OMN - 2002, Introduction to wavelet analysis
1
Introduction to wavelet analysis
or How to catch wild lions
Ole Nielsen Center for Mathematics and its Applications, ANU
Outline
• Today: – Introduction to wavelets – properties – applications • Next week: – Wavelet Algorithms – Wavelets and Fourier Analysis – Wavelets and Partial Differential Equations
OMN - 2002, Introduction to wavelet analysis
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Basis functions and Decompositions
Systems of basis functions
• Polynomials (Taylor series, Chebyshev expansions) • Trigonometric (Fourier series) • Hat functions (Finite Elements) • Hierarchical bases, wavelets
How to use.
Decompose general function into a sum of basis functions.
Why decompose
• Compression • Denoising/Datafitting • Detection of special features (motion detection, edge finding) • Modification (e.g transposition of sound wave) • Solving PDEs
OMN - 2002, Introduction to wavelet analysis
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A simple example 37 35 28 28 58 18 21 15
Original signal 60 55 50 45 40 35 30 25 20 15 1 2 3 4 5 6 7 8
OMN - 2002, Introduction to wavelet analysis
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Separate averages from differences (Haar filters)
Decomposition (analysis)
Averaging (x + y)/2 37 35 28 28 58 18 21 15 36
↓
Differencing (x − y)/2 37 35 28 28 58 18 21 15 1
↓ ↓ ↓ ↓ ↓
28
↓
38
↓
18
0
20
3
Reconstruction (synthesis)
36 28 38 18 1 0 20 3
+ 37
− 35
+ 28
− 28
+ 58
− 18
+ 21
− 15
OMN - 2002, Introduction to wavelet analysis
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Full transform
37 35 28 28 58 18 21 15 36 28 38 18 32 28 30 30 2 2 4 10 1 0 20 3 4 10 1 0 20 3
Complexity O(N )
OMN - 2002, Introduction to wavelet analysis
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Threshold = 2 30 30 2 0 4 10
Truncate
1 0 20 3
4 10 0 0 20 3 ↓ 30 30 4 10 0 0 20 3 ↓ 34 26 40 20 0 0 20 3 ↓ 34 34 26 26 60 20 23 17
Threshold = 2 60 55 50 45 40 35 30 25 20 15 1 2 Original Thresholded 3 4 5 6 7 8
OMN - 2002, Introduction to wavelet analysis
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Threshold = 3 30 30 2 0 4 10
Truncate
1 0 20 3
4 10 0 0 20 0 ↓ 30 30 4 10 0 0 20 0 ↓ 34 26 40 20 0 0 20 0 ↓ 34 34 26 26 60 20 20 20
Threshold = 3 60 55 50 45 40 35 30 25 20 15 1 2 Original Thresholded 3 4 5 6 7 8
OMN - 2002, Introduction to wavelet analysis
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Threshold = 4 30 30 2 0 4 10
Truncate
1 0 20 3
0 10 0 0 20 0 ↓ 30 30 0 10 0 0 20 0 ↓ 30 30 40 20 0 0 20 0 ↓ 30 30 30 30 60 20 20 20
Threshold = 4 60 55 50 45 40 35 30 25 20 15 1 2 Original Thresholded 3 4 5 6 7 8
OMN - 2002, Introduction to wavelet analysis
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Orthogonal Multiresolution Analysis
V0 ⊂ V1 ⊂ · · · ⊂ L2(IR), f ∈ Vj ⇔ f (2·) ∈ Vj+1 Define Wj : Vj+1 = Vj ⊕ Wj , V j ⊥ Wj
j
Vj = L2(IR)
{φ(x − k)}k∈Z is an orthonormal basis for V0 {ψ(x − k)}k∈Z is an orthonormal basis for W0
W0 V0 V1 V2
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Wavelet decomposition
Translations and dilations generate entire system
Scaling functions: Wavelets: φj,k (x) = 2j/2φ(2j x − k) ψj,k (x) = 2j/2ψ(2j x − k) Vj = span{φj,k }k=0,1,...,2j −1 Wj = span{ψj,k }k=0,1,...,2j −1
Orthogonality relations
−∞ ∞ ∞
φj,k (x)φj,l (x) dx = δk,l ψi,k (x)ψj,l (x) dx = δi,j δk,l φi,k (x)ψj,l (x) dx = 0, j≥i
−∞ ∞
−∞
where
δm,n =
1 m=n 0 m=n
Periodised version ([0, 1]
φjk (x) = ψjk (x) =
∞ l=−∞ ∞ l=−∞
φjk (x − l) ψjk (x − l)
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Example: Daubechies-4 bases (periodized)
φ2,l, l = 0, 1, 2, 3
φ
2,0
(x)
φ
2,1
(x)
φ
2,2
(x)
φ
2,3
(x)
0.1 0.08 0.06 0.04 0.02 0 −0.02 −0.04 0
0.1 0.08 0.06 0.04 0.02 0 −0.02 −0.04 0
0.1 0.08 0.06 0.04 0.02 0 −0.02 −0.04 0
0.1 0.08 0.06 0.04 0.02 0 −0.02 −0.04 0
1
1
1
1
ψ2,l, l = 0, 1, 2, 3
ψ
2,0
(x)
ψ
2,1
(x)
ψ
2,2
(x)
ψ
2,3
(x)
0.15
0.15
0.15
0.15
0.1
0.1
0.1
0.1
0.05
0.05
0.05
0.05
0
0
0
0
−0.05
−0.05
−0.05
−0.05
−0.1 0
1
−0.1 0
1
−0.1 0
1
−0.1 0
1
ψ3,l, l = 0, 1, 2, 3
ψ
3,0
(x)
ψ
3,1
(x)
ψ
3,2
(x)
ψ
3,3
(x)
0.2 0.15 0.1 0.05 0 −0.05 −0.1 −0.15 0
0.2 0.15 0.1 0.05 0 −0.05 −0.1 −0.15 0
0.2 0.15 0.1 0.05 0 −0.05 −0.1 −0.15 0
0.2 0.15 0.1 0.05 0 −0.05 −0.1 −0.15 0
1
1
1
1
ψ3,l, l = 4, 5, 6, 7
ψ
3,4
(x)
ψ
3,5
(x)
ψ
3,6
(x)
ψ
3,7
(x)
0.2 0.15 0.1 0.05 0 −0.05 −0.1 −0.15 0
0.2 0.15 0.1 0.05 0 −0.05 −0.1 −0.15 0
0.2 0.15 0.1 0.05 0 −0.05 −0.1 −0.15 0
0.2 0.15 0.1 0.05 0 −0.05 −0.1 −0.15 0
1
1
1
1
OMN - 2002, Introduction to wavelet analysis
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Projections
Wavelet decomposition (J ≥ J0) for f ∈ L2 P V J f = P V J0 f +
2j −1 k=0 2j −1 k=0
J−1 j=J0
PW j f
PVj f (x) = PWj f (x) =
cj,k φj,k (x) dj,k ψj,k (x)
Expansion coefficients
cj,l = dj,l =
−∞ ∞ ∞
f (x)φj,l (x) dx f (x)ψj,l (x) dx
−∞
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Dilation equations
Dilation equation
Express φ(x) ∈ V0 in terms of basis functions in V1 (φ1k ) φ(x) = where ak =
D−1 k=0
√ ak φ1,k (x) = 2
∞
k
ak φ(2x − k)
−∞
φ(x)φ1,k (x) dx
Wavelet equation
Express ψ(x) ∈ W0 in terms of basis functions in V1 (φ1k ) √ D−1 bk φ1,k (x) = 2 bk φ(2x − k) ψ(x) =
k=0 k
where bk =
−∞
∞
ψ(x)φ1,k (x) dx
D nonzero coefficients for compactly supported wavelets.
suppφ = suppψ = [0, D − 1]
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Dilation equation with two coefficients
√ 1 φ(x) = 2 (a0φ(2x) + a1φ(2x − 1)) , a0 = a1 = √ 2
φ(x)
φ(2x)
φ(2x − 1)
-
-
0
1
0
1
φ(x) = φ(2x) + φ(2x − 1)
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Dilation equation with three coefficients
1 1 φ(x) = φ(2x) + φ(2x − 1) + φ(2x − 2) 2 2
φ(2x − 1)
1 φ(2x) 2 1 φ(2x 2
− 2)
-
0
1
2
φ(x)
Note: This case cannot be made orthogonal
OMN - 2002, Introduction to wavelet analysis
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The Haar system
φ(x)
φ(2x)
φ(2x − 1)
-
-
0
1
0
1
φ(x) = φ(2x) + φ(2x − 1) ψ(x) = φ(2x) − φ(2x − 1)
φ(x)
ψ(x)
-
-
0
1
0
1
φ(x) =
1, x ∈ [0, 1] 0, otherwise
1 1, x ∈ [0, 2 ] 1 ψ(x) = −1, x ∈ ] 2 , 1] 0, otherwise
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Example - the Haar system
A strange function
4
3
2
1
0
−1
−2 −4
−3
−2
−1
0
1
2
3
4
Projection on V7 using Haar basis
Projection on V7 Haar space 4
3
2
1
0
−1
−2 −4
−3
−2
−1
0
1
2
3
4
OMN - 2002, Introduction to wavelet analysis
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Haar wavelet decomposition
smallosc.dat Wavelet decomposition (individual plots), D=2 5 0 −5 0.2 0 −0.2 0.5 0 −0.5 0.5 0 −0.5 1 0 −1 1 0 −1 4 2 0 −2 −4 W5 W
6
V
10
W9
W8
W
7
−3
−2
−1
0
1
2
3
4
V5
OMN - 2002, Introduction to wavelet analysis
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Vanishing Moments
Haar system:
• Can represent piecewise constants exactly • 1 vanishing moment of Haar wavelet
In general:
P vanishing moments: Exact reproduction of xp for p = 0, . . . , P − 1 x = µp = k
p ∞ k=−∞ ∞
µp φ(x − k) k
xpφ(x − k) dx −∞
Number of filter coefficients is related to VM
D = 2P
All algorithms are cast in terms of filter coefficients
OMN - 2002, Introduction to wavelet analysis
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Construction of smoother wavelets (Daubechies Family)
Orthonormality
δ0n = = =
−∞ ∞
∞
φ(x)φ(x − n) dx
−∞ k
k
ak ak−2n
ak φ1,k (x) al φ1,l (x − n) dx
l
Conservation of area (from dilation equation)
φ(x) =
∞ D−1 k=0
ak φ1,k (x)
−∞ ∞
1 D−1 ak φ(x) dx = √ −∞ 2 k=0 √ D−1 ak = 2
k=0
φ(y) dy
Smoothness (from P Vanishing moments)
−∞ D−1 k=0 ∞
xpψ(x) = 0 for p = 0, 1, . . . , P − 1 k p bk = 0 for p = 0, 1, . . . , P − 1
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Daubechies-4
• 4 coefficients a0, a1, a2, a3 • 2 vanishing moments
Conditions on filter coefficients
Normalization: Orthogonality: Conservation of area a 2 + a2 + a2 + a2 = 1 3 2 1 0 a 0 a2 + a 1 a3 = 0 √ a 0 + a1 + a2 + a3 = 2
Vanishing moment (0): a3 − a2 + a1 − a0 = 0 Vanishing moment (1): 0a3 − 1a2 + 2a1 − 3a0 = 0
a0 a1 a 2 a3
√ 1 + √3 1 3+ 3 = √ √ 3− 3 4 2 √ 1− 3
Basic scaling function φ
1 1
Basic Wavelet ψ
0
0
0
3
0
3
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Example - Daubechies 4 basis
The function
4
3
2
1
0
−1
−2 −4
−3
−2
−1
0
1
2
3
4
Projection on V7 using Daubechies 4 wavelets
Projection on V7 using D4 wavelets 4
3
2
1
0
−1
−2 −4
−3
−2
−1
0
1
2
3
4
OMN - 2002, Introduction to wavelet analysis
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Daubechies 4 wavelet decomposition
smallosc.dat Wavelet decomposition (individual plots), D=4 5 0 −5 0.1 0 −0.1 0.5 0 −0.5 1 0 −1 1 0 −1 0.5 0 −0.5 5 0 −5 −4 −3 −2 −1 0 1 2 3 4 V5 W5 W
6
V
10
W9
W8
W
7
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Decay of Wavelet coefficients
P vanishing moments
xp =
∞ k=−∞
µp φ(x − k), k
p = 0, . . . , P − 1
From orthogonality (φ ⊥ ψ)
−∞ ∞
xpψ(x − k) dx = 0
Decay of wavelet coefficients
dj,k =
−∞ ∞
f (x)ψj,k (x) dx
|dj,k | ≤ C 2−jP max f (P )(x) x x ∈ supp{ψj,k } • Exponential decay where f has P continuous derivatives in supp{ψ j,k }. • dj,k zero where f ∈ span{x0, x1, . . . , xP −1} • Discontinuitiues in f remain local in coefficient space
OMN - 2002, Introduction to wavelet analysis
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Fourier (cosine) analysis
Orthogonal basis of cosines {cos(nx)}∞ n=0
1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0 x 10
−3
200
400
600
800
1000
1200
f periodic L2 function
Cosine expansion: f (x) = where ak =
π −π ∞ k=0
ak cos(kx)
f (x) cos(kx) dx for k = 0, 1, . . . , N − 1
Fourier coefficients have global decay
|ak | ≤ Ck −P max f (P )(x) for all x (f assumed to have P continuous derivatives).
OMN - 2002, Introduction to wavelet analysis
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The Fast Wavelet Transform (FWT)
Two expansions for f ∈ Vj = Vj−1 ⊕ Wj−1
2j −1
f (x) = f (x) =
l=0 2j−1 −1 l=0
cj,l φj,l (x) cj−1,l φj−1,l (x) +
2j−1 −1 l=0
dj−1,l ψj−1,l (x)
The FWT recursion
cj−1,l = dj−1,l =
D−1 k=0 D−1 k=0
ak cj,k+2l bk cj,k+2l
j = J, J − 1, . . . , J0 + 1 l = 0, 1, . . . , 2j−1 − 1
Matrix formulation of FWT
d = W c, c = W T d
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Fast Wavelet Transform - example
Expansions for f ∈ V4
f (x) =
15 k=0
c4,k φ4,k (x)
3 2j −1 j=0 k=0
f (x) = c0,0 +
dj,k ψj,k (x)
The FWT: y = W x
x c4,0 c4,1 c4,2 c4,3 c4,4 c4,5 c4,6 c4,7 c4,8 c4,9 c4,10 c4,11 c4,12 c4,13 c4,14 c4,15 → c3,0 c3,1 c3,2 c3,3 c3,4 c3,5 c3,6 c3,7 d3,0 d3,1 d3,2 d3,3 d3,4 d3,5 d3,6 d3,7 → c2,0 c2,1 c2,2 c2,3 d2,0 d2,1 d2,2 d2,3 d3,0 d3,1 d3,2 d3,3 d3,4 d3,5 d3,6 d3,7 → c1,0 c1,1 d1,0 d1,1 d2,0 d2,1 d2,2 d2,3 d3,0 d3,1 d3,2 d3,3 d3,4 d3,5 d3,6 d3,7 → y c0,0 d0,0 d1,0 d1,1 d2,0 d2,1 d2,2 d2,3 d3,0 d3,1 d3,2 d3,3 d3,4 d3,5 d3,6 d3,7
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Complexity of the Fast Wavelet Transform
n: Number of data points involved in partial transform D: Wavelet genus.
One step of the FWT
2·D·n
Complexity of FWT
FF W T (N ) = N 2i i=0 = 4D(N − 1) = O(N ) 2D
log2 N −1
Complexity of FFT
FF F T (N ) = O(N log N )
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Compression using wavelets or cosines
1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0
200
400
600
800
1000
1200
Cosine transform
200 150 100
50
0
−50
−100 0
200
400
600
800
1000
1200
Wavelet transform
3 2.5 2
1.5
1
0.5
0
−0.5 0
200
400
600
800
1000
1200
Discard small elements
Threshold 10−4 10−2 Remaining coefficients Wavelets Cosines Original 12 100 99 2 11 29
OMN - 2002, Introduction to wavelet analysis
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Signal reconstruction
Reconstruction from 29 % orig. coefficients
1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0
200
400
600
800
1000
1200
Reconstruction from 11 % Fourier coefficients
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −0.1 0
200
400
600
800
1000
1200
OMN - 2002, Introduction to wavelet analysis
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Signal reconstruction (continued)
Reconstruction from 12 % WL coefficients
1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0
200
400
600
800
1000
1200
Reconstruction from 2 % WL coefficients
1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0
200
400
600
800
1000
1200
OMN - 2002, Introduction to wavelet analysis
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Datafitting/Denoising example
The problem
M observations (xi, yi ), i = 0, 1, . . . , M − 1 N linearly independent functions fj (x), j = 0, 1, . . . , N − 1 Minimize y−F where F (x) =
2
=
M −1 i=0 N −1 j=0
|yi − F (xi)|2 cj fj (x)
N << M
OMN - 2002, Introduction to wavelet analysis
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Fit with 30 polynomials
Original signal 4
2
0
−2 −4 4
−3
−2
−1
0 Signal with noise
1
2
3
4
2
0
−2 −4 4
−3
−2
−1
0
1
2
3
4
Fit with 30 Forsythe polynomials
2
0
−2 −4 1.5 1 0.5 0 −0.5 −1 −4
−3
−2
−1
0 Residues
1
2
3
4
−3
−2
−1
0
1
2
3
4
OMN - 2002, Introduction to wavelet analysis
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Fit with 70 polynomials
Original signal 4
2
0
−2 −4 4
−3
−2
−1
0 Signal with noise
1
2
3
4
2
0
−2 −4 4
−3
−2
−1
0
1
2
3
4
Fit with 70 Forsythe polynomials
2
0
−2 −4 1 0.5 0 −0.5 −1 −4
−3
−2
−1
0 Residues
1
2
3
4
−3
−2
−1
0
1
2
3
4
OMN - 2002, Introduction to wavelet analysis
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Fit with trigonometric functions
fk (x) = ei2πkx y(x) = ck = • FFT • Orthogonal decomposition • Global functions
1 −1 N k=0
ck ei2πkx
y(x)e−i2πkx dx
OMN - 2002, Introduction to wavelet analysis
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Fit with 30 Fourier bases
Original signal 4
2
0
−2 −4 4
−3
−2
−1
0 Signal with noise
1
2
3
4
2
0
−2 −4 4
−3
−2
−1
0
1
2
3
4
Fit with 30 Complex exponentials
2
0
−2 −4 1 0.5 0 −0.5 −1 −4
−3
−2
−1
0 Residues
1
2
3
4
−3
−2
−1
0
1
2
3
4
OMN - 2002, Introduction to wavelet analysis
37
Fit with 160 Fourier bases
Original signal 4
2
0
−2 −4 4
−3
−2
−1
0 Signal with noise
1
2
3
4
2
0
−2 −4 4
−3
−2
−1
0
1
2
3
4
Fit with 160 Complex exponentials
2
0
−2 −4 0.5
−3
−2
−1
0 Residues
1
2
3
4
0
−0.5
−1 −4
−3
−2
−1
0
1
2
3
4
OMN - 2002, Introduction to wavelet analysis
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Fit with Scaling functions φ(x)
• Local like splines • Ortonormal • Abscissae must be equidistant dyadic rationals • N must be power of 2 • Increased N ⇒ more details F (x) =
2J −1 l=0
cjl φjl (x)
OMN - 2002, Introduction to wavelet analysis
39
Fit with 32 scaling functions
Original signal 4
2
0
−2 −4 4
−3
−2
−1
0 Signal with noise
1
2
3
4
2
0
−2 −4 4
−3
−2
−1
0
1
2
3
4
Fit with 32 Scaling functions
2
0
−2 −4 1 0.5 0 −0.5 −1 −4
−3
−2
−1
0 Residues
1
2
3
4
−3
−2
−1
0
1
2
3
4
OMN - 2002, Introduction to wavelet analysis
40
Fit with 256 scaling functions
Original signal 4
2
0
−2 −4 4
−3
−2
−1
0 Signal with noise
1
2
3
4
2
0
−2 −4 4
−3
−2
−1
0
1
2
3
4
Fit with 256 Scaling functions
2
0
−2 −4 1
−3
−2
−1
0 Residues
1
2
3
4
0.5
0
−0.5 −4
−3
−2
−1
0
1
2
3
4
OMN - 2002, Introduction to wavelet analysis
41
Wavelet decomposition - original signal (D=16)
smallosc.dat Wavelet decomposition (individual plots), D=4 5 0 −5 0.1 0 −0.1 0.5 0 −0.5 1 0 −1 1 0 −1 0.5 0 −0.5 5 0 −5 −4 −3 −2 −1 0 1 2 3 4 V5 W
5
V10
W
9
W8
W7
W6
OMN - 2002, Introduction to wavelet analysis
42
Wavelet decomposition - noisy signal (D=16)
smallosc1.dat Wavelet decomposition (individual plots), D=16 5 0 −5 0.1 0 −0.1 0.2 0 −0.2 1 0 −1 0.5 0 −0.5 0.5 0 −0.5 −1 4 2 0 −2 −4 W6 W7 W8 W9 V10
W5
−3
−2
−1
0
1
2
3
4
V5
OMN - 2002, Introduction to wavelet analysis
43
Fit with Wavelets ψ(x)
F (x) =
2J0 −1 l=0
cJ0l φJ0l (x) +
J
2J−1 −1
j=J0 l=J0
dj,l ψj,l (x)
• Need all M scaling function coefficients (interpolation) • Compute all wavelet coefficients • Choose N << M largest wavelet coefficients • Transform back • Local information is well represented • Method is automatic
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44
Truncation algorithm
Wavelet Spectrum 15 10 5 0 −5 −10 −15 −20 0 200 400 600 Sorted Wavelet Spectrum 20 800 1000 1200
15
10
5
0 0
200
400
600
800
1000
1200
Truncated Wavelet Spectrum 15 10 5 0 −5 −10 −15 −20 0 200 400 600 800 1000 1200
OMN - 2002, Introduction to wavelet analysis
45
Hierarchy - noisy signal (D=16)
Hierarchy of coefficients (normalized) 12
10
(*2) (*2)
8
(*5) (*2)
6
(*6) (*11)
4
(*34) (*27)
2
(*21) (*30)
0 −4
(*21) −3 −2 −1 0 1 2 3 4
OMN - 2002, Introduction to wavelet analysis
46
Hierarchy - truncated (D=16)
Hierarchy of coefficients (normalized) 12
10
(*0) (*2)
8
(*5) (*2)
6
(*6) (*11)
4
(*34) (*27)
2
(*21) (*30)
0 −4
(*21) −3 −2 −1 0 1 2 3 4
OMN - 2002, Introduction to wavelet analysis
47
Wavelet fit
Original signal 4
2
0
−2 −4 4
−3
−2
−1
0 Signal with noise
1
2
3
4
2
0
−2 −4 4
−3
−2
−1
0 Fit with 30 Wavelets
1
2
3
4
2
0
−2 −4 1
−3
−2
−1
0 Residues
1
2
3
4
0.5
0
−0.5 −4
−3
−2
−1
0
1
2
3
4
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Detection of singularities
x3/6, 0≤x<1 2 f (x) = 3 1 2 x /6 − x /2 + x/2 − 1/8, 2 ≤x < 1
1 x, 0≤x<2 f (x) = x − 1, 1 ≤ x < 1 2
f
0
0.5 x
1
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Mathematical microscope
Wavelet decomposition (D = 8)
V9
W8
W
7
W
6
V
6
0
0.5
1
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2D smoothing spline
Jα (f ) =
n i=1
(f (x(i)) − y (i) )2 + α
2 1 0 −1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5
Ω
|ν|=2
2 ν
(Dν f (x))2dx
0.6
0.7
0.8
0.9
1
2D wavelet decomposition
Cf Df
PVJ ⊕VJ f = PVJ−1⊕VJ−1 f + PVJ−1⊕WJ−1 + PWJ−1⊕VJ−1 + PWJ−1⊕WJ−1 f
Df
1 0.9 0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
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2D image compression
Original image X
Wavelet spectrum Y
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Compression by 1:100
Reconstruction from 1 % Wavelet coefficients
Reconstruction from 1 % Fourier coefficients
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Why wavelets?
FOURIER Frequency localization: Perfect Time localization: Poor Manipulation: Easy Differential operators: Diag Complexity: O(N log N ) Sparsity: Smooth signals
WAVELETS Good Good Flexible Sparse O(N ) Piecewise smooth signals
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Next week
• Wavelet Algorithms • Wavelets and Fourier Analysis • Wavelets and Partial Differential Equations