CSE489-02 &CSE589-02 Multimedia Processing

Lecture 8. Wavelet Transform

Spring 2009

Wavelet Definition

“The wavelet transform is a tool that cuts up data, functions or operators into different frequency components, and then studies each component with a resolution matched to its scale”

Dr. Ingrid Daubechies, Lucent, Princeton U.

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Fourier vs. Wavelet FFT, basis functions: sinusoids

Wavelet transforms: small waves, called wavelet

FFT can only offer frequency information

Wavelet: frequency + temporal information

Fourier analysis doesn’t work well on discontinuous, “bursty” data music, video, power, earthquakes,…

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Fourier vs. Wavelet

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► Fourier Loses time (location) coordinate completely Analyses the whole signal Short pieces lose “frequency” meaning

► Wavelets Localized time-frequency analysis Short signal pieces also have significance Scale = Frequency band

Fourier transform

Fourier transform:

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Wavelet Transform

Scale and shift original waveform

Compare to a wavelet

Assign a coefficient of similarity

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Scaling-- value of “stretch”

Scaling a wavelet simply means stretching (or compressing) it.

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f(t) = sin(t)

scale factor1

Scaling-- value of “stretch”

Scaling a wavelet simply means stretching (or compressing) it.

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f(t) = sin(2t)scale factor 2

Scaling-- value of “stretch”

Scaling a wavelet simply means stretching (or compressing) it.

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f(t) = sin(3t)scale factor 3

More on scaling It lets you either narrow down the frequency band of

interest, or determine the frequency content in a narrower time interval

Scaling = frequency band

Good for non-stationary data

Low scaleCompressed wavelet Rapidly changing detailsHigh frequency

High scale Stretched wavelet Slowly changing, coarse features Low frequency

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Scale is (sort of) like frequency

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Small scale-Rapidly changing details, -Like high frequency

Large scale-Slowly changing details-Like low frequency

Scale is (sort of) like frequency

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The scale factor works exactly the same with wavelets. The smaller the scale factor, the more "compressed" the wavelet.

Shifting

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Shifting a wavelet simply means delaying (or hastening) its onset. Mathematically, delaying a function  f(t)   by k is represented by f(t-k)

Shifting

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C = 0.0004

C = 0.0034

Five Easy Steps to a Continuous Wavelet Transform

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1. Take a wavelet and compare it to a section at the start of the original signal.

2. Calculate a correlation coefficient c

Five Easy Steps to a Continuous Wavelet Transform

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3. Shift the wavelet to the right and repeat steps 1 and 2 until you've covered the whole signal.

4. Scale (stretch) the wavelet and repeat steps 1 through 3.

5. Repeat steps 1 through 4 for all scales.

Coefficient Plots

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Discrete Wavelet Transform “Subset” of scale and position based on power of

two rather than every “possible” set of scale and

position in continuous wavelet transform

Behaves like a filter bank: signal in, coefficients out

Down-sampling necessary (twice as much data as original signal)

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Discrete Wavelet transform

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signal

filters

Approximation(a)

Details(d)

lowpass highpass

Results of wavelet transform — approximation and details Low frequency:

approximation (a)

High frequency details (d)

“Decomposition” can be performed iteratively

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Wavelet synthesis

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•Re-creates signal from coefficients

•Up-sampling required

Multi-level Wavelet Analysis

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Multi-level wavelet decomposition tree Reassembling original signal

2-D 4-band filter bank

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Approximation

Vertical detail

Horizontal detail

Diagonal details

An Example of One-level Decomposition

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An Example of Multi-level Decomposition

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Wavelet Series Expansions

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0 0

0

0

2

, ,

Wavelet series expansion of function ( ) ( )

relative to wavelet ( ) and scaling function ( )

( ) ( ) ( ) ( ) ( )

where ,

( ) : approximation and/or scaling coefficients

j j k j j kk j j k

j

f x L

x x

f x c k x d k x

c k

d

( ) : detail and/or wavelet coefficientsj k

Wavelet Series Expansions

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0 0 0, ,

, ,

( ) ( ), ( ) ( ) ( )

( ) ( ), ( ) ( ) ( )

j j k j k

j j k j k

c k f x x f x x dx

and

d k f x x f x x dx

Wavelet Transforms in Two Dimensions

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( , ) ( ) ( )

( , ) ( ) ( )

( , ) ( ) ( )

( , ) ( ) ( )

H

V

D

x y x y

x y x y

x y x y

x y x y

/2, ,

/2, ,

( , ) 2 (2 ,2 )

( , ) 2 (2 ,2 )

{ , , }

j j jj m n

i j i j jj m n

x y x m y n

x y x m y n

i H V D

0

1 1

0 , ,0 0

1( , , ) ( , ) ( , )

M N

j m nx y

W j m n f x y x yMN

1 1

, ,0 0

1( , , ) ( , ) ( , )

, ,

M Ni i

j m nx y

W j m n f x y x yMN

i H V D

Inverse Wavelet Transforms in Two Dimensions

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0

0

0 , ,

, ,, ,

1( , ) ( , , ) ( , )

1 ( , , ) ( , )

j m nm n

i ij m n

i H V D j j m n

f x y W j m n x yMN

W j m n x yMN