Embed Size (px)
Citation preview
WAVELET TRANSFORMWAVELET TRANSFORM
OVERVIEW• Wavelet
– A small wave• Wavelet Transforms
– 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”
– Convert a signal into a series of wavelets– Provide a way for analyzing waveforms, bounded in both
frequency and duration– Allow signals to be stored more efficiently than by Fourier
transform– Be able to better approximate real-world signals– Well-suited for approximating data with sharp
discontinuities• “Microscope”
– Notice gross features with a large "window“– Notice small features with a small
Historical Development• Pre-1930
– Joseph Fourier (1807) with his theories of frequency analysis
– Haar (1909)• The 1950 - 1970s
– Using scale-varying basis functions; computing the energy of a function
– George Zweig, Jean Morlet, Alex Grossmann, Guido WeisS
• Post-1980– Yves Meyer, Stephane Mallat, Ingrid Daubechies, – Ronald Coifman, Victor Wickerhauser, Martin Vetterli, PP
Vaidyanathan, …
Application Domain
• Science– Geology (Morlet is a geophysicist working on
oil prospection)– Physics (I. Daubecies’ original motivation)
• Engineering– Signal processing, computer vision/graphics,
network modeling, signal modulation, financial engineering, chemical engineering
WAVELET APPLICATIONS• Typical Application Fields
– Astronomy, acoustics, nuclear engineering, sub-band coding, signal and image processing, neurophysiology, music, magnetic resonance imaging, speech discrimination, optics, fractals, turbulence, earthquake-prediction, radar, human vision, and pure mathematics applications
• Sample Applications– Identifying pure frequencies– De-noising signals– Detecting discontinuities and breakdown points– Detecting self-similarity– Compressing images
Mathematical Transformation
• Why– To obtain a further information from the signal that is
not readily available in the raw signal.• Raw Signal
– Normally the time-domain signal• Processed Signal
– A signal that has been "transformed" by any of the available mathematical transformations
• Fourier Transformation– The most popular transformation
FREQUENCY ANALYSIS
• Frequency Spectrum– Be basically the frequency components (spectral
components) of that signal– Show what frequencies exists in the signal
• Fourier Transform (FT)– One way to find the frequency content– Tells how much of each frequency exists in a signal
( ) ( ) knN
N
nWnxkX ⋅+=+ ∑
−
=
1
011
( ) ( ) knN
N
k
WkXN
nx −−
=∑ ⋅+=+
1
0
111
⎟⎠⎞
⎜⎝⎛−
= Nj
N ewπ2
( ) ( ) dtetxfX ftjπ2−∞
∞−
⋅= ∫
( ) ( ) dfefXtx ftjπ2⋅= ∫∞
∞−
STATIONARITY OF SIGNAL
• Stationary Signal– Signals with frequency content unchanged in
time– All frequency components exist at all times
• Non-stationary Signal– Frequency changes in time– One example: the “Chirp Signal”
STATIONARITY OF SIGNAL
0 0 . 2 0 . 4 0 . 6 0 . 8 1- 3
- 2
- 1
0
1
2
3
0 5 1 0 1 5 2 0 2 50
1 0 0
2 0 0
3 0 0
4 0 0
5 0 0
6 0 0
Time
Mag
nitu
de M
agni
tud
e
Frequency (Hz)
2 Hz + 10 Hz + 20Hz
Stationary
0 0 . 5 1- 1
- 0 . 8
- 0 . 6
- 0 . 4
- 0 . 2
0
0 . 2
0 . 4
0 . 6
0 . 8
1
0 5 1 0 1 5 2 0 2 50
5 0
1 0 0
1 5 0
2 0 0
2 5 0
Time
Mag
nitu
de Mag
nitu
de
Frequency (Hz)
Non-Stationary
0.0-0.4: 2 Hz + 0.4-0.7: 10 Hz + 0.7-1.0: 20Hz
CHIRP SIGNALS
0 0.5 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 250
50
100
150
Time
Mag
nitu
de Mag
nitu
de
Frequency (Hz)0 0.5 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 250
50
100
150
Time
Mag
nitu
de Mag
nitu
de
Frequency (Hz)
Different in Time DomainFrequency: 2 Hz to 20 Hz Frequency: 20 Hz to 2 Hz
Same in Frequency Domain
At what time the frequency components occur? FT can not tell!At what time the frequency components occur? FT can not tell!
NOTHING MORE, NOTHING LESS
• FT Only Gives what Frequency Components Exist in the Signal
• The Time and Frequency Information can not be Seen at the Same Time
• Time-frequency Representation of the Signal is Needed
ONE EARLIER SOLUTION: SHORT-TIME FOURIER TRANSFORM (STFT)
Most of Transportation Signals are Non-Stationary.(We need to know Whether and also When an incident was
happened)
SFORT TIME FOURIER TRANSFORM (STFT)
• Dennis Gabor (1946) Used STFT– To analyze only a small section of the signal at a time
-- a technique called Windowing the Signal.• The Segment of Signal is Assumed Stationary • A 3D transform
( )( ) ( ) ( )[ ] dtetttxft ftj
t
π−ω •′−ω•=′ ∫ 2*X ,STFT
( ) functionwindowthe:tω
A function of time and frequency
STFT - Disadvantages•• Heisenberg Principle :Heisenberg Principle :--
One can not get infinite time and frequency resolution beyond Heisenberg’s Limit
•• Trade offs :Trade offs :• Wider window
Good frequency resolution , Poor time resolution
• Narrower windowGood time resolution , Poor frequency resolution
π41. ≥ΔΔ ft
DRAWBACKS OF STFT• Unchanged Window• Dilemma of Resolution
– Narrow window -> poor frequency resolution – Wide window -> poor time resolution
• Heisenberg Uncertainty Principle– Cannot know what frequency exists at what time intervals
Via Narrow Window Via Wide Window
Source: Bhushan D Patil, Indian Institute of Technology, Bombay, India.
Heisenberg Uncertainty• In quantum mechanics, the Heisenberg uncertainty
principle states that certain pairs of physical properties, like position and momentum, cannot both be known to arbitrary precision. That is, the more precisely one property is known, the less precisely the other can be known. This statement has been interpreted in two different ways. According to Heisenberg its meaning is that it is impossible to determine simultaneously both the position and velocity of an electron or any other particle with any degree of accuracy or certainty.
• Dalam mekanika kuantum, prinsip ketidakpastian Heisenberg menyatakan bahwa pasangan tertentu dari sifat fisik, sepertiposisi dan momentum, tidak dapat diketahui baik secarapasti. Artinya, lebih tepat satu properti yang diketahui, yang kurang tepat dapat diketahui lain. Pernyataan ini telahditafsirkan dalam dua cara yang berbeda. MenurutHeisenberg artinya bahwa tidak mungkin untuk menentukansecara simultan baik posisi dan kecepatan elektron ataupartikel lainnya dengan tingkat ketepatan atau kepastian.
MULTIRESOLUTION ANALYSIS (MRA)
• Wavelet Transform– An alternative approach to the short time Fourier
transform to overcome the resolution problem – Similar to STFT: signal is multiplied with a function
• Multiresolution Analysis– Analyze the signal at different frequencies with
different resolutions– Good time resolution and poor frequency resolution at
high frequencies– Good frequency resolution and poor time resolution at
low frequencies– More suitable for short duration of higher frequency;
and longer duration of lower frequency components
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
• Mother wavelet 1st derivative of Gaussian function (DOG)– WT of signal with Gaussian wavelet ψ(t) is the derivative of
signal smoothed by Gaussian window θ(t).– Zero-crossings in WT maxima or minima signal.– Maxima or minima in WT point of inflection in signal
Continuous Wavelet transform
τσ
τψτσ
σ dtftWf )()(1),( ∫∞
∞−
−=
2=σ
4=σ
8=σ
16=σ
32=σ
ECG
Source:ECG/Wavelet/HMM/WTSign/T&R/Conc, IDEE, Mastricht Inst.
Continuous Wavelet transform
time
time
freq
uen
cy
Continuous Wavelet transform
Absolute value of wavelet coefficient
Captures frequency changes through timeSource:Morten Maruf, Technical University of Denmark, 2006.
epoc
h
chan
nel
time-frequency
epoc
h
chan
nel
time
epochtimechannel ××X epochfrequencytimechannel ×−×X
Continuous Wavelet transform
Source:Morten Maruf, Technical University of Denmark, 2006.
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
ttxs
ss xx ⎟⎠⎞
⎜⎝⎛ τ−
ψ•=τΨ=τ ∫ψψ *1 , ,CWT
Translation(The location of
the window)
Scale
Mother Wavelet
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
• In order to become a wavelet a function must satisfy the above two conditions
∞<Ψ
=Ψ
∫
∫∞
∞−
∞
∞−
dtt
dtt
2|)(|
0)(
DEFINITION OF CONTINUOUS WAVELET TRANSFORM
Inverse wavelet transform
∫
∫
∫ ∫
∞
∞−
∞
∞−
∞
∞−
∞
∞−
=Ψ
Ψ=
Ψ=
0)(
|||)(|
)(),(11)( ,2
dtt
provided
dwwwC
where
dadbtbaWaC
tx ba
DEFINITION OF CONTINUOUS WAVELET TRANSFORM
COMPUTATION OF CWT
( ) ( ) ( ) dts
ttxs
ss xx ⎟⎠⎞
⎜⎝⎛ τ−
ψ•=τΨ=τ ∫ψψ *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, andintegrated over all times; then multiplied by ψ;
Step 3: Shift the wavelet to t= ε, and get the transform value at t= ε 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.
RESOLUTION OF TIME & FREQUENCY
Time
Frequency
Better time resolution;Poor frequency resolution
Better frequency resolution;Poor time resolution
• Each box represents a equal portion • Resolution in STFT is selected once for entire analysis
COMPARSION OF TRANSFORMATIONS
From http://www.cerm.unifi.it/EUcourse2001/Gunther_lecturenotes.pdf, p.10
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 (dyadic).• 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
• Discrete wavelets
• In reality, we often choose • In the discrete case, the wavelets can be generated
from dilation equations, for example,φ(t) = [h(0)φ(2t) + h(1)φ(2t-1) + h(2)φ(2t-2) + h(3)φ(2t-3)] ...(2)
• Solving equation (2), one may get the so called scaling function φ(t).
• Use different sets of parameters h(i)one may get different scaling functions.
),( 02
0, ktaa jj
kj −= ψψ
., Zkj ∈.20 =a
2
DISCRETIZATION OF CWT
• The corresponding wavelet can be generated by the following equation
ψ (t)= [h(3)φ(2t) - h(2)φ(2t-1) + h(1)φ(2t-2) - h(0)φ(2t-3)] …. (3)
• When and equation (3) generates the D4 (Daubechies) wavelets.
2
,24/)31()0( +=h ,24/)33()1( +=h ,24/)33()2( −=h24/)31()3( −=h
EXAMPLE : DISCRETIZATION OF CWT
• In general, consider h(n) as a low pass filter and g(n) as a high-pass filter where
• g is called the mirror filter of h. g and h are called quadrature mirror filters (QMF).
• Redefine– Scaling function
).1()1()( nNhng n −−−=
).2()(2)( nxnhxn
−= ∑ φφ
DISCRETIZATION OF CWT
– Wavelet function
• Decomposition and reconstruction of a signal by the QMF.
where and
is down-sampling and is up-sampling
).2()(2)( nxngxn
−= ∑ φψ
2
2
2
2
f(n) +
f(n)
)(ng
)(nh )(nh
)(ng
)()( nhnh −= ).()( ngng −=
DISCRETIZATION OF CWT
Multi Resolution Analysis• Analyzing a signal both in time domain and
frequency domain is needed many a times– But resolutions in both domains is limited by
Heisenberg uncertainty principle
• Analysis (MRA) overcomes this , how?– Gives good time resolution and poor frequency
resolution at high frequencies and good frequency resolution and poor time resolution at low frequencies
– This helps as most natural signals have low frequency content spread over long duration and high frequency content for short durations
SUBBABD CODING ALGORITHM• Halves the Time Resolution
– Only half number of samples resulted• Doubles the Frequency Resolution
– The spanned frequency band halved
0-1000 Hz
D2: 250-500 Hz
D3: 125-250 Hz
Filter 1
Filter 2
Filter 3
D1: 500-1000 Hz
A3: 0-125 Hz
A1
A2
X[n]512
256
128
64
64
128
256SS
A1
A2 D2
A3 D3
D1
Source: Bhushan D Patil, Indian Institute of Technology, Bombay, 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
Multiresolution Approximation• A measurable, square-integrable one-dimensional function
f(x) can be approximated in multiresolution.• Let A2j be the operator which approximates a signal f(x) at
a resolution 2j. We expect it has following properties:1. ∀g(x) ∈ V2j, |g(x) – f(x)| ≥ |A2jf(x) – f(x)|2. ∀j ∈ Z, V2j ⊂ V2j+1
3. ∀j ∈ Z, f(x) ∈ V2j ⇔ f(2x) ∈ V2j+1
4. ∃Isomorphism I from V1 to I2(Z)5. ∀k ∈ Z, A1fk(x) = A1f(x-k), where fk(x) = f(x-k)6. I(A1f(x)) = (αi)i∈Z ⇔ I(A1fk(x)) = (αi-k)i∈Z
7.8.
Υ+∞
−∞=+∞→
=j
jjj VV 22lim
is dense in L2(R)
}0{22lim ==+∞
−∞=−∞→Ι
jjjj VV
is dense in L2(R)
Multiresolution ApproximationWe call any set of vector spaces (V2j)j∈Z which satisfies the properties 2-8 a multiresolution approximation of L2(R). The associated set of operators A2j satisfying 1-6 give the approximation of any L2(R) function at a resolution 2j.How to numerically characterize the operator A2j?
Theorem 1: Let (V2j)j∈Z be a multiresolutionapproximation of L2(R). There exists a unique function Φ(x) ∈ L2(R), called a scaling function, such that is we set Φ2j(x) = 2jΦ(2jx) for j ∈ Z, (the dilation of Φ(x) by 2j), then
Znjj nxj ∈
−− −Φ ))2(2( 2
)2()2(),(2)(),()( 2222 nxnuufxfARLxf jj
n
jjjj
−−+∞
−∞=
− −Φ⟩−Φ⟨=∈∀ ∑
is an orthonormal basis of V2j
Znjd nuuffA jj ∈
− ⟩−Φ⟨= ))2(),((22
Multiresolution Transform
)2()2(),2(2)2( 12
122
12 11 kxkununx jjj
k
jjjjjj
−−−−−+∞
−∞=
−−− −Φ•⟩−Φ−Φ⟨=−Φ ++∑
⟩−−ΦΦ⟨=⟩−Φ−Φ⟨ −+−−−−− ))2((),()2(),2(2 11 21
221 nkuukunu jjj
jj
⟩−Φ⟨•⟩−−ΦΦ⟨=⟩−Φ⟨ −−+∞
−∞=
−+−∑ )2(),())2((),()2(),( 1
222 11 kuufnkuunuuf j
k
jjj
∀j ∈ Z, V2j ⊂ V2j+1
⟩−ΦΦ⟨=∈∀ − )(),()(, 12 nuunhZn
⟩−Φ⟨•−=⟩−Φ⟨ −−+∞
−∞=
−+∑ )2(),()2(~)2(),( 1
22 1 kuufknhnuuf j
k
jjj
)()(~ nhnh −=
Fourier Transform of a scaling function
∑+∞
−∞=
−=n
inenhH ωω )()(
Therem 2: Let Φ(x) be a scaling function, and let H be a discrete filter with impulse response h(n). Let H(ω) be the Fourier series defined by
H(ω) satisfies the following two properties:
|H(0)| = 1 and h(n) = O(n-2) at infinity
|H(ω)|2 + |H(ω+π)|2 = 1
Conversely let H(ω) be a Fourier series satisfying above two equations and such that
|H(ω)| ≠ 0 for ω∈[0, π/2] ∏+∞
=
−=Φ1
)2()(ˆp
pH ωω
Wavelet Representation
)2
(ˆ)2
()(ˆ ωωω Φ=Ψ G
2),(2 ))2(2 Zjnjj nxj ∈
−− −Ψ
is an orthonormal basis of L2(R) and Ψ(x) is called an orthogonal wavelet.
The difference of information between 2j+1 and 2j is called the detail signal at the resolution 2j. Let O2j be orthogonal complement of V2j, i.e.
O2j is orthogonal to V2j
O2j ⊕ V2j = V2j+1
Theorem 3: Let (V2j)j∈Z be a multiresolution vector space sequence, Φ(x) the scaling function, and H the corresponding conjugate filter. Let Ψ(x) be a function whose Fourier transform is given by
Let Ψ2j(x) = 2jΨ(2jx) denote the dilation of Ψ(x) by 2j. Then
Wavelet Representation
Wavelet Representation
))(,( 122 −≤≤−− jJd fDfA jJ
For any J > 0, the original discrete signal A1f measured at the resolution 1 can be represented by
This set of discrete signals is called an orthogonal wavelet representation, and consists of the reference signal at a coarse resolution A2-Jf and the detail signals at the resolution 2j for –J ≤ j ≤ -1.
Znj nuuffD jj ∈
− ⟩−Ψ⟨= )2(),((22
Wavelet Representation
)2()2(),2(2)2( 12
122
12 11 kxkununx jjj
k
jjjjjj
−−−−−+∞
−∞=
−−− −Φ•⟩−Φ−Ψ⟨=−Ψ ++∑
⟩−−ΦΨ⟨=⟩−Φ−Ψ⟨ −+−−−−− ))2((),()2(),2(2 11 2
122
1 nkuukunu jjjjj
⟩−Φ⟨•⟩−−ΦΨ⟨=⟩−Ψ⟨ −−+∞
−∞=
−+−∑ )2(),())2((),()2(),( 1
222 11 kuufnkuunuuf j
k
jjj
⟩−ΦΨ⟨=∈∀ − )(),()(, 12 nuungZn
⟩−Φ⟨•−=⟩−Ψ⟨ −−+∞
−∞=
−+∑ )2(),()2(~)2(),( 1
22 1 kuufkngnuuf j
k
jjj
)()(~ ngng −=
Wavelet Representation
Laplacian Pyramid and FWT
FWT
Znjjjj nxnx jj ∈
−−−− −Ψ−Φ ))2(2),2(2(22
∑
∑∞+
−∞=
−−−−−
+∞
−∞=
−−−−−−−
−Ψ•⟩−Φ−Ψ⟨+
−Φ•⟩−Φ−Φ⟨=−Φ
+
++
k
jjjj
k
jjjjj
kxnuku
kxnukunx
jjj
jjjj
)2()2(),2(2
)2()2(),2(2)2(
21
22
21
221
2
1
11
is an orthonormal basis of V2j+1, so
O2j is the orthogonal complement of V2j in V2j+1,
FWT
∑
∑∞+
−∞=
−−−−−
+∞
−∞=
−−−−−−−
⟩−Ψ⟨•⟩−Φ−Ψ⟨+
⟩−Φ⟨•⟩−Φ−Φ⟨=⟩−Φ⟨
+
++
k
jjjj
k
jjjjj
kuufnuku
kuufnukunxuf
jjj
jjjj
)2(),()2(),2(2
)2(),()2(),2(2)2(),(
21
22
21
221
2
1
11
∑
∑∞+
−∞=
−
+∞
−∞=
−−−
⟩−Ψ⟨•−+
⟩−Φ⟨•−=⟩−Φ⟨ +
k
j
k
jj
kuufkng
kuufknhnxuf
j
jj
)2(),()2(2
)2(),()2(2)2(),(
2
21
2 1
FWT
⎪⎩
⎪⎨⎧
=02
1)(nh
g(n) = (-1)1-nh(1-n)
In signal processing, G and H are called quadrature mirror filters.
Generally, we design the H first and then calculate the G, Φand Ψ.
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
−=
02
12
1
)(ngn=0,1
Others
n=0
n=1
Others
Ψ(x) is Haar function.
Extension to Image
12
122 jjj VVV ⊗=
2),(2))2,2(2(
Zmnjjj mynxj ∈
−−− −−Φ
)()(),( yxyx ΦΦ=Φ
forms an orthonormal basis of V2j
22 ),(22),(2 ))2()2(2())2,2(2( Zmnjjj
Zmnjjj mynxmynx jjj ∈
−−−∈
−−− −Φ−Φ=−−Φ
Extension to Image
3),(32
22
12
))2,2(2),2,2(2),2,2(2(Zmn
jjjjjjjjj mynxmynxmynx jjj ∈−−−−−−−−− −−Ψ−−Ψ−−Ψ
)()(),(1 yxyx ΨΦ=Ψ
is an orthonormal basis of O2j
)()(),(2 yxyx ΦΨ=Ψ
)()(),(3 yxyx ΨΨ=Ψ
2
2
2
2
),(2232
),(2222
),(2212
),(222
))2,2))(()(),(((
))2,2))(()(),(((
))2,2))(()(),(((
))2,2))(()(),(((
Zmnjj
Zmnjj
Zmnjj
Zmnjjd
mnyxyxffD
mnyxyxffD
mnyxyxffD
mnyxyxffA
jjj
jjj
jjj
jjj
∈−−
∈−−
∈−−
∈−−
−Ψ−Ψ∗=
−Φ−Ψ∗=
−Ψ−Φ∗=
−Φ−Φ∗=
Extension to Image
Extension to Image
Application
• Compact Coding of Wavelet Image Representation• Texture Discrimination and Fractal Analysis
REFERENCES• Morten Maruf, “An introduction to
ERPWAVELAB”, Technical University of Denmark, 2006.
• Bhushan D Patil, “Introduction of Wavelet Transform”, Department of Electrical Engineering Indian Institute of Technology, Bombay, India, 2006.
• Yuelong Jiang, “Wavelet Representation”, 2000.
Some References of Wavelet• Mathworks, Inc. Matlab Toolbox
http://www.mathworks.com/access/helpdesk/help/toolbox/wavelet/wavelet.html• Robi Polikar, The Wavelet Tutorial, http://users.rowan.edu/~polikar/WAVELETS/WTpart1.html• Robi Polikar, Multiresolution Wavelet Analysis of Event Related Potentials for the Detection of Alzheimer's
Disease, Iowa State University, 06/06/1995• Amara Graps, An Introduction to Wavelets, IEEE Computational Sciences and Engineering, Vol. 2, No 2,
Summer 1995, pp 50-61.• Resonance Publications, Inc. Wavelets. http://www.resonancepub.com/wavelets.htm• R. Crandall, Projects in Scientific Computation, Springer-Verlag, New York, 1994, pp. 197-198, 211-212. • Y. Meyer, Wavelets: Algorithms and Applications, Society for Industrial and Applied Mathematics,
Philadelphia, 1993, pp. 13-31, 101-105. • G. Kaiser, A Friendly Guide to Wavelets, Birkhauser, Boston, 1994, pp. 44-45. • W. Press et al., Numerical Recipes in Fortran, Cambridge University Press, New York, 1992, pp. 498-499,
584-602. • M. Vetterli and C. Herley, "Wavelets and Filter Banks: Theory and Design," IEEE Transactions on Signal
Processing, Vol. 40, 1992, pp. 2207-2232. • I. Daubechies, "Orthonormal Bases of Compactly Supported Wavelets," Comm. Pure Appl. Math., Vol 41,
1988, pp. 906-966. • V. Wickerhauser, Adapted Wavelet Analysis from Theory to Software, AK Peters, Boston, 1994, pp. 213-
214, 237, 273-274, 387. • M.A. Cody, "The Wavelet Packet Transform," Dr. Dobb's Journal, Vol 19, Apr. 1994, pp. 44-46, 50-54. • J. Bradley, C. Brislawn, and T. Hopper, "The FBI Wavelet/Scalar Quantization Standard for Gray-scale
Fingerprint Image Compression," Tech. Report LA-UR-93-1659, Los Alamos Nat'l Lab, Los Alamos, N.M. 1993.
• D. Donoho, "Nonlinear Wavelet Methods for Recovery of Signals, Densities, and Spectra from Indirect and Noisy Data," Different Perspectives on Wavelets, Proceeding of Symposia in Applied Mathematics, Vol 47, I. Daubechies ed. Amer. Math. Soc., Providence, R.I., 1993, pp. 173-205.
• B. Vidakovic and P. Muller, "Wavelets for Kids," 1994, unpublished. Part One, and Part Two.
