Download - Jiquan Chen University of Toledo
OVERVIEW
• What is wavelet?
– Wavelets are mathematical functions
• What does it do?
– Cut up data into different frequency components , and then study each component with a resolution matched to its scale
• Why it is needed?
– Analyzing discontinuities and sharp spikes of the signal
– Applications as image compression, human vision, radar, and earthquake prediction
by : Tilottama Goswami
5 Dec. 2000 15-859B - Introduction to Scientific
Computing
3
Wavelet History
• 1805 Fourier analysis developed
• 1965 Fast Fourier Transform (FFT) algorithm
…
• 1980’s beginnings of wavelets in physics, vision, speech processing (ad hoc)
• … little theory … why/when do wavelets work?
• 1986 Mallat unified the above work
• 1985 Morlet & Grossman continuous wavelet transform … asking: how can you get perfect reconstruction without redundancy?
Author: Paul Heckbert
5 Dec. 2000 15-859B - Introduction to Scientific
Computing
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• 1985 Meyer tried to prove that no orthogonal wavelet other than Haar exists, found one by trial and error!
• 1987 Mallat developed multiresolution theory, DWT, wavelet construction techniques (but still noncompact)
• 1988 Daubechies added theory: found compact, orthogonal wavelets with arbitrary number of vanishing moments!
• 1990’s: wavelets took off, attracting both theoreticians and engineers
Author: Paul Heckbert
Wavelet History
• Pre-1930 – Joseph Fourier (1807) with his theories of frequency analysis
• The 1930s
– Using scale-varying basis functions; computing the energy of a function
• 1960-1980 – Guido Weiss and Ronald R. Coifman; Grossman and Morlet
• Post-1980 – Stephane Mallat; Y. Meyer; Ingrid Daubechies; wavelet
applications today
Bhushan D Patil, IIT, India
Wavelet History
What existed before this technique?
• Approximation using superposition of functions has existed since the early 1800's
• Joseph Fourier discovered that he could superpose sines and cosines to represent other functions , to approximate choppy signals
• These functions are non-local (and stretch out to infinity)
• Do a very poor job in approximating sharp spikes
by : Tilottama Goswami
What’s wrong with Fourier?
• Solve differential equations.
• Accurately store and transmit data.
• Obtain a different perspective for a signal.
Well what is it used for? To:
What need do wavelets satisfy?
Author Unknown
What’s wrong with Fourier?
Very helpful for this signal!
What need do wavelets satisfy?
The Fourier Transform (Series) method is used to decompose a signal into its global frequency components. EXAMPLE 1:
Author Unknown
What’s wrong with Fourier?
Not quite as helpful here.
What need do wavelets satisfy?
The Fourier Transform (Series) method is used to decompose a signal into its global frequency components. EXAMPLE 2:
Author Unknown
What’s wrong with Fourier?.....
What need do wavelets satisfy?
ANSWER: 1.) The Fourier Transform is unable to pick out local frequency content.
2.) It has a “hard time” representing functions that are oscillatory.
Author Unknown
The STFT
So how can we get at the local frequency information? Most obvious solution: use a window!
What’s wrong with Fourier?.....
Author Unknown
The STFT (aka WFT, Windowed Fourier Transform)
The STFT
Example of what happens when you window a function, with a given window function g(t-5).
STEP 1: Window the function using g(t – t0)
Author Unknown
STEP 2: Take the Fourier Transform of the resulting signal
Complex
conjugate
The STFT
Author Unknown
The STFT (aka WFT, Windowed Fourier Transform)
STEP 2: Take the Fourier Transform of the resulting signal
Complex conjugate
The STFT
A function of two variables.
Author Unknown
The STFT (aka WFT, Windowed Fourier Transform)
Summary of the STFT
• 1.) To get local frequency information, a window function, g(t), is first chosen.
• 2.) The STFT is taken as follows:
The STFT
3.) The STFT is a function of two variables, time and frequency. It allows one to observe the time-frequency distribution of the energy of a signal.
4.) With each window function, g, is associated both a time and a frequency minimum resolution. Once g is chosen, these resolutions are fixed and are the same at all frequencies and times. These resolutions are related by an uncertainty relation.
Author Unknown
Our first wavelet transform:
The CWT
Note that as alpha increases: the frequency decreases, and the window function expands.
This is just here for weighting.
Author Unknown
Our first wavelet transform:
The CWT
This is an inner product:
The basis is called a “wavelet” basis and consists of all the translated and dilated versions of the “mother” wavelet.
The mother Morlet
wavelet.
Two other Morlet
wavelets
Author Unknown
The General CWT:
The CWT
Can use any mother wavelet, h(t), you want. Some typical examples:
Pic from wikipedia.org
Author Unknown
The General CWT:
The CWT
Pic from wikipedia.org
The task of determining which wavelet basis to use for a given application or signal is tough. One should consult with the Adapted Wavelet literature on this.
WHICH
ONE?!
Author Unknown
5 Dec. 2000 15-859B - Introduction to Scientific
Computing
21
What Are Wavelets?
In general, a family of representations using:
• hierarchical (nested) basis functions
• finite (“compact”) support
• basis functions often orthogonal
• fast transforms, often linear-time
Author: Paul Heckbert
• Discrete Wavelet Transform
When your signal is in vector form (or pixel form), the discrete wavelet transform may be applied. The idea of scale becomes slightly more difficult to define here.
Discrete Versions
Author Unknown
Moving on: Discrete Versions
Moving on: Discrete Versions
• Discrete Wavelet Transform
Effectively, the DWT is nothing but a system of filters. There are two filters involved, one is the “wavelet filter”, and the other is the “scaling filter”. The wavelet filter, is a high pass filter, while the scaling filter is a low pass filter.
Scaling Filter ~ Averaging Filter
Wavelet Filter ~ Details Filter
Pic from wikipedia.org
Discrete Versions
Author Unknown
• Discrete Wavelet Transform
Example calculation: the Haar Wavelet. Observe: 1.) how
the “scale” is
changed 2.) the high
pass is the QMF of
the low pass
(quadrature mirror
filter.)
Pic from wikipedia.org
Discrete Versions
Author Unknown
Moving on: Discrete Versions
• Discrete Wavelet Transform For the DWT, it is difficult to construct orthogonal,
continuous filters. Luckily other people have already done it for us. Ingrid Daubechies developed many such sets.
Pic from wikipedia.org
Discrete Versions
Author Unknown
Moving on: Discrete Versions
PRINCIPLES OF WAELET TRANSFORM
• Split Up the Signal into a Bunch of Signals
• Representing the Same Signal, but all Corresponding to Different Frequency Bands
• Only Providing What Frequency Bands Exists at What Time Intervals
Bhushan D Patil, IIT, India
Fourier Transforms
• Fourier transform have single set of basis functions
– Sines
– Cosines
• Time-frequency tiles
• Coverage of the time-frequency plane
by : Tilottama Goswami
Wavelet Transforms
• Wavelet transforms have a infinite set of basis functions
• Daubechies wavelet basis functions
• Time-frequency tiles
• Coverage of the time-frequency plane
by : Tilottama Goswami
Some Applications
WAVELAB
“Most of the basic wavelet theory has been done… The future of wavelets lies in the as-yet uncharted territory of applications.” – Amara Graps, 1995
Available at:
http://www.stat.stanford.edu/~wavelab/
Created by David Donoho et. al. (including Ofer Levi)
Pic from wavelab site Applications
Author Unknown
From www.eetimes.com
Thresholding for file size reduction.
(Pioneered by David Donoho)
Applications
Author Unknown
Some Applications
From D. Donoho via Amara Graps
Wavelet Introduction
Thresholding for noise reduction.
(Pioneered by David Donoho)
Applications
Author Unknown
Some Applications
Image Enhancement.
From: “Wavelet-based image enhancement in
x-ray imaging and tomography”, by Bronnikov
and Duifhuis
Applications
Author Unknown
Some Applications
How do wavelets look like?
• Trade-off between how compactly the basis functions are localized in space and how smooth they are.
• Classified by number of vanishing moments
• Filter or Coefficients
– smoothing filter (like a moving average)
– data's detail information
by : Tilottama Goswami
DEFINITION OF CONTINUOUS WAVELET TRANSFORM
• Wavelet – Small wave
– Means the window function is of finite length
• Mother Wavelet – A prototype for generating the other window functions
– All the used windows are its dilated or compressed and shifted versions
dts
ttx
sss xx
*1 , ,CWT
Translati
on (The location of
the window)
Scale
Mother Wavelet
Bhushan D Patil, IIT, India
SCALE
• Scale
– S>1: dilate the signal
– S<1: compress the signal
• Low Frequency -> High Scale -> Non-detailed Global View of Signal -> Span Entire Signal
• High Frequency -> Low Scale -> Detailed View Last in Short Time
• Only Limited Interval of Scales is Necessary
Bhushan D Patil, IIT, India
COMPUTATION OF CWT
dts
ttx
sss xx
*1 , ,CWT
Step 1: The wavelet is placed at the beginning of the signal, and set s=1 (the most compressed wavelet); Step 2: The wavelet function at scale “1” is multiplied by the signal, and integrated over all times; then multiplied by ; Step 3: Shift the wavelet to t= , and get the transform value at t= and s=1; Step 4: Repeat the procedure until the wavelet reaches the end of the signal; Step 5: Scale s is increased by a sufficiently small value, the above procedure is repeated for all s; Step 6: Each computation for a given s fills the single row of the time-scale plane; Step 7: CWT is obtained if all s are calculated.
Bhushan D Patil, IIT, India
DISCRETIZATION OF CWT
• It is Necessary to Sample the Time-Frequency (scale) Plane. • At High Scale s (Lower Frequency f ), the Sampling Rate N can be Decreased. • The Scale Parameter s is Normally Discretized on a Logarithmic Grid. • The most Common Value is 2. • The Discretized CWT is not a True Discrete Transform • Discrete Wavelet Transform (DWT)
– Provides sufficient information both for analysis and synthesis – Reduce the computation time sufficiently – Easier to implement – Analyze the signal at different frequency bands with different resolutions – Decompose the signal into a coarse approximation and detail information
Bhushan D Patil, IIT, India
RECONSTRUCTION
• What – How those components can be assembled back into
the original signal without loss of information? – A Process After decomposition or analysis. – Also called synthesis
• How – Reconstruct the signal from the wavelet coefficients – Where wavelet analysis involves filtering and down
sampling, the wavelet reconstruction process consists of up sampling and filtering
Bhushan D Patil, IIT, India
Small-Block Pine Pine-Oak-Aspen Forest
Large-Block Pine Oak Pine Barrens
Landscape Level (Four Landscapes)
Do differences among landscape-level disturbance regimes influence patterns of understory plant diversity or composition?
Length: 3000+ m n=600+ plots Plot size: 1x1 m
Transect Measurements
•percent cover by species
•canopy cover (%)
•litter cover (%)
•litter depth (cm)
•cwd (%)
•duff depth (cm)
•species, dbh, % cover overstory trees
•patch type
Select Species Distributions
OP
B
OPB BOPB OPB PA SPB CC YA2 H2
H1 JPO SPB
MA
YA
1
0 5
0 100
0
1
0 20
0
15 3
0
0 40 80
0 3
6
0
1
2
1000 0 2000 3000
Pe
rce
nt C
ove
r
Distance (m)
Pteridium aquilinum Amelanchier arborea Hieracium aurantiacum Conyza canadensis Trientalis borealis Trifolium pratense
Cumulative Species Richness
0
40
80
120
160
0 1000 2000 3000
RR
P
MP
TRP60 RJP RP12
OC
C
RP
7
RP
7
OC
C
OC
C
TRP60
C C
0
40
80
120
160
MA
BO
PB
OPB
PA
SPB CC YA2
H2
H1
JPO SPB
OP
B
YA
2
OC
C
OC
C
OBCC OCC H H H F F2
JP
O
RP5 MP
NC
C
PO
A
RP
15
OR
P15
0 1000 2000 3000
H2
MP
F2
H H H H H F2
F H2
C
RP
60
Distance (m)
Nu
mb
er
of
Sp
ec
ies
Pine Barrens Large-Block PO
Small-Block Pine POA Forest
Old
Harvest
Landing
Sand Road-
Mod. Use
Sand Road-
Light Use
Clearing
Access Road
ATV Trail
Dry
Streambed
Grassy
Roadside
Brosofske et al. 2006
Wavelet Analysis of Shannon Diversity (H’) 1500
Old Harvest
Landing
Sand Road-
Mod. Use
Sand Road-
Light Use
Clearing
Access Road
0
1000
500
Re
so
luti
on
(m
)
OPB BOPB OPB PA SPB CC YA2 H2
H1 JPO SPB
OP
B M
A
YA
2
1000 0 2000 3000 Distance (m)
H’
0
1.0
2.0
1000
1500
500
0
0 0.6 1.2
W. Var.
Sca
le
W E
Wavelet Analysis Comparison
1000 500 0 1500 2000 2500 3000
BOPB OPB PA
SPB CC
YA2 H2
H1
JPO SPB O
PB
M
A
YA
2
OC
C
OC
C
OBCC OCC H H H F F2 JP
O
RP5 MP
NC
C
PO
A
RP
15
OR
P1
5
1000 500 0 1500 2000 2500 3000
0
500
1000
1500
0
500
1000
1500
Re
so
luti
on
(m
)
Distance (m)
Pine
Barrens
POA
Forest
Small-
Block
Pine
Large-
Block
PO
MP F2
H H H H H
F2 H
2
F H2
C TRP60 RJP RP12 RRP
OC
C
RP
7
RP
7
RP
60
OC
C
MP
OC
C TRP60
C C
W E
Old
Harvest
Landing
Sand Road-
Mod. Use
Sand Road-
Light Use
Clearing
Access Road
ATV Trail
Dry
Streambed
Grassy
Roadside
Cross-Correlations Between Wavelet Transform Values
Pine Barrens Large-Block Pine-Oak
Small-Block Pine Pine-Oak-Aspen Forest
COARSE
WOODY
DEBRIS
OV
ER
AL
L S
PE
CIE
S
RIC
HN
ES
S
EX
OT
IC S
PE
CIE
S
RIC
HN
ES
S
CANOPY
COVER
DUFF
DEPTH
SLOPE
STEEPNESS
DISTANCE-TO-
NEAREST-EDGE
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Corr
ela
tion
0 250 500 750 1000Scale (m)
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Corr
ela
tion
0 250 500 750 1000Scale (m)
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Corr
ela
tion
0 250 500 750 1000Scale (m)
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Corr
ela
tion
0 250 500 750 1000Scale (m)
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Corr
ela
tion
0 250 500 750 1000Scale (m)
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Corr
ela
tion
0 250 500 750 1000Scale (m)
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Corr
ela
tion
0 250 500 750 1000Scale (m)
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Corr
ela
tion
0 250 500 750 1000Scale (m)
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Corr
ela
tion
0 250 500 750 1000Scale (m)
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Corr
ela
tion
0 250 500 750 1000Scale (m)
Exercise with R and Matlab
Data1=xlsread(‘H:/fc_data.xls’);
Load (“x.dat’);
Ts=Data1(:,6);
Save Ts;
Fc=Data1(:,7);
Save Fc;
Re=Data1(:,8);
Save Re;
GPP=data1(:,9);
Save GPP;
Help wcoher
Wcoher (Fc, Ts, 1:4, ‘Haar’, ‘plot’, ‘all’);
Subplot ()
Diary()
Wavemenu
Wavedemo
Help wavelet
Qs:
• Wavelet variance? • Exporting data and individual
figures? • Cross-wavelets, >3? • 2-D, 3-D wavelet? • Different families? • …