basis function for which the corresponding dual func-tion lies entirely in the region in which f is polynomial will have a zerocoefficient. The smaller the support of ψ is, the more zero coefficients wewill obtain. More importantly, edges produce large wavelet coefficients. Thelarger ψ is, the more likely it is to overlap an edge. Hence it is importantthat our wavelets have reasonably small support.

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1. Wavelet-based Image Coding: An Overview 31

There is a tradeoff between wavelet support and the regularity and accu-racy of approximation.Wavelets with short support have strong constraintson their regularity and accuracy of approximation, but as the support isincreased they can be made to have arbitrary degrees of smoothness andnumbers of vanishing moments. This limitation on support is equivalent tokeeping the analysis filters short. Limiting filter length is also an importantconsideration in the subband coding literature, because long filters lead toringing artifacts around edges.

5 A Basic Wavelet Image Coder

State-of-the-art wavelet coders are all derived from the transform coderparadigm. There are three basic components that underly current waveletcoders: a decorrelating transform, a quantization procedure, and an entropycoding procedure. Considerable current research is being performed on allthree of these components. Before we discuss state-of-the-art coders in thenext sections, we will describe a basic wavelet transform coder and discussoptimized versions of each of the components.7

5.1 Choice of Wavelet Basis

Deciding on the optimal wavelet basis to use for image coding is a difficultproblem. A number of design criteria, including smoothness, accuracy ofapproximation, size of support, and filter frequency selectivity are knownto be important. However, the best combination of these features is notknown.The simplest form of wavelet basis for images is a separable basis formed

from translations and dilations of products of one dimensional wavelets. Us-ing separable transforms reduces the problem of designing efficient waveletsto a one-dimensional problem, and almost all current coders employ sep-arable transforms. Recent work of Sweldens and Kovacevic [36] simplifiesconsiderably the design of non-separable bases, and such bases may provemore efficient than separable transforms.The prototype basis functions for separable transforms are φ(x)φ(y),

φ(x)ψ(y), ψ(x)φ(y), and ψ(x)ψ(y). Each step of the transform for suchbases involves two frequency splits instead of one. Suppose we have anN × N image. First each of the N rows in the image is split into a low-pass half and a high pass half. The result is an N × N

2sub-image and an

N × N2 high-pass sub-image. Next each column of the sub-images is split

into a low-pass and a high-pass half. The result is a four-way partition

7C++ source code for a coder that implements these components is available fromthe web site http://www.cs.dartmouth.edu/∼gdavis/wavelet/wavelet.html.

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32 Geoffrey M. Davis, Aria Nosratinia

of the image into horizontal low-pass/vertical low-pass, horizontal high-pass/vertical low-pass, horizontal low-pass/vertical high-pass, and horizon-tal high-pass/vertical high-pass sub-images. The low-pass/low-pass sub-image is subdivided in the same manner in the next step as is illustratedin Figure 17.Unser [35] shows that spline wavelets are attractive for coding appli-

cations based on approximation theoretic considerations. Experiments byRioul [32] for orthogonal bases indicate that smoothness is an importantconsideration for compression. Experiments by Antonini et al [37] find thatboth vanishing moments and smoothness are important, and for the filterstested they found that smoothness appeared to be slightly more impor-tant than the number of vanishing moments. Nonetheless, Vetterli andHerley [38] state that “the importance of regularity for signal processingapplications is still an open question.” The bases most commonly usedin practice have between one and two continuous derivatives. Additionalsmoothness does not appear to yield significant improvements in codingresults.Villasenor et al [39] have systematically examined all minimum order

biorthogonal filter banks with lengths ≤ 36. In addition to the criteriaalready mentioned, [39] also examines measures of oscillatory behavior andof the sensitivity of the coarse-scale approximations Ajf(x) to translationsof the function f(x). The best filter found in these experiments was a 7/9-tap spline variant with less dissimilar lengths from [37], and this filter isone of the most commonly used in wavelet coders.There is one caveat with regard to the results of the filter evaluation

in [39]. Villasenor et al compare peak signal to noise ratios generated by asimple transform coding scheme. The bit allocation scheme they use workswell for orthogonal bases, but it can be improved upon considerably inthe biorthogonal case. This inefficient bit allocation causes some promisingbiorthogonal filter sets to be overlooked.For biorthogonal transforms, the squared error in the transform domain

is not the same as the squared error in the original image. As a result,the problem of minimizing image error is considerably more difficult thanin the orthogonal case. We can reduce image-domain errors by performingbit allocation using a weighted transform-domain error measure that wediscuss in section 5.5. A number of other filters yield performance com-parable to that of the 7/9 filter of [37] provided that we do bit allocationwith a weighted error measure. One such basis is the Deslauriers-Dubucinterpolating wavelet of order 4 [40, 41], which has the advantage of havingfilter taps that are dyadic rationals. Both the spline wavelet of [37] and theorder 4 Deslauriers-Dubuc wavelet have 4 vanishing moments in both ψ(x)andψ(x), and the basis functions have just under 2 continuous derivativesin the L2 sense.One new very promising set of filters has been developed by Balasingham

and Ramstad [42]. Their design procedure combines classical filter design

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1. Wavelet-based Image Coding: An Overview 33

x=0

x

Dead zone

FIGURE 15. Dead-zone quantizer, with larger encoder partition around x = 0(dead zone) and uniform quantization elsewhere.

techniques with ideas from wavelet constructions and yields filters thatperform significantly better than the popular 7/9 filter set from [37].

5.2 Boundaries

Careful handling of image boundaries when performing the wavelet trans-form is essential for effective compression algorithms. Naive techniques forartificially extending images beyond given boundaries such as periodizationor zero-padding lead to significant coding inefficiencies. For symmetricalwavelets an effective strategy for handling boundaries is to extend the im-age via reflection. Such an extension preserves continuity at the boundariesand usually leads to much smaller wavelet coefficients than if discontinuitieswere present at the boundaries. Brislawn [43] describes in detail proceduresfor non-expansive symmetric extensions of boundaries. An alternative ap-proach is to modify the filter near the boundary. Boundary filters [44, 45]can be constructed that preserve filter orthogonality at boundaries. Thelifting scheme [46] provides a related method for handling filtering near theboundaries.

5.3 Quantization

Most current wavelet coders employ scalar quantization for coding. Thereare two basic strategies for performing the scalar quantization stage. Ifwe knew the distribution of coefficients for each subband in advance, theoptimal strategy would be to use entropy-constrained Lloyd-Max quantizersfor each subband. In general we do not have such knowledge, but we canprovide a parametric description of coefficient distributions by sending sideinformation. Coefficients in the high pass subbands of a wavelet transformare known a priori to be distributed as generalized Gaussians [27] centeredaround zero.A much simpler quantizer that is commonly employed in practice is a

uniform quantizer with a dead zone. The quantization bins, as shown inFigure 15, are of the form [n∆, (n+ 1)∆) for n ∈ Z except for the centralbin [−∆,∆). Each bin is decoded to the value at its center in the simplestcase, or to the centroid of the bin. In the case of asymptotically high rates,uniform quantization is optimal [47]. Although in practical regimes thesedead-zone quantizers are suboptimal, they work almost as well as Lloyd-

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34 Geoffrey M. Davis, Aria Nosratinia

Max coders when we decode to the bin centroids [48]. Moreover, dead-zonequantizers have the advantage that of being very low complexity and robustto changes in the distribution of coefficients in source. An additional ad-vantage of these dead-zone quantizers is that they can be nested to producean embedded bitstream following a procedure in [49].

5.4 Entropy Coding

Arithmetic coding provides a near-optimal entropy coding for the quan-tized coefficient values. The coder requires an estimate of the distributionof quantized coefficients. This estimate can be approximately specified byproviding parameters for a generalized Gaussian or a Laplacian density.Alternatively the probabilities can be estimated online. Online adaptiveestimation has the advantage of allowing coders to exploit local changesin image statistics. Efficient adaptive estimation procedures are discussedin [50] and [51].Because images are not jointly Gaussian random processes, the transform

coefficients, although decorrelated, still contain considerable structure. Theentropy coder can take advantage of some of this structure by conditioningthe encodings on previously encoded values. A coder of [49] obtains modestperformance improvements using such a technique.

5.5 Bit Allocation

The final question we need to address is that of how finely to quantize eachsubband. As we discussed in Section 3.2, the general idea is to determinethe number of bits bj to devote to coding subband j so that the total dis-tortion

∑j Dj(bj) is minimized subject to the constraint that

∑j bj ≤ b.

Here Dj(b) is the amount of distortion incurred in coding subband j withb bits. When the functions Dj(b) are known in closed form we can solvethe problem using the Kuhn-Tucker conditions. One common practice isto approximate the functions Dj(b) with the rate-distortion function for aGaussian random variable. However, this approximation is not very accu-rate at low bit rates. Better results may be obtained by measuring Dj(b)for a range of values of b and then solving the constrained minimizationproblem using integer programming techniques. An algorithm of Shohamand Gersho [52] solves precisely this problem.For biorthogonal wavelets we have the additional problem that squared

error in the transform domain is not equal to squared error in the invertedimage. Moulin [53] has formulated a multiscale relaxation algorithm whichprovides an approximate solution to the allocation problem for this case.Moulin’s algorithm yields substantially better results than the naive ap-proach of minimizing squared error in the transform domain.A simpler approach is to approximate the squared error in the image by

weighting the squared errors in each subband. The weight wj for subband

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1. Wavelet-based Image Coding: An Overview 35

j is obtained as follows: we set a single coefficient in subband j to 1 andset all other wavelet coefficients to zero. We then invert this transform.The weight wj is equal to the sum of the squares of the values in the re-sulting inverse transform. We allocate bits by minimizing the weighted sum∑

j wjDj(bj) rather than the sum∑

j Dj(bj). Further details may be foundin Naveen and Woods [54]. This weighting procedure results in substantialcoding improvements when using wavelets that are not very close to be-ing orthogonal, such as the Deslauriers-Dubuc wavelets popularized by thelifting scheme [46]. The 7/9 tap filter set of [37], on the other hand, hasweights that are all nearly 1, so this weighting provides little benefit.

5.6 Perceptually Weighted Error Measures

Our goal in lossy image coding is to minimize visual discrepancies betweenthe original and compressed images. Measuring visual discrepancy is a dif-ficult task. There has been a great deal of research on this problem, butbecause of the great complexity of the human visual system, no simple,accurate, and mathematically tractable measure has been found.Our discussion up to this point has focused on minimizing squared er-

ror distortion in compressed images primarily because this error metric ismathematically convenient. The measure suffers from a number of deficits,however. For example, consider two images that are the same everywhereexcept in a small region. Even if the difference in this small region is largeand highly visible, the mean squared error for the whole image will be smallbecause the discrepancy is confined to a small region. Similarly, errors thatare localized in straight lines, such as the blocking artifacts produced bythe discrete cosine transform, are much more visually objectionable thansquared error considerations alone indicate.There is evidence that the human visual system makes use of a multireso-

lution image representation; see [55] for an overview. The eye is much moresensitive to errors in low frequencies than in high. As a result, we can im-prove the correspondence between our squared error metric and perceivederror by weighting the errors in different subbands according to the eye’scontrast sensitivity in a corresponding frequency range. Weights for thecommonly used 7/9-tap filter set of [37] have been computed by Watson etal in [56].

6 Extending the Transform Coder Paradigm

The basic wavelet coder discussed in Section 5 is based on the basic trans-form coding paradigm, namely decorrelation and compaction of energy intoa few coefficients. The mathematical framework used in deriving the wavelettransform motivates compression algorithms that go beyond the traditional

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36 Geoffrey M. Davis, Aria Nosratinia

mechanisms used in transform coding. These important extensions are atthe heart of modern wavelet coding algorithms of Sections 7 and 9. We takea moment here to discuss these extensions.Conventional transform coding relies on energy compaction in an ordered

set of transform coefficients, and quantizes those coefficients with a priorityaccording to their order. This paradigm, while quite powerful, is based onseveral assumptions about images that are not always completely accurate.In particular, the Gaussian assumption breaks down for the joint distribu-tions across image discontinuities. Mallat and Falzon [57] give the followingexample of how the Gaussian, high-rate analysis breaks down at low ratesfor non-Gaussian processes.Let Y [n] be a random N -vector defined by

Y [n] =

X if n = PX if n = P + 1(modN)0 otherwise

(1.29)

Here P is a random integer uniformly distributed between 0 and N−1 andX is a random variable that equals 1 or -1 each with probability 1

2. X and

P are independent. The vector Y has zero mean and a covariance matrixwith entries

E{Y [n]Y [m]} =

2N for n = m1N for |n−m| ∈ {1, N − 1}0 otherwise

(1.30)

The covariance matrix is circulant, so the KLT for this process is the simplythe Fourier transform. The Fourier transform of Y is a very inefficient repre-sentation for coding Y . The energy at frequency k will be |1+e2πi k

N |2 whichmeans that the energy of Y is spread out over the entire low-frequency halfof the Fourier basis with some spill-over into the high-frequency half. TheKLT has “packed” the energy of the two non-zero coefficients of Y intoroughly N

2 coefficients. It is obvious that Y was much more compact in itsoriginal form, and could be coded better without transformation: Only twocoefficients in Y are non-zero, and we need only specify the values of thesecoefficients and their positions.As suggested by the example above, the essence of the extensions to

traditional transform coding is the idea of selection operators. Instead ofquantizing the transform coefficients in a pre-determined order of prior-ity, the wavelet framework lends itself to improvements, through judiciouschoice of which elements to code. This is made possible primarily becausewavelet basis elements are spatially as well as spectrally compact. In partsof the image where the energy is spatially but not spectrally compact (likethe example above) one can use selection operators to choose subsets ofthe wavelet coefficients that represent that signal efficiently. A most no-table example is the Zerotree coder and its variants (Section 7).

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1. Wavelet-based Image Coding: An Overview 37

More formally, the extension consists of dropping the constraint of lin-ear image approximations, as the selection operator is nonlinear. The workof DeVore et al. [58] and of Mallat and Falzon [57] suggests that at lowrates, the problem of image coding can be more effectively addressed as aproblem in obtaining a non-linear image approximation. This idea leads tosome important differences in coder implementation compared to the linearframework. For linear approximations, Theorems 3.1 and 3.3 in Section 3.1suggest that at low rates we should approximate our images using a fixedsubset of the Karhunen-Loeve basis vectors. We set a fixed set of transformcoefficients to zero, namely the coefficients corresponding to the smallesteigenvalues of the covariance matrix. The non-linear approximation idea,on the other hand, is to approximate images using a subset of basis func-tions that are selected adaptively based on the given image. Informationdescribing the particular set of basis functions used for the approximation,called a significance map, is sent as side information. In Section 7 we de-scribe zerotrees, a very important data structure used to efficiently encodesignificance maps.Our example suggests that a second important assumption to relax is

that our images come from a single jointly Gaussian source. We can obtainbetter energy packing by optimizing our transform to the particular imageat hand rather than to the global ensemble of images. The KLT providesefficient variance packing for vectors drawn from a single Gaussian source.However, if we have a mixture of sources the KLT is considerably less effi-cient. Frequency-adaptive and space/frequency-adaptive coders decomposeimages over a large library of different bases and choose an energy-packingtransform that is adapted to the image itself. We describe these adaptivecoders in Section 8.Trellis coded quantization represents a more drastic departure from the

transform coder framework. While TCQ coders operate in the transformdomain, they effectively do not use scalar quantization. Trellis coded quan-tization captures not only correlation gain and fractional bitrates, but alsothe packing gain of VQ. In both performance and complexity, TCQ is es-sentially VQ in disguise.The selection operator that characterizes the extension to the transform

coder paradigm generates information that needs to be conveyed to thedecoder as “side information”. This side information can be in the formof zerotrees, or more generally energy classes. Backward mixture estima-tion represents a different approach: it assumes that the side informationis largely redundant and can be estimated from the causal data. By cut-ting down on the transmitted side information, these algorithms achieve aremarkable degree of performance and efficiency.For reference, Table 1.1 provides a comparison of the peak signal to

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38 Geoffrey M. Davis, Aria Nosratinia

TABLE 1.1. Peak signal to noise ratios in decibels for coders discussedin the paper. Higher values indicate better performance.

Lena (bits/pixel) Barbara (bits/pixel)Type of Coder 1.0 0.5 0.25 1.0 0.5 0.25

JPEG [59] 37.9 34.9 31.6 33.1 28.3 25.2Optimized JPEG [60] 39.6 35.9 32.3 35.9 30.6 26.7

Baseline Wavelet [61] 39.4 36.2 33.2 34.6 29.5 26.6

Zerotree (Shapiro) [62] 39.6 36.3 33.2 35.1 30.5 26.8Zerotree (Said & Pearlman) [63] 40.5 37.2 34.1 36.9 31.7 27.8Zerotree (R/D optimized) [64] 40.5 37.4 34.3 37.0 31.3 27.2

Frequency-adaptive [65] 39.3 36.4 33.4 36.4 31.8 28.2Space-frequency adaptive [66] 40.1 36.9 33.8 37.0 32.3 28.7Frequency-adaptive + zerotrees [67] 40.6 37.4 34.4 37.7 33.1 29.3

TCQ subband [68] 41.1 37.7 34.3 – – –Bkwd. mixture estimation (EQ) [69] 40.9 37.7 34.6 – – –

noise ratios for the coders we discuss in the paper.8 The test images arethe 512×512 Lena image and the 512×512 Barbara image. Figure 16 showsthe Barbara image as compressed by JPEG, a baseline wavelet transformcoder, and the zerotree coder of Said and Pearlman [63]. The Barbara imageis particularly difficult to code, and we have compressed the image at a lowrate to emphasize coder errors. The blocking artifacts produced by the dis-crete cosine transform are highly visible in the image on the top right. Thedifference between the two wavelet coded images is more subtle but quitevisible at close range. Because of the more efficient coefficient encoding (tobe discussed below), the zerotree-coded image has much sharper edges andbetter preserves the striped texture than does the baseline transform coder.

7 Zerotree Coding

The rate-distortion analysis of the previous sections showed that optimalbitrate allocation is achieved when the signal is divided into subbands suchthat each subband contains a “white” signal. It was also shown that fortypical signals of interest, this leads to narrower bands in the low frequen-cies and wider bands in the high frequencies. Hence, wavelet transformshave very good energy compaction properties.This energy compaction leads to efficient utilization of scalar quantizers.

However, a cursory examination of the transform in Figure 17 shows thata significant amount of structure is present, particularly in the fine scalecoefficients. Wherever there is structure, there is room for compression, and

8More current numbers may be found on the web athttp://www.icsl.ucla.edu/∼ipl/psnr results.html

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1. Wavelet-based Image Coding: An Overview 39

FIGURE 16. Results of different compression schemes for the 512 × 512 Barbaratest image at 0.25 bits per pixel. Top left: original image. Top right: baselineJPEG, PSNR = 24.4 dB. Bottom left: baseline wavelet transform coder [61],PSNR = 26.6 dB. Bottom right: Said and Pearlman zerotree coder, PSNR =27.6 dB.

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40 Geoffrey M. Davis, Aria Nosratinia

advanced wavelet compression algorithms all address this structure in thehigher frequency subbands.One of the most prevalent approaches to this problem is based on ex-

ploiting the relationships of the wavelet coefficients across bands. A directvisual inspection indicates that large areas in the high frequency bands havelittle or no energy, and the small areas that have significant energy are sim-ilar in shape and location, across different bands. These high-energy areasstem from poor energy compaction close to the edges of the original image.Flat and slowly varying regions in the original image are well-described bythe low-frequency basis elements of the wavelet transform (hence leading tohigh energy compaction). At the edge locations, however, low-frequency ba-sis elements cannot describe the signal adequately, and some of the energyleaks into high-frequency coefficients. This happens similarly at all scales,thus the high-energy high-frequency coefficients representing the edges inthe image have the same shape.Our a priori knowledge that images of interest are formed mainly from

flat areas, textures, and edges, allows us to take advantage of the resultingcross-band structure. Zerotree coders combine the idea of cross-band cor-relation with the notion of coding zeros jointly (which we saw previouslyin the case of JPEG), to generate very powerful compression algorithms.The first instance of the implementation of zerotrees is due to Lewis

and Knowles [70]. In their algorithm the image is represented by a tree-structured data construct (Figure 18). This data structure is implied by adyadic discrete wavelet transform (Figure 19) in two dimensions. The rootnode of the tree represents the scaling function coefficient in the lowestfrequency band, which is the parent of three nodes. Nodes inside the treecorrespond to wavelet coefficients at a scale determined by their heightin the tree. Each of these coefficients has four children, which correspondto the wavelets at the next finer scale having the same location in space.These four coefficients represent the four phases of the higher resolutionbasis elements at that location. At the bottom of the data structure lie theleaf nodes, which have no children.Note that there exist three such quadtrees for each coefficient in the low

frequency band. Each of these three trees corresponds to one of three filter-ing orderings: there is one tree consisting entirely of coefficients arising fromhorizontal high-pass, vertical low-pass operation (HL); one for horizontallow-pass, vertical high-pass (LH), and one for high-pass in both directions(HH).The zerotree quantization model used by Lewis and Knowles was arrived

at by observing that often when a wavelet coefficient is small, its childrenon the wavelet tree are also small. This phenomenon happens because sig-nificant coefficients arise from edges and texture, which are local. It is notdifficult to see that this is a form of conditioning. Lewis and Knowles tookthis conditioning to the limit, and assumed that insignificant parent nodesalways imply insignificant child nodes. A tree or subtree that contains (or

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1. Wavelet-based Image Coding: An Overview 41

FIGURE 17. Wavelet transform of the image “Lena.”

LL3 LH3

HL3 HH3LH2

HL2

HH2

LH1

HL1

HH1

Typical Wavelet tree

FIGURE 18. Space-frequency structure of wavelet transform

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42 Geoffrey M. Davis, Aria Nosratinia

2H0

2H1

2H0

2H1

2H0

2H1

Low frequencyComponent

DetailComponents

InputSignal

FIGURE 19. Filter bank implementing a discrete wavelet transform

is assumed to contain) only insignificant coefficients is known as a zerotree.Lewis and Knowles used the following algorithm for the quantization

of wavelet coefficients: Quantize each node according to an optimal scalarquantizer for the Laplacian density. If the node value is insignificant ac-cording to a pre-specified threshold, ignore all its children. These ignoredcoefficients will be decoded as zeros at the decoder. Otherwise, go to eachof its four children and repeat the process. If the node was a leaf node anddid not have a child, go to the next root node and repeat the process.Aside from the nice energy compaction properties of the wavelet trans-

form, the Lewis and Knowles coder achieves its compression ratios by jointcoding of zeros. For efficient run-length coding, one needs to first find aconducive data structure, e.g. the zig-zag scan in JPEG. Perhaps the mostsignificant contribution of this work was to realize that wavelet domaindata provide an excellent context for run-length coding: not only are largerun lengths of zeros generated, but also there is no need to transmit thelength of zero runs, because they are assumed to automatically terminateat the leaf nodes of the tree. Much the same as in JPEG, this is a form ofjoint vector/scalar quantization. Each individual (significant) coefficient isquantized separately, but the symbols corresponding to small coefficientsin fact are representing a vector consisting of that element and the zerorun that follows it to the bottom of the tree.While this compression algorithm generates subjectively acceptable im-

ages, its rate-distortion performance falls short of baseline JPEG, which atthe time was often used for comparison purposes. The lack of sophisticationin the entropy coding of quantized coefficients somewhat disadvantages thiscoder, but the main reason for its mediocre performance is the way it gen-erates and recognizes zerotrees. As we have noted, whenever a coefficientis small, it is likely that its descendents are also insignificant. However, theLewis and Knowles algorithm assumes that small parents always have smalldescendents, and therefore suffers large distortions when this does not holdbecause it zeros out large coefficients. The advantage of this method is thatthe detection of zerotrees is automatic: zerotrees are determined by mea-suring the magnitude of known coefficients. No side information is requiredto specify the locations of zerotrees, but this simplicity is obtained at thecost of reduced performance. More detailed analysis of this tradeoff gave

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1. Wavelet-based Image Coding: An Overview 43

rise to the next generation of zerotree coders.

7.1 The Shapiro and Said-Pearlman Coders

The Lewis and Knowles algorithm, while capturing the basic ideas inherentin many of the later coders, was incomplete. It had all the intuition thatlies at the heart of more advanced zerotree coders, but did not efficientlyspecify significance maps, which is crucial to the performance of waveletcoders.A significance map is a binary function whose value determines whether

each coefficient is significant or not. If not significant, a coefficient is as-sumed to quantize to zero. Hence a decoder that knows the significance mapneeds no further information about that coefficient. Otherwise, the coeffi-cient is quantized to a non-zero value. The method of Lewis and Knowlesdoes not generate a significance map from the actual data, but uses oneimplicitly, based on a priori assumptions on the structure of the data. Onthe infrequent occasions when this assumption does not hold, a high priceis paid in terms of distortion. The methods to be discussed below make useof the fact that, by using a small number of bits to correct mistakes in ourassumptions about the occurrences of zerotrees, we can reduce the codedimage distortion considerably.The first algorithm of this family is due to Shapiro [71] and is known

as the embedded zerotree wavelet (EZW) algorithm. Shapiro’s coder wasbased on transmitting both the non-zero data and a significance map. Thebits needed to specify a significance map can easily dominate the coderoutput, especially at lower bitrates. However, there is a great deal of re-dundancy in a general significance map for visual data, and the bitratesfor its representation can be kept in check by conditioning the map valuesat each node of the tree on the corresponding value at the parent node.Whenever an insignificant parent node is observed, it is highly likely thatthe descendents are also insignificant. Therefore, most of the time, a “ze-rotree” significance map symbol is generated. But because p, the probabilityof this event, is close to 1, its information content, −p logp, is very small.So most of the time, a very small amount of information is transmitted,and this keeps the average bitrate needed for the significance map relativelysmall.Once in a while, one or more of the children of an insignificant node will

be significant. In that case, a symbol for “isolated zero” is transmitted.The likelihood of this event is lower, and thus the bitrate for conveyingthis information is higher. But it is essential to pay this price to avoidlosing significant information down the tree and therefore generating largedistortions.In summary, the Shapiro algorithm uses three symbols for significance

maps: zerotree, isolated zero, or significant value. But using this structure,and by conditionally entropy coding these symbols, the coder achieves very

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44 Geoffrey M. Davis, Aria Nosratinia

MSB

LSB

Wavelet Coefficients in scan order

1 1 1

1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 0 0 0 0 0 0 0 0 0 0 0 0

....

....

1 1 1 0 0 1 01 0 0 0 0 ....

1 1 11 1 1 0 0 ....

x x x

x x x x x x x x x x

x x x x x x

......

......

......

......

......

......

......

......

......

FIGURE 20. Bit plane profile for raster scan ordered wavelet coefficients.

good rate-distortion performance.In addition, Shapiro’s coder also generates an embedded code. Coders

that generate embedded codes are said to have the progressive transmissionor successive refinement property. Successive refinement consists of firstapproximating the image with a few bits of data, and then improving theapproximation as more and more information is supplied. An embeddedcode has the property that for two given rates R1 > R2, the rate-R2 codeis a prefix to the rate-R1 code. Such codes are of great practical interestfor the following reasons:

• The encoder can easily achieve a precise bitrate by continuing tooutput bits when it reaches the desired rate.

• The decoder can cease decoding at any given point, generating animage that is the best representation possible with the decoded num-ber of bits. This is of practical interest for broadcast applicationswhere multiple decoders with varying computational, display, andbandwidth capabilities attempt to receive the same bitstream. Withan embedded code, each receiver can decode the passing bitstreamaccording to its particular needs and capabilities.

• Embedded codes are also very useful for indexing and browsing, whereonly a rough approximation is sufficient for deciding whether theimage needs to be decoded or received in full. The process of screeningimages can be speeded up considerably by using embedded codes:after decoding only a small portion of the code, one knows if thetarget image is present. If not, decoding is aborted and the next imageis requested, making it possible to screen a large number of imagesquickly. Once the desired image is located, the complete image isdecoded.

Shapiro’s method generates an embedded code by using a bit-slice ap-proach (see Figure 20). First, the wavelet coefficients of the image are

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1. Wavelet-based Image Coding: An Overview 45

indexed into a one-dimensional array, according to their order of impor-tance. This order places lower frequency bands before higher frequencybands since they have more energy, and coefficients within each band ap-pear in a raster scan order. The bit-slice code is generated by scanningthis one-dimensional array, comparing each coefficient with a threshold T .This initial scan provides the decoder with sufficient information to recoverthe most significant bit slice. In the next pass, our information about eachcoefficient is refined to a resolution of T/2, and the pass generates anotherbit slice of information. This process is repeated until there are no moreslices to code.Figure 20 shows that the upper bit slices contain a great many zeros

because there are many coefficients below the threshold. The role of zerotreecoding is to avoid transmitting all these zeros. Once a zerotree symbolis transmitted, we know that all the descendent coefficients are zero, sono information is transmitted for them. In effect, zerotrees are a cleverform of run-length coding, where the coefficients are ordered in a way togenerate longer run lengths (more efficient) as well as making the runsself-terminating, so the length of the runs need not be transmitted.The zerotree symbols (with high probability and small code length) can

be transmitted again and again for a given coefficient, until it rises above thesinking threshold, at which point it will be tagged as a significant coefficient.After this point, no more zerotree information will be transmitted for thiscoefficient.To achieve embeddedness, Shapiro uses a clever method of encoding the

sign of the wavelet coefficients with the significance information. There arealso further details of the priority of wavelet coefficients, the bit-slice cod-ing, and adaptive arithmetic coding of quantized values (entropy coding),which we will not pursue further in this review. The interested reader isreferred to [71] for more details.Said and Pearlman [72] have produced an enhanced implementation of

the zerotree algorithm, known as Set Partitioning in Hierarchical Trees(SPHIT). Their method is based on the same premises as the Shapiro algo-rithm, but with more attention to detail. The public domain version of thiscoder is very fast, and improves the performance of EZW by 0.3-0.6 dB.This gain is mostly due to the fact that the original zerotree algorithmsallow special symbols only for single zerotrees, while in reality, there areother sets of zeros that appear with sufficient frequency to warrant spe-cial symbols of their own. In particular, the Said-Pearlman coder providessymbols for combinations of parallel zerotrees.Davis and Chawla [73] have shown that both the Shapiro and the Said

and Pearlman coders are members of a large family of tree-structured sig-nificance mapping schemes. They provide a theoretical framework that ex-plains in more detail the performance of these coders and describe an al-gorithm for selecting a member of this family of significance maps that isoptimized for a given image or class of images.

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46 Geoffrey M. Davis, Aria Nosratinia

7.2 Zerotrees and Rate-Distortion Optimization

In the previous coders, zerotrees were used only when they were detectedin the actual data. But consider for the moment the following hypotheticalexample: assume that in an image, there is a wide area of little activity, sothat in the corresponding location of the wavelet coefficients there exists alarge group of insignificant values. Ordinarily, this would warrant the useof a big zerotree and a low expenditure of bitrate over that area. Suppose,however, that there is a one-pixel discontinuity in the middle of the area,such that at the bottom of the would-be zerotree, there is one significantcoefficient. The algorithms described so far would prohibit the use of azerotree for the entire area.Inaccurate representation of a single pixel will change the average dis-

tortion in the image only by a small amount. In our example we can gainsignificant coding efficiency by ignoring the single significant pixel so thatwe can use a large zerotree. We need a way to determine the circumstancesunder which we should ignore significant coefficients in this manner.The specification of a zerotree for a group of wavelet coefficient is a form

of quantization. Generally, the values of the pixels we code with zerotreesare non-zero, but in using a zerotree we specify that they be decoded as ze-ros. Non-zerotree wavelet coefficients (significant values) are also quantized,using scalar quantizers. If we saves bitrate by specifying larger zerotrees,as in the hypothetical example above, the rate that was saved can be as-signed to the scalar quantizers of the remaining coefficients, thus quantizingthem more accurately. Therefore, we have a choice in allocating the bitrateamong two types of quantization. The question is, if we are given a unit ofrate to use in coding, where should it be invested so that the correspondingreduction in distortion is maximized?This question, in the context of zerotree wavelet coding, was addressed

by Xiong et al. [74], using well-known bit allocation techniques [1]. The cen-tral result for optimal bit allocation states that, in the optimal state, theslope of the operational rate-distortion curves of all quantizers are equal.This result is intuitive and easy to understand. The slope of the opera-tional rate-distortion function for each quantizer tells us how many unitsof distortion we add/eliminate for each unit of rate we eliminate/add. Ifone of the quantizers has a smaller R-D slope, meaning that it is givingus less distortion reduction for our bits spent, we can take bits away fromthis quantizer (i.e. we can reduce its step size) and give them to the other,more efficient quantizers. We continue to do so until all quantizers have anequal slope.Obviously, specification of zerotrees affects the quantization levels of non-

zero coefficients because total available rate is limited. Conversely, specify-ing quantization levels will affect the choice of zerotrees because it affectsthe incremental distortion between zerotree quantization and scalar quanti-zation. Therefore, an iterative algorithm is needed for rate-distortion opti-

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1. Wavelet-based Image Coding: An Overview 47

mization. In phase one, the uniform scalar quantizers are fixed, and optimalzerotrees are chosen. In phase two, zerotrees are fixed and the quantizationlevel of uniform scalar quantizers is optimized. This algorithm is guaranteedto converge to a local optimum [74].There are further details of this algorithm involving prediction and de-

scription of zerotrees, which we leave out of the current discussion. Theadvantage of this method is mainly in performance, compared to bothEZW and SPHIT (the latter only slightly). The main disadvantages of thismethod are its complexity, and perhaps more importantly, that it does notgenerate an embedded bitstream.

8 Frequency, space-frequency adaptive coders

8.1 Wavelet Packets

The wavelet transform does a good job of decorrelating image pixels in prac-tice, especially when images have power spectra that decay approximatelyuniformly and exponentially. However, for images with non-exponentialrates of spectral decay and for images which have concentrated peaks inthe spectra away from DC, we can do considerably better.Our analysis of Section 3.5 suggests that the optimal subband decompo-

sition for an image is one for which the spectrum in each subband is ap-proximately flat. The octave-band decomposition produced by the wavelettransform produces nearly flat spectra for exponentially decaying spectra.The Barbara test image shown in Figure 16 contains a narrow-band com-ponent at high frequencies that comes from the tablecloth and the stripedclothing. Fingerprint images contain similar narrow-band high frequencycomponents.The best basis algorithm, developed by Coifman and Wickerhauser [75],

provides an efficient way to find a fast, wavelet-like transform that providesa good approximation to the Karhunen-Loeve transform for a given image.As with the wavelet transform, we start by assuming that a given signalcorresponds to a sum of fine-scale scaling functions. The transform performsa change of basis, but the new basis functions are not wavelets but ratherwavelet packets [76].Like wavelets, wavelet packets are formed from translated and dilated

linear combinations of scaling functions. However, the recurrence relationsthey satisfy are different, and the functions form an overcomplete set. Con-sider a signal of length 2N . The wavelet basis for such a signal consists ofa scaling function and 2N − 1 translates and dilates of the wavelet ψ(x).Wavelet packets are formed from translates and dilates of 2N different pro-totype functions, and there are N2N different possible functions that canbe used to form a basis.Wavelet packets are formed from recurrence relations similar to those for

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48 Geoffrey M. Davis, Aria Nosratinia

wavelets and generalize the theoretical framework of wavelets. The simplestwavelet packet π0(x) is the scaling function φ(x). New wavelet packets πj(x)for j > 0 are formed by the recurrence relations

π2j(x) =∑

k

hkπj(2x− k) (1.31)

π2j+1(x) =∑

k

gkπj(2x− k). (1.32)

where the hk and gk are the same as those in the recurrence equations( 4.3) and ( 4.3).The idea of wavelet packets is most easily seen in the frequency do-

main. Recall from Figure 14 that each step of the wavelet transform splitsthe current low frequency subband into two subbands of equal width, onehigh-pass and one low-pass. With wavelet packets there is a new degree offreedom in the transform. Again there are N stages to the transform for asignal of length 2N , but at each stage we have the option of splitting thelow-pass subband, the high-pass subband, both, or neither. The high andlow pass filters used in each case are the same filters used in the wavelettransform. In fact, the wavelet transform is the special case of a waveletpacket transform in which we always split the low-pass subband. With thisincreased flexibility we can generate 2N possible different transforms in 1-D. The possible transforms give rise to all possible dyadic partitions of thefrequency axis. The increased flexibility does not lead to a large increasein complexity; the worst-case complexity for a wavelet packet transform isO(N logN).

8.2 Frequency Adaptive Coders

The best basis algorithm is a fast algorithm for minimizing an additive costfunction over the set of all wavelet packet bases. Our analysis of transformcoding for Gaussian random processes suggests that we select the basis thatmaximizes the transform coding gain. The approximation theoretic argu-ments of Mallat and Falzon [57] suggest that at low bit rates the basis thatmaximizes the number of coefficients below a given threshold is the bestchoice. The best basis paradigm can accommodate both of these choices.See [77] for an excellent introduction to wavelet packets and the best basisalgorithm. Ramchandran and Vetterli [65] describe an algorithm for findingthe best wavelet packet basis for coding a given image using rate-distortioncriteria.An important application of this wavelet-packet transform optimization

is the FBI Wavelet/Scalar Quantization Standard for fingerprint compres-sion. The standard uses a wavelet packet decomposition for the transformstage of the encoder [78]. The transform used is fixed for all fingerprints,however, so the FBI coder is a first-generation linear coder.

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1. Wavelet-based Image Coding: An Overview 49

The benefits of customizing the transform on a per-image basis dependconsiderably on the image. For the Lena test image the improvement inpeak signal to noise ratio is modest, ranging from 0.1 dB at 1 bit per pixelto 0.25 dB at 0.25 bits per pixel. This is because the octave band partitionsof the spectrum of the Lena image are nearly flat. The Barbara image (seeFigure 16), on the other hand, has a narrow-band peak in the spectrum athigh frequencies. Consequently, the PSNR increases by roughly 2 dB overthe same range of bitrates [65]. Further impressive gains are obtained bycombining the adaptive transform with a zerotree structure [67].

8.3 Space-Frequency Adaptive Coders

The best basis algorithm is not limited only to adaptive segmentation of thefrequency domain. Related algorithms permit joint time and frequency seg-mentations. The simplest of these algorithms performs adapted frequencysegmentations over regions of the image selected through a quadtree decom-position procedure [79, 80]. More complicated algorithms provide combina-tions of spatially varying frequency decompositions and frequency varyingspatial decompositions [66]. These jointly adaptive algorithms work partic-ularly well for highly nonstationary images.The primary disadvantage of these spatially adaptive schemes are that

the pre-computation requirements are much greater than for the frequencyadaptive coders, and the search is also much larger. A second disadvan-tage is that both spatial and frequency adaptivity are limited to dyadicpartitions. A limitation of this sort is necessary for keeping the complexitymanageable, but dyadic partitions are not in general the best ones.

9 Utilizing Intra-band Dependencies

The development of the EZW coder motivated a flurry of activity in thearea of zerotree wavelet algorithms. The inherent simplicity of the zerotreedata structure, its computational advantages, as well as the potential forgenerating an embedded bitstream were all very attractive to the codingcommunity. Zerotree algorithms were developed for a variety of applica-tions, and many modifications and enhancements to the algorithm weredevised, as described in Section 7.With all the excitement incited by the discovery of EZW, it is easy

to automatically assume that zerotree structures, or more generally inter-band dependencies, should be the focal point of efficient subband imagecompression algorithms. However, some of the best performing subbandimage coders known today are not based on zerotrees. In this section, weexplore two methods that utilize intra-band dependencies. One of themuses the concept of Trellis Coded Quantization (TCQ). The other uses both

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50 Geoffrey M. Davis, Aria Nosratinia

D D D D D D D D0 1 2 3 0 1 2 3

D D D D0 2 0 2

D D D D1 3 1 3

S

S

0

1

FIGURE 21. TCQ sets and supersets

inter- and intra-band information, and is based on a recursive estimation ofthe variance of the wavelet coefficients. Both of them yield excellent codingresults.

9.1 Trellis coded quantization

Trellis Coded Quantization (TCQ) [81] is a fast and effective method ofquantizing random variables. Trellis coding exploits correlations betweenvariables. More interestingly, it can use non-rectangular quantizer cells thatgive it quantization efficiencies not attainable by scalar quantizers. Thecentral ideas of TCQ grew out of the ground-breaking work of Ungerboeck[82] in trellis coded modulation. In this section we describe the operationalprinciples of TCQ, mostly through examples. We will briefly touch uponvariations and improvements on the original idea, especially at the lowbitrates applicable in image coding. In Section 9.2, we review the use ofTCQ in multiresolution image compression algorithms.The basic idea behind TCQ is the following: Assume that we want to

quantize a stationary, memoryless uniform source at the rate of R bitsper sample. Performing quantization directly on this uniform source wouldrequire an optimum scalar quantizer with 2N reproduction levels (symbols).The idea behind TCQ is to first quantize the source more finely, with 2R+k

symbols. Of course this would exceed the allocated rate, so we cannot havea free choice of symbols at all times.In our example we take k = 1. The scalar codebook of 2R+1 symbols is

partitioned into subsets of 2R−1 symbols each, generating four sets. In ourexample R = 2; see Figure 21. The subsets are designed such that each ofthem represents reproduction points of a coarser, rate-(R − 1) quantizer.The four subsets are designated D0, D1, D2, and D3. Also, define S0 =D0

⋃D2 and S1 = D1

⋃D3, where S0 and S1 are known as supersets.

Obviously, the rate constraint prohibits the specification of an arbitrarysymbol out of 2R+1 symbols. However, it is possible to exactly specify, withR bits, one element out of either S0 or S1. At each sample, assuming weknow which one of the supersets to use, one bit can be used to determinethe active subset, and R − 1 bits to specify a codeword from the subset.The choice of superset is determined by the state of a finite state machine,described by a suitable trellis. An example of such a trellis, with eight

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1. Wavelet-based Image Coding: An Overview 51

D0D2

D1D3

D0D2

D1D3

D0D2

D0D2

D1D3

D1D3

FIGURE 22. An 8-state TCQ trellis with subset labeling. The bits that specify thesets within the superset also dictate the path through the trellis.

states, is given in Figure 22. The subsets {D0, D1, D2, D3} are also used tolabel the branches of the trellis, so the same bit that specifies the subset(at a given state) also determines the next state of the trellis.Encoding is achieved by spending one bit per sample on specifying the

path through the trellis, while the remaining R−1 bits specify a codewordout of the active subset. It may seem that we are back to a non-optimalrate-R quantizer (either S0 or S1). So why all this effort? The answeris that we have more codewords than a rate-R quantizer, because thereis some freedom of choosing from symbols of either S0 or S1. Of coursethis choice is not completely free: the decision made at each sample islinked to decisions made at past and future sample points, through thepermissible paths of the trellis. But it is this additional flexibility thatleads to the improved performance. Availability of both S0 and S1 meansthat the reproduction levels of the quantizer are, in effect, allowed to “slidearound” and fit themselves to the data, subject to the permissible pathson the trellis.Before we continue with further developments of TCQ and subband cod-

ing, we should note that in terms of both efficiency and computationalrequirements, TCQ is much more similar to VQ than to scalar quantiza-tion. Since our entire discussion of transform coding has been motivated byan attempt to avoid VQ, what is the motivation for using TCQ in subbandcoding, instead of standard VQ? The answer lies in the recursive struc-ture of trellis coding and the existence of a simple dynamic programmingmethod, known as the Viterbi algorithm [83], for finding the TCQ code-words. Although it is true that block quantizers, such as VQ, are asymptot-ically as efficient as TCQ, the process of approaching the limit is far fromtrivial for VQ. For a given realization of a random process, the code vectorsgenerated by the VQ of size N − 1 have no clear relationship to those withvector dimension N . In contrast, the trellis encoding algorithm increases

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52 Geoffrey M. Davis, Aria Nosratinia

the dimensionality of the problem automatically by increasing the lengthof the trellis.The standard version of TCQ is not particularly suitable for image cod-

ing, because its performance degrades quickly at low rates. This is duepartially to the fact that one bit per sample is used to encode the trellisalone, while interesting rates for image coding are mostly below one bit persample. Entropy constrained TCQ (ECTCQ) improves the performance ofTCQ at low rates. In particular, a version of ECTCQ due to Marcellin [84]addresses two key issues: reducing the rate used to represent the trellis(the so-called “state entropy”), and ensuring that zero can be used as anoutput codeword with high probability. The codebooks are designed usingthe algorithm and encoding rule from [85].

9.2 TCQ subband coders

In a remarkable coincidence, at the 1994 International Conference in ImageProcessing, three research groups [86, 87, 88] presented similar but inde-pendently developed image coding algorithms. The main ingredients of thethree methods are subband decomposition, classification and optimal rateallocation to different subsets of subband data, and entropy-constrainedTCQ. These works have been brought together in [68]. We briefly discussthe main aspects of these algorithms.Consider a subband decomposition of an image, and assume that the sub-

bands are well represented by a non-stationary random process X, whosesamples Xi are taken from distributions with variances σ2

i . One can com-pute an “average variance” over the entire random process and performconventional optimal quantization. But better performance is possible bysending overhead information about the variance of each sample, and quan-tizing it optimally according to its own p.d.f.This basic idea was first proposed by Chen and Smith [89] for adaptive

quantization of DCT coefficients. In their paper, Chen and Smith proposedto divide all DCT coefficients into four groups according to their “activ-ity level”, i.e. variance, and code each coefficient with an optimal quan-tizer designed for its group. The question of how to partition coefficientsinto groups was not addressed, however, and [89] arbitrarily chose to formgroups with equal population.9

However, one can show that equally populated groups are not a always

9If for a moment, we disregard the overhead information, the problem of partitioning

the coefficients bears a strong resemblance to the problem of best linear transform.Both operations, namely the linear transform and partitioning, conserve energy. The

goal in both is to minimize overall distortion through optimal allocation of a finite rate.Not surprisingly, the solution techniques are similar (Lagrange multipliers), and they

both generate sets with maximum separation between low and high energies (maximumarithmetic to geometric mean ratio).

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1. Wavelet-based Image Coding: An Overview 53

a good choice. Suppose that we want to classify the samples into J groups,and that all samples assigned to a given class i ∈ {1, ..., J} are grouped intoa source Xi. Let the total number of samples assigned to Xi be Ni, and thetotal number of samples in all groups be N . Define pi = Ni/N to be theprobability of a sample belonging to the source Xi. Encoding the sourceXi at rate Ri results in a mean squared error distortion of the form [90]

Di(Ri) = ε2i σ2i 2

−2Ri (1.33)

where εi is a constant depending on the shape of the pdf. The rate allocationproblem can now be solved using a Lagrange multiplier approach, much inthe same way as was shown for optimal linear transforms, resulting in thefollowing optimal rates:

Ri =R

J+12log2

ε2i σ2i∏J

j=1(ε2j σ2

j )pj

(1.34)

where R is the total rate and Ri are the rates assigned to each group.Classification gain is defined as the ratio of the quantization error of theoriginal signal X, divided by that of the optimally bit-allocated classifiedversion.

Gc =ε2 σ2∏J

j=1(ε2j σ2

j )pj

(1.35)

One aims to maximize this gain over {pi}. It is not unexpected thatthe optimization process can often yield non-uniform {pi}, resulting inunequal population of the classification groups. It is noteworthy that non-uniform populations not only have better classification gain in general, butalso lower overhead: Compared to a uniform {pi}, any other distributionhas smaller entropy, which implies smaller side information to specify theclasses.The classification gain is defined for Xi taken from one subband. A gen-

eralization of this result in [68] combines it with the conventional codinggain of the subbands. Another refinement takes into account the side in-formation required for classification. The coding algorithm then optimizesthe resulting expression to determine the classifications. ECTCQ is thenused for final coding.Practical implementation of this algorithm requires attention to a great

many details, for which the interested reader is referred to [68]. For example,the classification maps determine energy levels of the signal, which arerelated to the location of the edges in the image, and are thus related indifferent subbands. A variety of methods can be used to reduce the overheadinformation (in fact, the coder to be discussed in the next section makes themanagement of side information the focus of its efforts) Other issues includealternative measures for classification, and the usage of arithmetic codedTCQ. The coding results of the ECTCQ based subband coding are some of

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54 Geoffrey M. Davis, Aria Nosratinia

the best currently available in the literature, although the computationalcomplexity of these algorithms is also considerably greater than the othermethods presented in this paper.

9.3 Mixture Modeling and Estimation

A common thread in successful subband and wavelet image coders is mod-eling of image subbands as random variables drawn from a mixture ofdistributions. For each sample, one needs to detect which p.d.f. of the mix-ture it is drawn from, and then quantize it according to that pdf. Sincethe decoder needs to know which element of the mixture was used for en-coding, many algorithms send side information to the decoder. This sideinformation becomes significant, especially at low bitrates, so that efficientmanagement of it is pivotal to the success of the image coder.All subband and wavelet coding algorithms discussed so far use this idea

in one way or another. They only differ in the constraints they put on sideinformation so that it can be coded efficiently. For example, zerotrees area clever way of indicating side information. The data is assumed from amixture of very low energy (zero set) and high energy random variables,and the zero sets are assumed to have a tree structure.The TCQ subband coders discussed in the last section also use the same

idea. Different classes represent different energies in the subbands, and aretransmitted as overhead. In [68], several methods are discussed to com-press the side information, again based on geometrical constraints on theconstituent elements of the mixture (energy classes).A completely different approach to the problem of handling information

overhead is explored in [69, 91]. These two works were developed simultane-ously but independently. The version developed in [69] is named EstimationQuantization (EQ) by the authors, and is the one that we present in thefollowing. The title of [91] suggests a focus on entropy coding, but in factthe underlying ideas of the two are remarkably similar. We will refer to thethe aggregate class as backward mixture-estimation encoding (BMEE).BMEE models the wavelet subband coefficients as non-stationary gener-

alized Gaussian, whose non-stationarity is manifested by a slowly varyingvariance (energy) in each band. Because the energy varies slowly, it can bepredicted from causal neighboring coefficients. Therefore, unlike previousmethods, BMEE does not send the bulk of mixture information as over-head, but attempts to recover it at the decoder from already transmitteddata, hence the designation “backward”. BMEE assumes that the causalneighborhood of a subband coefficient (including parents in a subband tree)has the same energy (variance) as the coefficient itself. The estimate of en-ergy is found by applying a maximum likelihood method to a training setformed by the causal neighborhood.Similar to other recursive algorithms that involve quantization, BMEE

has to contend with the problem of stability and drift. Specifically, the

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1. Wavelet-based Image Coding: An Overview 55

decoder has access only to quantized coefficients, therefore the estimator ofenergy at the encoder can only use quantized coefficients. Otherwise, theestimates at the encoder and decoder will vary, resulting in drift problems.This presents the added difficulty of estimating variances from quantizedcausal coefficients. BMEE incorporates the quantization of the coefficientsinto the maximum likelihood estimation of the variance.The quantization itself is performed with a dead-zone uniform quantizer

(see Figure 15). This quantizer offers a good approximation to entropy con-strained quantization of generalized Gaussian signals. The dead-zone andstep sizes of the quantizers are determined through a Lagrange multiplieroptimization technique, which was introduced in the section on optimalrate allocation. This optimization is performed offline, once each for a va-riety of encoding rates and shape parameters, and the results are stored ina look-up table. This approach is to be credited for the speed of the algo-rithm, because no optimization need take place at the time of encoding theimage.Finally, the backward nature of the algorithm, combined with quantiza-

tion, presents another challenge. All the elements in the causal neighbor-hood may sometimes quantize to zero. In that case, the current coefficientwill also quantize to zero. This degenerate condition will propagate throughthe subband, making all coefficients on the causal side of this degeneracyequal to zero. To avoid this condition, BMEE provides for a mechanism tosend side information to the receiver, whenever all neighboring elementsare zero. This is accomplished by a preliminary pass through the coeffi-cients, where the algorithm tries to “guess” which one of the coefficientswill have degenerate neighborhoods, and assembles them to a set. Fromthis set, a generalized Gaussian variance and shape parameter is computedand transmitted to the decoder. Every time a degenerate case happens, theencoder and decoder act based on this extra set of parameters, instead ofusing the backward estimation mode.The BMEE coder is very fast, and especially in the low bitrate mode

(less than 0.25 bits per pixel) is extremely competitive. This is likely tomotivate a re-visitation of the role of side information and the mechanismof its transmission in wavelet coders.

10 Future Trends

Current research in image coding is progressing along a number of fronts.At the most basic level, a new interpretation of the wavelet transformhas appeared in the literature. This new theoretical framework, called thelifting scheme [41], provides a simpler and more flexible method for design-ing wavelets than standard Fourier-based methods. New families of non-separable wavelets constructed using lifting have the potential to improve

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56 Geoffrey M. Davis, Aria Nosratinia

coders. One very intriguing avenue for future research is the exploration ofthe nonlinear analogs of the wavelet transform that lifting makes possible.The area of classification and backward estimation based coders is an

active one. Several research groups are reporting promising results [92, 93].One very promising research direction is the development of coded images

that are robust to channel noise via joint source and channel coding. Seefor example [94], [95] and [96].The adoption of wavelet based coding to video signals presents special

challenges. One can apply 2-D wavelet coding in combination to temporalprediction (motion estimated prediction), which will be a direct counterpartof current DCT-based video coding methods. It is also possible to considerthe video signal as a three-dimensional array of data and attempt to com-press it with 3-D wavelet analysis. This approach presents difficulties thatarise from the fundamental properties of the discrete wavelet transform.The discrete wavelet transform (as well as any subband decomposition) isa space-varying operator, due to the presence of decimation and interpo-lation. This space variance is not conducive to compact representation ofvideo signals, as described below.Video signals are best modeled by 2-D projections whose position in

consecutive frames of the video signal varies by unknown amounts. Becausevast amounts of information are repeated in this way, one can achieveconsiderable gain by representing the repeated information only once. Thisis the basis of motion compensated coding. However, since the waveletrepresentation of the same 2-D signal will vary once it is shifted10, thisredundancy is difficult to reproduce in the wavelet domain. A frequencydomain study of the difficulties of 3-D wavelet coding of video is presentedin [97], and leads to the same insights. Some attempts have also been madeon applying 3-D wavelet coding on the residual 3-D data after motioncompensation, but have met with indifferent success.

11 Summary and Conclusion

Image compression is governed by the general laws of information theoryand specifically rate-distortion theory. However, these general laws are non-constructive and the more specific techniques of quantization theory areneeded for the actual development of compression algorithms.Vector quantization can theoretically attain the maximum achievable

coding efficiency. However, VQ has three main impediments: computationalcomplexity, delay, and the curse of dimensionality. Transform coding tech-niques, in conjunction with entropy coding, capture important gains of VQ,

10Unless the shift is exactly by a correct multiple of M samples, where M is thedownsampling rate

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1. Wavelet-based Image Coding: An Overview 57

while avoiding most of its difficulties.Theoretically, the Karhunen-Loeve transform is optimal for Gaussian

processes. Approximations to the K-L transform, such as the DCT, haveled to very successful image coding algorithms such as JPEG. However,even if one argues that image pixels can be individually Gaussian, theycannot be assumed to be jointly Gaussian, at least not across the imagediscontinuities. Image discontinuities are the place where traditional codersspend the most rate, and suffer the most distortion. This happens becausetraditional Fourier-type transforms (e.g., DCT) disperse the energy of dis-continuous signals across many coefficients, while the compaction of energyin the transform domain is essential for good coding performance.The discrete wavelet transform provides an elegant framework for signal

representation in which both smooth areas and discontinuities can be rep-resented compactly in the transform domain. This ability comes from themulti-resolution properties of wavelets. One can motivate wavelets throughspectral partitioning arguments used in deriving optimal quantizers forGaussian processes. However, the usefulness of wavelets in compressiongoes beyond the Gaussian case.State of the art wavelet coders assume that image data comes from a

source with fluctuating variance. Each of these coders provides a mechanismto express the local variance of the wavelet coefficients, and quantizes thecoefficients optimally or near-optimally according to that variance. Theindividual wavelet coders vary in the way they estimate and transmit thisvariances to the decoder, as well as the strategies for quantizing accordingto that variance.Zerotree coders assume a two-state structure for the variances: either

negligible (zero) or otherwise. They send side information to the decoderto indicate the positions of the non-zero coefficients. This process yieldsa non-linear image approximation rather than the linear truncated KLT-based approximation motivated by our Gaussian model. The set of zero co-efficients are expressed in terms of wavelet trees (Lewis & Knowles, Shapiro,others) or combinations thereof (Said & Pearlman). The zero sets are trans-mitted to the receiver as overhead, as well as the rest of the quantized data.Zerotree coders rely strongly on the dependency of data across scales of thewavelet transform.Frequency-adaptive coders improve upon basic wavelet coders by adapt-

ing transforms according to the local inter-pixel correlation structure withinan image. Local fluctuations in the correlation structure and in the variancecan be addressed by spatially adapting the transform and by augmentingthe optimized transforms with a zerotree structure.Other wavelet coders use dependency of data within the bands (and

sometimes across the bands as well). Coders based on Trellis Coded Quan-tization (TCQ) partition coefficients into a number of groups, according totheir energy. For each coefficient, they estimate and/or transmit the groupinformation as well as coding the value of the coefficient with TCQ, ac-

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58 Geoffrey M. Davis, Aria Nosratinia

cording to the nominal variance of the group. Another newly developedclass of coders transmit only minimal variance information while achievingimpressive coding results, indicating that perhaps the variance informationis more redundant than previously thought.While some of these coders may not employ what might strictly be called

a wavelet transform, they all utilize a multi-resolution decomposition, anduse concepts that were motivated by wavelet theory. Wavelets and the ideasarising from wavelet analysis have had an indelible effect on the theory andpractice of image compression, and are likely to continue their dominantpresence in image coding research in the near future.

Acknowledgments: G. Davis thanks the Digital Signal Processing group atRice University for their generous hospitality during the writing of thispaper. This work has been supported in part by a Texas Instruments Vis-iting Assistant Professorship at Rice University and an NSF MathematicalSciences Postdoctoral Research Fellowship.

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1. Wavelet-based Image Coding: An Overview 63

[63] A. Said and W. A. Pearlman, “A new, fast, and efficient image codecbased on set partitioning in hierarchichal trees,” IEEE Trans. Circuitsand Systems for Video Technology, vol. 6, pp. 243–250, June 1996.

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[67] Z. Xiong, K. Ramchandran, M. Orchard, and K. Asai, “Waveletpackets-based image coding using joint space-frequency quantization,”Preprint, 1996.

[68] R. L. Joshi, H. Jafarkhani, J. H. Kasner, T. R. Fisher, N. Farvardin,and M. W. Marcellin, “Comparison of different methods of classifi-cation in subband coding of images,” IEEE Transactions on ImageProcessing, submitted.

[69] S. M. LoPresto, K. Ramchandran, and M. T. Orchard, “Image cod-ing based on mixture modeling of wavelet coefficients and a fastestimation-quantization framework,” in Proc. Data Compression Con-ference, (Snowbird, Utah), pp. 221–230, 1997.

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64 Geoffrey M. Davis, Aria Nosratinia

[74] Z. Xiong, K. Ramchandran, and M. T. Orchard, “Space-frequencyquantization for wavelet image coding,” IEEE Transactions on ImageProcessing, vol. 6, pp. 677–693, May 1997.

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[77] M. V. Wickerhauser, Adapted Wavelet Analysis from Theory to Soft-ware. Wellesley, MA: A. K. Peters, 1994.

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[79] C. Herley, J. Kovacevic, K. Ramchandran, and M. Vetterli, “Tilings ofthe time-frequency plane: Construction of arbitrary orthogonal basesand fast tiling algorithms,” IEEE Transactions on Signal Processing,vol. 41, pp. 3341–3359, Dec. 1993.

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[82] G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEETransactions on Information Theory, vol. IT-28, pp. 55–67, January1982.

[83] G. D. Forney, “The Viterbi algorithm,” Proceedings of the IEEE,vol. 61, pp. 268–278, March 1973.

[84] M. W. Marcellin, “On entropy-constrained trellis coded quantization,”IEEE Transactions on Communications, vol. 42, pp. 14–16, January1994.

[85] P. A. Chou, T. Lookabaugh, and R. M. Gray, “Entropy-constrainedvector quantization,” IEEE Transactions on Information Theory,vol. 37, pp. 31–42, January 1989.

[86] H. Jafarkhani, N. Farvardin, and C. C. Lee, “Adaptive image codingbased on the discrete wavelet transform,” in Proc. IEEE Int. Conf. Im-age Proc. (ICIP), vol. 3, (Austin, TX), pp. 343–347, November 1994.

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1. Wavelet-based Image Coding: An Overview 65

[87] R. L. Joshi, T. Fischer, and R. H. Bamberger, “Optimum classificationin subband coding of images,” in Proc. IEEE Int. Conf. Image Proc.(ICIP), vol. 2, (Austin, TX), pp. 883–887, November 1994.

[88] J. H. Kasner and M. W. Marcellin, “Adaptive wavelet coding of im-ages,” in Proc. IEEE Int. Conf. Image Proc. (ICIP), vol. 3, (Austin,TX), pp. 358–362, November 1994.

[89] W. H. Chen and C. H. Smith, “Adaptive coding of monochrome andcolor images,” IEEE Transactions on Communications, vol. COM-25,pp. 1285–1292, November 1977.

[90] N. S. Jayant and P. Noll, Digital Coding of waveforms. EnglewoodCliffs, NJ: Prentice-Hall, 1984.

[91] C. Chrysafis and A. Ortega, “Efficient context based entropy cod-ing for lossy wavelet image compression,” in Proc. Data CompressionConference, (Snowbird, Utah), pp. 241–250, 1997.

[92] Y. Yoo, A. Ortega, and B. Yu, “Progressive classification and adaptivequantization of image subbands.” Preprint, 1997.

[93] D. Marpe and H. L. Cycon, “Efficient pre-coding techniques forwavelet-based image compression.” Submitted to PCS, Berlin, 1997.

[94] P. G. Sherwood and K. Zeger, “Progressive image coding on noisychannels,” in Proc. Data Compression Conference, (Snowbird, UT),pp. 72–81, Mar. 1997.

[95] S. L. Regunathan, K. Rose, and S. Gadkari, “Multimode image codingfor noisy channels,” in Proc. Data Compression Conference, (Snow-bird, UT), pp. 82–90, Mar. 1997.

[96] J. Garcia-Frias and J. D. Villasenor, “An analytical treatment ofchannel-induced distortion in run length coded image subbands,” inProc. Data Compression Conference, (Snowbird, UT), pp. 52–61, Mar.1997.

[97] A. Nosratinia and M. T. Orchard, “A multi-resolution framework forbackward motion compensation,” in Proc. SPIE Symposium on Elec-tronic Imaging, (San Jose, CA), February 1995.

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Tutorial on Hidden Markov Models

Copyright c© 1995, 2002 Javier R. Movellan.

This is an open source document. Permission is granted to copy, distribute and/or modifythis document under the terms of the GNU Free Documentation License, Version 1.2 orany later version published by the Free Software Foundation; with no Invariant Sections,no Front-Cover Texts, and no Back-Cover Texts.

Endorsements

This document is endorsed by its copyright holder: Javier R. Movellan. Modified versionsof this document should delete this endorsement.

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1 Notation for discrete HMM

S = {S1, ..., SN}. Set of possible hidden states.

N. Number of distinct hidden states.

V = {v1, ..., vM}. Set of possible external observations.

M. Number of distinct external observations.

o = (o1, ..., oK). A sample of sequences of external observations (the training sample).Each element in o is an entire sequence of observations (e.g., a word).

K. Number of sequences (e.g., words) in the training sample.

o = (o1, ..., oT ). A sequence of external observations (e.g., a word). If there is more thana sequence I use a superscript l.

ot. A variable representing the external observation at time t. If there is more than asequence of interest, I use a superscript (e.g., o4

2 = 3 means that in sequence number 4, attime 2 we observe v3).

T . The number of time steps in the sequence .

q = (q1, ..., qK). Collection of sequences of internal states (internal state sequences thatmay correspond to each sequence in the training sequence).

q = (q1, ..., qT ). A sequence of internal states. If there is more than a sequence of interestI use a superscript to denote the sequence of interest.

qt. A variable representing the internal state at time t. If there is more than a sequence Iuse a superscript (e.g., q4

2 = 3 means that the system is in state S3 at time 2 in sequencenumber 4 of the training sample).

λ = (A, B, π). A hidden Markov model as defined by its A, B and π matrices.

aij = Pλ(qt+1 = j|qt = i). State transition probability for model λ.

A = {aij}. The NxN matrix of transition probabilities.

bj(k) = Pλ(ot = k|qt = j). Emission probability for observation vk by state Sj .

B = {bj(k)}. The MxN matrix of state to observations probabilities.

πi = Pλ(q1 = i). Initial state probability for model λ.

π = {πi}. The vector of initial state probabilities.

Q(λ, λ) =∑

q Pλ(q|0)log Pλ(qo). The auxiliary function maximized by the E-M algo-rithm. λ represents the current model, λ represents the new model under consideration.

Ql(λ, λ) The E-M function restricted to the sequence ol from the training sample.

αt(i) = Pλ(o1...ot qt = i). The forward variable for the sequence o at time t for state i. Ifthere is more than one sequence of interest I use a superscript to denote the sequence.

βt(i) = Pλ(ot+1...oT |qt = i). The scaled backward variable for the sequence o at time tfor state i. If there is more than one sequence of interest I use a superscript to denote thesequence.

αt(i). The scaled forward variable for the sequence o at time t for state i. If there is morethan one sequence of interest I use a superscript to denote the sequence.

βt(i). The scaled backward variable for the sequence o at time t for state i. If there is more

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than one sequence of interest I use a superscript to denote the sequence.

c1...cT . The scaling coefficients in the scaled forward and backward algorithm for thesequence ol. If there is more than one sequence of interest I use a superscript to denote thesequence.

γt(i) = Pλ(qt = i|o) If there is more than one sequence of interest I use a superscript todenote the sequence.

ξt(i, j) = Pλ(qt = i qt+1 = j|o). If there is more than one sequence of interest I use asuperscript to denote the sequence.

2 EM training with Discrete Observation Models

In this section we review two methods for training standard HMM models with discreteobservations: E-M training and Viterbi training.

2.1 The E-M auxiliary function

Let λ represent the current model and λ represent a candidate model. Our objective is tomake Pλ(o) ≥ Pλ(o), or equivalently log Pλ(o) ≥ log Pλ(o).

Due to the presence of stochastic constraints (e.g., aij ≥ and∑

j aij = 1) it turns out tobe easier to maximize an auxiliary function Q(·) rather than to directly maximize log Pλ.The E-M auxiliary function is defined as follows:

Q(λ, λ) =∑

q

Pλ(q|o)log Pλ(qo) (1)

Here we show that Q(λ, λ) ≥ Q(λ, λ) → log Pλ(o) ≥ log Pλ(o).

For any model λ or λ it must be true that

Pλ(o) =Pλ(oq)

Pλ(q|o)(2)

or logPλ(o) = logPλ(oq) − logPλ(q|o).

Also,logPλ(o) =

q

Pλ(q|o)logPλ(o) (3)

since logPλ(o) is a constant. Thus, from equation 2 it follows that

log Pλ(o) =∑

q

Pλ(q|o)logPλ(oq) −∑

q

Pλ(q|o)logPλ(q|o) (4)

= Q(λ, λ) −∑

q

Pλ(q|o)logPλ(q|o) (5)

Applying the Q(·, ·) function to (λ, λ) and to (λ, λ), it follows that,

Q(λ, λ) − Q(λ, λ) = log Pλ(o) − log Pλ(o) − KL(λ, λ) (6)

where KL(·, ·) is the Kullback-Leibler criterion (relative entropy) of the probability distri-bution Pλ(q|o) with respect to the probability distribution Pλ(q|o)

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KL(λ, λ) =∑

q

Pλ(q|o)logPλ(q|o)

Pλ(q|o)(7)

Rearranging terms,

log Pλ(o) − log Pλ(o) = Q(λ, λ) − Q(λ, λ) + KL(λ, λ) (8)

and since the KL criterion is always positive, it follows that if

Q(λ, λ) − Q(λ, λ) ≥ 0 (9)

thenlog Pλ(o)log Pλ(o) ≥ 0 (10)

2.2 The overall training sample E-M function

We defined the overall E-M function

Q(λ, λ) =∑

q

Pλ(q|o)log Pλ(qo) (11)

with o including the entire set of sequences in the training sample. Assuming that thesequences are independent, it follows that

Q(λ, λ) =∑

q

Pλ(q|o)∑

l

log Pλ(qlol) =∑

l

q

(log Pλ(qlol))Pλ(q|o) (12)

And since each state sequence ql depends only on the corresponding observation sequenceol it follows that

Q(λ, λ) =∑

l

ql

q−ql

(log Pλ(qlol))

K∏

m=1

Pλ(qm|o) (13)

where q − ql = (q1, ..., ql−1, ql+1, ..., qK) represents an entire collection of sequencesexcept for the sequence ql. Thus

Q(λ, λ) =∑

l

ql

q−ql

(log Pλ(qlol))Pλ(ql|ol)

K∏

m6=l

Pλ(qm|o) = (14)

=∑

l

ql

(log Pλ(qlol))Pλ(ql|ol)∑

q−ql

m6=l

Pλ(qm|o) (15)

and since∑

q−ql

K∏

m6=l

Pλ(qm|o) = 1 (16)

it follows that the E-M function can be decomposed into additive E-M functions, one perobservation sequence in the training sample:

Q(λ, λ) =∑

l

ql

Pλ(ql|ol) log Pλ(qlol) =∑

l

Ql(λ, λ) (17)

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2.3 Maximizing the E-M function

For simplicity let us start with the case in which there is a single sequence. The resultseasily generalize to multiple sequences. Since we work with a single sequence we maydrop the l superscript.

Q(λ, λ) =∑

q

Pλ(q|o)log Pλ(qo) (18)

And since,Pλ(qo) = πq1

bq1(o1)aq1q2

bq2(o2)aq2q3

... (19)

it follows that,Q(λ, λ) =

q

Pλ(q|o)log πq1+ (20)

+

Tl∑

t=1

q

Pλ(q|o)log bqt(ot)+ (21)

+

Tl−1∑

t=1

q

Pλ(q|o)log aqtqt+1(22)

The first term can be expressed as follows∑

q

Pλ(q|o)log πq1=

j

log πj

q

Pλ(q|o)δ(j, q1) (23)

where δ(j, q1) tells us to include only those cases in which q1 = j. Therefore,

q

Pλ(q|o)δ(j, q1) = Pλ(q1 = j|o) = γ1(j) (24)

The second term can be expressed as follows

Tl∑

t=1

q

Pλ(q|o)log bqt(ot) =

Tl∑

t=1

i

j

log bi(j)∑

q

Pλ(q|o)δ(i, qt)δ(j, ot) (25)

where δ(i, qt)δ(j, ot) tells us to include only those cases for which qt = i and ot = j.Therefore,

q

Pλ(q|o)δ(i, qt)δ(j, ot) = Pλ(qt = i|o)δ(ot, j) = γt(i)δ(ot, j) = (26)

The third term can be expressed as follows

Tl−1∑

t=1

q

Pλ(q|o)log aqtqt+1=

Tl−1∑

t=1

i

j

log aij

q

Pλ(q|o)δ(i, qt)δ(j, qt+1) (27)

where δ(i, qt)δ(j, qt+1) tells us to include only those cases for which qt = i and qt+1 = j.Therefore,

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q

Pλ(q|o)δ(i, qt)δ(j, qt+1) = Pλ(qt = i qt+1 = j|o) = ξt(i, j) (28)

Putting it together

Q(λ, λ) =∑

j

γ1(j) log πj+ (29)

+∑

i

j

log bi(j) (

Tl∑

t=1

γt(i)δ(ot, j))+ (30)

+∑

i

j

log aij (

Tl−1∑

t=1

ξt(i, j)) (31)

When there is more than a sequence, we just need to add up over sequences to obtain theoverall Q(·, ·) function

Q(λ, λ) =∑

j

log πj(∑

l

γl1(j))+ (32)

+∑

i

j

log bi(j) (∑

l

Tl∑

t=1

γlt(i)δ(o

lt, j))+ (33)

+∑

i

j

log aij(∑

l

Tl−1∑

t=1

ξlt(i, j)) (34)

Note that the part of the overall Q(λ, λ) function dependent on πj is of the form wj log xj

with xj = πj and

wj =

K∑

l=1

γl1(j) (35)

with constraints∑

j xj = 1, and xj ≥ 0.

Equivalently, the part of the overall Q(λ, λ) function dependent of bi(j) is of the formwj log xj with xj = bi(j) and

wj =K

l=1

Tl∑

t=1

γlt(j)δ(o

lt, j) (36)

with constraints∑

j xj = 1, and xj ≥ 0.

Finally, the part of the overall Q(λ, λ) function dependent of aij is also of the formwj log xj with xj = aij and

wj =

K∑

l=1

Tl−1∑

t=1

ξlt(i, j) (37)

with constraints∑

j xj = 1, and xj ≥ 0.

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It is easy to show that the maximum of a function of the form wj log xj with constraintsthat xj ≥ 0 and

j xj = 1 is achieved for

xj =wj

j wj(38)

The parameters of the new model λ that maximize the overall Q(λ, λ) function easilyfollow:

πi =PK

l=1γ1(i)

K

bi(j) =PK

l=1

PTlt=1

γt(i)δ(olt,j)

P

Kl=1

PTlt=1

γlt(i)

aij =PK

l=1

PTl−1

t=1ξt(i,j)

P

Kl=1

PTl−1

t=1γl

t(i)

(39)

2.4 Obtaining the E-M parameters from the scaled forward and backwardalgorithms

Section under construction

2.5 MM Training

MM training (Maximization Maximization) may be seen as an approximation to EM train-ing. In MM training we basically substitute the Expected value operation by a Max oper-ation thus the MM name. Other names for these algorithms are: Viterbi training (becausewe use the Viterbi algorithm to do the Max operation) and segmented K-means (K-meansis a classical clustering method that belongs to the MM family).

In Viterbi-based decoding, the degree of match between a model λ and an observationsequence ol is defined as ρ(λ, ol) = maxql log Pλ(ql ol) = log Pλ(ql ol). Wehave seen how for a fixed model λ, the Viterbi recurrence can be used to find ql andρ(λ, ol). When there is more than one training sequences they are assumed independentand ρ(λ, o) =

∑Kl=1 ρ(λ, ol). Thus, the optimal states can be found by applying Viterbi

decoding independently for each of the training sequences.

In MM training the objective is to find an optimal point of log Pλ(ql ol) with the model λand the state sequence q as optimizing variables.

To begin with, assume that Viterbi decoding has found the best sequence of hidden statesfor the current λ model: q = (q1, ..., qK). Once we have q we find a new model λ suchthat log Pλ(o q) ≥ log Pλ(o q). To do so simply define a dummy model λ such thatPλ(qo) = δ(q, o), thus the dummy model is such that only state sequence q can co-occurwith observation sequence o.

For such model, Pλ(q1 = j|ol) = δ(i, ql), Pλ(qt = i|ol) = δ(i, qt), and Pλ(qt = i qt+1 =

j|ol) = δ(i, qt)δ(j, qlt+1). Also note that Q(λ, λ) = log Pλ(o q), the function we want

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to optimize. Thus, substituting the λ parameters in the standard E-M formulas guaranteesthat Q(λ, λ) ≥ Q(λ, λ) or log Pλ(o q) ≥ log Pλ(o q)

Thus, the MM training rules are as follows:

πi =PK

l=1δ(i,ql

1)

K

bi(j) =PK

l=1

PTlt=1

δ(i,qlt)δ(j,o

lt)

PKl=1

PTlt=1

δ(i,qlt)

aij =PK

l=1

PTl−1

t=1δ(i,ql

t)δ(j,qlt+1)

PKl=1

PTl−1

t=1δ(i,ql

t)

(40)

Note that Viterbi decoding maximizes log Pλ(q, o) with respect to q for a fixed model (thefirst M step) then we maximizes log Pλ(q, o) with respect to λ, for a fixed q (the secondM step). Since we are always maximizing with respect to some variables, log Pλ(q o) canonly increase and convergence to a local maximum is guaranteed.

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3 Notation for Continuous Density Models

The notation for the continuous case is the same as the discrete case with the followingadditional terms.

P Number of dimensions per observation (e.g. cepstral coefficients): ot = (ot1 ...otP ).

M. Number of clusters within a state.

V = {v11, ..., v1M ..., vN1..., vNM}. Set of possible clusters of external observations (Mclusters per state). Each cluster vij is identified by a state index i and a cluster index j.

mt variable identifying the cluster index of the cluster that occurred at time t. For exampleif cluster vik occurred at time t then qt = i and mt = k.

m = (m1...mt). A sequence of cluster indexes, one per time step.

gik = Pλ(mt = k|qt = j). Emission probability (gain) of cluster kvik by state Sj .

φ(·). A kernel function (e.g. Gaussian) to model the probability density of a cluster ofobservations produced by a state.

bj(ot) = Pλ(ot|qt = j).

µik = (µik1, ..., µikP ). The centroid or prototype of cluster vik .

Σik =

σ2ik1 σik12 ... σik1p

σik21 σ2ik2 ... σik2p

...σikp1 σikp2 ... σ2

ikp

The covariance matrix of cluster vik .

σ2ikn. The variance of the lth dimension (e.g. cepstral) within cluster vik , a diagonal ele-

ment of Σik.

σiknm, The covariance between dimension l and dimension m within the cluster vik anoff-diagonal elements of Σik. Usually assumed zero.

4 Continuous Observation Models

In this section we study the E-M and Viterbi training procedures for continuous observationHMMs.

4.1 Mixtures of Densities

In the discrete case the observations are discrete, represented by an integer. In the con-tinuous case the observations at each time step are P-dimensional real-valued vectors (e.g.cepstrals coefficients). In our notation ot = (ot1, ..., otP ).

The continuous observation model produces sequences of observations in the followingway: At each time step the system generates a hidden state qt according to a a state to statetransition probability distribution aqt−1qt . Once qt has been generated, the system gener-ates a hidden cluster mt according to a state to cluster emission probability distributiongqtmt . Once the hidden cluster has been determined, an observation vector is producedprobabilistically according to some kernel probability distribution (e.g. Multivariate Gaus-sian). We can think of the clusters as low level hidden states embedded within high levelhidden states qt. For example, the high level hidden states may represent phonemes andthe low-level hidden clusters may represent acoustic categories within the same phoneme.For simplicity each state is assumed to have the same number of clusters (M) but the set ofclusters is different from state to state. Thus, there is a total of NxM clusters, M per state.

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We represent cluster k of state Sj as vjk and we use the variable mt to identify the clusternumber within a state at time t. Thus if qt = i and mt = k it means that at time t clustervjk occurred.

If we know the cluster at time t, the probability density of a vector of continuous obser-vations (e.g. cepstral coefficients) is modeled by a kernel function, usually a multivariateGaussian

Pλ(ot|vjk) = Pλ(ot|qt = j mt = k) = φ(ot, µjk , Σjk) (41)

Where φ(·) is the kernel function (e.g. multivariate Gaussian) µjk = (µjk1, ..., µjkP ) isa centroid or prototype that determines the position of the cluster in P-dimensional space,and

Σjk =

σ2jk1 σjk12 ... σjk1P

σjk21 σ2jk2 ... σjk2P

...σjkP1 σjkP2 ... σ2

jkP

(42)

is a covariance matrix that determines the width and tilt of the cluster in P-dimensionalspace. The diagonal terms {σ2

jk1...σ2jkP } are the cluster variances for each dimension.

They determine the spread of the cluster on each dimension. The off-diagonal elements areknown as the cluster covariances and they determine the tilt of the cluster. For the Gaussiancase, the kernel function is

φ(ot, µjk, Σjk) =1

(2π)P/2|Σjk |1/2e−d(ot,µjk) (43)

where |Σjk | is the determinant of the variance matrix, and d(ot, µjk) is known as the Ma-halanobis distance between the observation and the kernel’s centroid.

d(ot, µjk) =1

2(ot − µjk)Σ−1

jk (ot − µjk)′ (44)

Since the covariance matrices Σjk are symmetric, each kernel is defined by P (P+3)2 param-

eters (P for the centroids and P (P + 1)/2 for the variances). In practice the off-diagonalvariances are assumed zero, reducing the number of parameters per kernel to 2P. In suchcase the determinant |Σjk| is just the product of P scalar variances |Σjk| =

∏Pl=1 σ2

jkl, andthe Mahalanobis distance becomes a scaled Euclidean distance:

d(ot, µjk) =1

2

P∑

l=1

(otl − µjkl)2

σ2jkl

(45)

Given a state Sj , the system randomly chooses one of its M possible clusters with state tocluster emission probability P (mt = k|qt = j). This probability is assumed independentof t and thus it can be represented by a parameter with no time index. In our notationP (mt = k|qt = j) = gjk, the gain of the kth cluster embedded in state Sj .

Thus, the overall probability density of the observations generated by a state Sj is given bya weighted mixture of kernel functions.

Pλ(ot|qt = j) =

M∑

l=1

P (mt = l|qt = j)P (ot|qt = j, mt = k) = (46)

or

bj(ot) =

M∑

k=1

gjkφ(ot, µjk, Σjk) (47)

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4.2 Forward and backward variables

The un-scaled and the scaled algorithms work the same as in the discrete case. Only nowthe emission probability terms bj(ot) are modeled by a mixture of densities.

bj(ot) =

M∑

k=1

gjkφ(ot, µjk, Σjk) (48)

4.3 EM Training

In the continuous case the clusters are low level hidden states mt embedded within highlevel hidden states qt. Thus, the E-M function is defined over all possible q m sequencesof high-level and low-level hidden states

Q(λ, λ) =∑

q

m

Pλ(qm|o)log Pλ(qmo) (49)

and since

Pλ(qmo) = πq1gq1m1

φ(o1, µq1m1, Σq1m1

), ..., aqT−1qT gqT mT φ(oT , µqT mT , ΣqT mT )(50)

it follows that

Q(λ, λ) =∑

q

m

Pλ(qm|o)log πq1+ (51)

T−1∑

t=1

q

m

Pλ(qm|o)log aqtqt+1+ (52)

+

T∑

t=1

q

m

Pλ(qm|o)log gqtmt (53)

+

T∑

t=1

q

m

Pλ(qm|o)log φ(ot, µqtmt , Σqtmt) (54)

Since the factors in the first two terms are independent of m they simplify into∑

q

m

Pλ(qm|o)log πq1=

q

Pλ(q|o)log πq1(55)

and

T−1∑

t=1

q

m

Pλ(qm|o)log aqtqt+1=

T−1∑

t=1

q

Pλ(q|o)log aqtqt+1(56)

These terms are identical as in the discrete case and thus the same training rules for initialstate probabilities and for state transition probabilities apply here.

To find the training formulas for the cluster gains, we focus on the part of Q(·) dependenton the gain terms. This part can be transformed as follows

T∑

t=1

q

m

Pλ(qm|o)log gqt(mt) = (57)

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=

T∑

t=1

N∑

i=1

M∑

k=1

q

m

Pλ(qm|o)log gikδ(i, qt)δ(j, mt) (58)

=

T∑

t=1

N∑

i=1

M∑

k=1

Pλ(qt = i mt = k|o)log gik (59)

Thus the part of Q(·) that depends on gik is of the form wj log xj with xj = gik and

wj =

T∑

t=1

Pλ(qt = i mt = k|o) (60)

with constraints∑

j xj = 1 and xj ≤ 0, with maximum achieved for

xj =wj

j wj(61)

Thus

gik =

∑Tt=1 Pλ(qt = i mt = k|o)

∑Tt=1

∑Mk=1 Pλ(qt = i mt = k|o)

=

∑Tt=1 Pλ(qt = i mt = k|o)∑T

t=1 Pλ(qt = i|o)(62)

To find the learning rules for the centroids and variances we focus on the part of Q(·) thatdepends on the cluster centroids and variances, which is given by the following expression:

T∑

t=1

q

m

Pλ(qm|o)log φ(ot, µqtmt , Σqtmt) (63)

This expression can be transformed as followsT

t=1

q

m

Pλ(qm|o)log φ(ot, µqtmt , Σqtmt) = (64)

=

T∑

t=1

N∑

i=1

M∑

k=1

q

m

Pλ(qm|o)log φ(ot, µik, Σik)δ(i, qt)δ(k, mt) (65)

=

T∑

t=1

N∑

i=1

M∑

k=1

Pλ(qt = i mt = k|o)log φ(ot, µik , Σik) (66)

Thus, at a maximum,

∂µikn

T∑

t=1

N∑

i=1

M∑

k=1

Pλ(qt = i mt = k|o)log φ(ot, µik , Σik) = 0 (67)

and since

∂log φ(ot, µik, Σik)

∂µikn) =

1

σ2ikl

(otl − µikn) (68)

it follows that at a maximumT

t=1

Pλ(qt = i mt = k|o)(otl − µikn) = 0 (69)

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or

µikn =

∑Tt=1 Pλ(qt = i mt = k|o) otn

∑Tt=1 Pλ(qt = i|o)

(70)

A similar argument can be made for the diagonal variance σ2ikl. In this case

∂log φ(ot, µik, Σik)

∂σ2ikn

= −1

2σ2ikn

(1 −(otn − µikn)2

σ2ikn

) (71)

Thus, at a maximum

T∑

t=1

Pλ(qt = i mt = k|o)(1 −(otl − µikn)2

σ2ikn

) = 0 (72)

from which the re-estimation formula easily follows:

σ2ikn =

∑Tt=1 Pλ(qt = i mt = k|o)(otl − µikn)2

∑Tt=1 Pλ(qt = i|o)

(73)

Training for the mixture gains, mixture centroids, and mixture variances requires theP (qt = jmt = k|o) terms, for t = 1...T , j = 1..N , k = 1...M . To obtain theseterms note the following:

P (qt = j mt = k|o) = P (qt = j|o)P (mt = k|qt = j o) = (74)

= P (qt = j|o)P (mt = k|qt = j ot) (75)

= P (qt = j|o)P (otmt = k|qt = j)

P (ot|qt = j)(76)

= P (qt = j|o)P (mt = k|qt = j)P (ot|qt = jmt = k)

P (ot|qt = j)(77)

Thus

P (qt = jmt = k|o) = P (qt = j|o)gjk φ(ot, µik, Σik)

bj(ot)(78)

As in the discrete case, the P (qt = j|o) can be obtained through the scaled feed-forwardalgorithm.

For the case with multiple training sequences the overall Q(·) decomposes into additiveQl(·), one per training sequence. As a consequence we have to add in the numerator anddenominator of the training formulas the effects of each training sequence.

Summarizing, the E-M learning rules for the mixture of Gaussian densities case with diag-onal covariance matrices are as follows:

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πi =PK

l=1Pλ(q1=j|ol)

K

aij =PK

l=1

PTl−1

t=1Pλ(qt=i qt+1=j|ol)

P

Kl=1

PTl−1

t=1Pλ(qt=i|ol)

gik =PK

l=1

PTlt=1

Pλ(qt=i mt=k|ol)PK

l=1

PTt=1

Pλ(qt=i|ol)

µikn =PK

l=1

PTlt=1

Pλ(qt=i mt=k|ol) oltn

P

Tt=1

Pλ(qt=i|ol)

σ2ikn =

PKl=1

PTlt=1

Pλ(qt=i mt=k|ol) (otn−µikn)2

PKl=1

PTlt=1

Pλ(qt=i|ol)

(79)

where l = 1, ..., K indexes the training sequence, t = 1, ..., Tl indexes the time withina training sequence, i = 1, ..., N and j = 1, ..., N index the hidden state, k = 1, ..., Mindexes the cluster embedded within a state, and n = 1, ..., P indexes the dimension of thecontinuous vector of observations (e.g. the cepstral coefficient). The P (qt = jmt = k|o),Pλ(qt = i|ol) and Pλ(qt = i qt+1 = j|ol) terms are obtained from the scaled forward-backward algorithms according to the following formulas:

P (qt = i, mt = k|o) = P (qt = i|o) gik φ(ot,µik,Σik)bi(ot)

bi(ot) =∑M

k=1 gikφ(ot, µik, Σik)

φ(ot, µik, Σik) = 1(2π)P/2

Q

Pn=1

σikne−d(ot,µik)

d(ot, µik) = 12

∑Pn=1(

otn−µikn

σikn)2

Pλ(qt = i|ol) =αl

t(i)βlt(i)

P

i αlt(i)β

lt(i)

Pλ(qt = i qt+1 = j|ol) = αt(i)aij βt+1(j)bj(olt+1)

(80)

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4.4 Viterbi decoding

We can use Viterbi decoding to find the best possible sequence of high level hidden statesq and low level clusters m. There are two approaches to this problem. One approachattempts to find simultaneously the best joint sequence of high-level and low-level states.Thus the goal is to find qm = arg maxqm Pλ(qm|o). The second approach first finds thebest possible sequence of high level states q = arg maxq Pλ(q|o) and once q has beenfound, m is defined as m = arg maxm Pλ(m|q o). The two approaches do not necessarilyyield the same results. The second approach is the standard in the literature.

For the second, most used version, of Viterbi decoding, the same Viterbi recurrence as inthe discrete case applies but using the continuous version of bi(ot). Once we have qt, thedesired mt is simply the cluster within Sqt which is closest (in Mahalanobis distance) toot.

4.5 Viterbi training

The objective in Viterbi training (also known as segmental k-means) is to find an opti-mal point (local maximum) of log Pλ(oqm) with q and λ being the optimizing vari-ables. It does not matter how qm are found as long as a consistent procedure is usedthroughout training. As in the discrete case we define a dummy model λ such thatPλ(ql ml|o) = δ(ql, ql)δ(ml, ml). For such model, Pλ(ql

1 = j|ol) = δ(i, ql1), Pλ(ql

t =

i|ol) = δ(i, qlt), Pλ(ql

t = i qlt+1 = j|ol) = δ(i, ql

t)δ(j, qlt+1), and Pλ(ql

t = i mlt = k|ol) =

δ(i, qlt)δ(k, ml

t).

As in the discrete case note that Q(λ, λ) = log Pλ(o qm), the function we want to op-timize. Thus, substituting the λ parameters in the standard E-M formulas guarantees thatQ(λ, λ) ≥ Q(λ, λ), and thus log Pλ(o qm) ≥ log Pλ(o qm)

Thus, the Viterbi training rules are as follows:

πi =PK

l=1δ(i,ql

1)

K

aij =PK

l=1

PTl−1

t=1δ(i,ql

t)δ(j,qlt+1)

P

Kl=1

PTl−1

t=1δ(i,ql

t)

gik =PK

l=1

PTlt=1

δ(i,qlt)δ(k,ml

t)PK

l=1

PTt=1

δ(i,qlt)

µikn =PK

l=1

PTlt=1

δ(i,qlt)δ(k,ml

t) oltn

P

Tt=1

δ(i,qlt)

σ2ikn =

PKl=1

PTlt=1

δ(i,qlt)δ(k,ml

t)(otn−µikn)2

PKl=1

PTlt=1

δ(i,qlt)

(81)

Since Viterbi decoding maximizes Pλ(q, m, o) with respect to q and m and Viterbi train-ing maximizes Pλ(q, m, o) with respect to λ, repeatedly applying Viterbi decoding and

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Viterbi training, can only make Pλ(q, m, o) increase and convergence to a local maximumis guaranteed.

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5 Factored Sampling Methods for Continuous State Models

Many recognition problems can be framed in terms of infering something about qt theinternal state of a system, based on a sequence of observations o = o1 · · · ot. These infer-ences are in many cases based on estimates of p(qt|o1 · · · ot). When the states are discreteand countable, these conditonal state probabilities can be obtained using the forwards algo-rithm. However, the algorithm cannot be used when the states are continuous. In such case,direct sampling methods are appropriate. Here is an example of how these methods work.We start with a sensor model: p(ot|qt) and a Markovian state dynamics model p(qt+1|qt).Our goal is to obtain estimates of p(qt|ot) for all t.

1. RecursionAssume we have an estimate p(qt|ot). Our goal is to update that estimate for thenext time step p(qt+1|ot+1).

(a) First we draw a random sample X from p(qt|ot). This sample will implicitelydefine a re-estimation of p(qt|ot) in terms of a mixture of delta functions:p(qt|ot) = 1

N

∑Ni=1 δ(qt, xi)

(b) For each observation xi we obtain another random observation yi using thestate dynamics p(qt+1|qt = xi). The new sample Y = {y1 · · · yn} im-plicitely defines our estimates of p(qt+1|ot).p(qt|ot) = 1

N

∑Ni=1 δ(qt+1, yi)

(c) We know that

p(qt+1|o1 · · · ot+1) =p(o1 · · · ot)

p(o1 · · · ot+1)p(qt+1ot+1|o1 · · · ot) = (82)

p(o1 · · · ot)

p(o1 · · · ot+1)p(qt+1|o1 · · · ot)p(ot+1|qt+1)

The fraction is a constant K(ot+1) independent of qt+1, we already have anestimate of p(qt+1|ot) so we just need to weight it by p(ot+1|qt+1).p(qt+1|ot+1) = K 1

N

∑Ni=1 δ(qt+1, yi)p(ot+1|qt+1) =

=1

(N)∑

i p(ot+1|yi)

N∑

i=1

δ(qt+1, yi)p(ot+1|yi)

We can now use p(qt+1|ot+1) to estimate parameters like the mean or thevariance of the distribution. More generally,ω =

dqt+1Q(p(qt+1|ot+1), qt+1)

2. InitializationThe initialization step is basically the same as the recursion step only that insteadof using the state transition probabilities we use the initial state probabilities

(a) Obtain a sample of N random states : X= {x1 · · ·xN} from the initial stateprobability function π(·). These N samples will implicitely work as ourestimate of the initial state probability.

p(q1) =1

N

N∑

i=1

δ(q1, yi) (83)

(b) Weight each observation by the sensor probability p(o1|q1 = yi). This de-fines our initial distribution estimates

p(q0|o1) =1

N∑N

i=1 p(o1|xi)δ(q0, yi)p(o0|yi) (84)

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