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Wavelets DS-GA 1013 / MATH-GA 2824 Mathematical Tools for Data Science Carlos Fernandez-Granda
Wavelets
DS-GA 1013 / MATH-GA 2824 Mathematical Tools for Data Science
Carlos Fernandez-Granda
Prerequisites
Linear algebra (basis, projection, orthogonal complement, direct sum)
Fourier series
Discrete Fourier transform
Short-time Fourier transform
Image
Vertical line (column 135)
0 100 200 300 400 500
0.2
0.4
0.6
0.8
1.0
Multiresolution analysis
Scale / resolution at which information is encoded is not uniform
Goal: Decompose signals into components at different resolutions
Challenge: Design basis of vectors to achieve this
If vectors are orthogonal, then we can just project onto them to separatecontributions of each scale
Father wavelet
We use a low-pass vector, called scaling vector or father wavelet, to extractcoarsest scale
Haar father wavelet
Approximation using Haar father wavelet
Approximation Coefficients
0 100 200 300 400 500
0.2
0.4
0.6
0.8
1.0DataApproximation
0.04 0.02 0.00 0.02 0.0402468
10121416
Mother wavelets
We use shifts and dilations of mother wavelet to capture information atdifferent scales
We can choose the shifts so that the basis vectors are all orthogonal
Haar mother wavelets
Father wavelet + coarsest mother wavelet
Approximation Coefficients
0 100 200 300 400 500
0.2
0.4
0.6
0.8
1.0DataApproximation
0.04 0.02 0.00 0.02 0.040.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Father wavelet + 2 coarsest mother wavelets
Approximation Coefficients
0 100 200 300 400 500
0.2
0.4
0.6
0.8
1.0DataApproximation
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
Father wavelet + 3 coarsest mother wavelets
Approximation Coefficients
0 100 200 300 400 500
0.2
0.4
0.6
0.8
1.0DataApproximation
0 1 2 3
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Father wavelet + 4 coarsest mother wavelets
Approximation Coefficients
0 100 200 300 400 500
0.2
0.4
0.6
0.8
1.0DataApproximation
0 2 4 6
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Father wavelet + 5 coarsest mother wavelets
Approximation Coefficients
0 100 200 300 400 500
0.2
0.4
0.6
0.8
1.0DataApproximation
0 5 10 150.2
0.0
0.2
0.4
0.6
0.8
Father wavelet + 6 coarsest mother wavelets
Approximation Coefficients
0 100 200 300 400 500
0.2
0.4
0.6
0.8
1.0DataApproximation
0 10 20 300.1
0.0
0.1
0.2
0.3
0.4
Father wavelet + 7 coarsest mother wavelets
Approximation Coefficients
0 100 200 300 400 500
0.2
0.4
0.6
0.8
1.0DataApproximation
0 20 40 60
0.05
0.00
0.05
0.10
0.15
0.20
0.25
Father wavelet + 8 coarsest mother wavelets
Approximation Coefficients
0 100 200 300 400 500
0.2
0.4
0.6
0.8
1.0DataApproximation
0 25 50 75 100 125
0.10
0.05
0.00
0.05
0.10
0.15
0.20
Father wavelet + 9 coarsest mother wavelets
Approximation Coefficients
0 100 200 300 400 500
0.2
0.4
0.6
0.8
1.0DataApproximation
0 50 100 150 200 2500.2
0.1
0.0
0.1
0.2
Multiresolution decomposition
Let N := 2K for some K , a multiresolution decomposition of RN is asequence of nested subspaces VK ⊂ VK−1 ⊂ . . . ⊂ V0 satisfying:
I V0 = RN
I If x ∈ Vk then x shifted by 2k is also in Vk (invariance to translations)
I Dilating x ∈ Vj yields vector in Vj+1
Example
Subspace Vk contains vectors that are constant on segments of length 2k
Satisfies conditions:
I V0 = RN
I If x ∈ Vk then x shifted by 2k is also in Vk (invariance to translations)
I Dilating x ∈ Vj yields vector in Vj+1
Spanned by shifts/dilations of Haar father wavelets
Problem: Basis vectors are not orthogonal (at all!)
Solution
Decompose the finer subspaces into a direct sum
Vk = Vk+1 ⊕Wk , 0 ≤ k ≤ K − 1,
Wk is the orthogonal complement of Vk+1 in Vk , so it capturesfinest resolution available at level k
We can then decompose RN into different scales
RN = V0 = V1 ⊕W1
= V2 ⊕W2 ⊕W1
= Vk ⊕Wk ⊕ · · · ⊕W2 ⊕W1
Haar multiresolution decomposition
V3
W3
W2
W1
PVk x is an approximation of x at scale 2k
Vertical line (column 135)
0 100 200 300 400 500
0.2
0.4
0.6
0.8
1.0
Projection onto V9
Projection Coefficients for V9
0 100 200 300 400 500
0.2
0.4
0.6
0.8
1.0DataApproximation
0.04 0.02 0.00 0.02 0.0402468
10121416
Projection onto V8
Projection Coefficients for W9
0 100 200 300 400 500
0.2
0.4
0.6
0.8
1.0DataApproximation
0.04 0.02 0.00 0.02 0.040.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Projection onto V7
Projection Coefficients for W8
0 100 200 300 400 500
0.2
0.4
0.6
0.8
1.0DataApproximation
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
Projection onto V6
Projection Coefficients for W7
0 100 200 300 400 500
0.2
0.4
0.6
0.8
1.0DataApproximation
0 1 2 3
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Projection onto V5
Projection Coefficients for W6
0 100 200 300 400 500
0.2
0.4
0.6
0.8
1.0DataApproximation
0 2 4 6
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Projection onto V4
Projection Coefficients for W5
0 100 200 300 400 500
0.2
0.4
0.6
0.8
1.0DataApproximation
0 5 10 150.2
0.0
0.2
0.4
0.6
0.8
Projection onto V3
Projection Coefficients for W4
0 100 200 300 400 500
0.2
0.4
0.6
0.8
1.0DataApproximation
0 10 20 300.1
0.0
0.1
0.2
0.3
0.4
Projection onto V2
Projection Coefficients for W3
0 100 200 300 400 500
0.2
0.4
0.6
0.8
1.0DataApproximation
0 20 40 60
0.05
0.00
0.05
0.10
0.15
0.20
0.25
Projection onto V1
Projection Coefficients for W2
0 100 200 300 400 500
0.2
0.4
0.6
0.8
1.0DataApproximation
0 25 50 75 100 125
0.10
0.05
0.00
0.05
0.10
0.15
0.20
Projection onto V0
Projection Coefficients for W1
0 100 200 300 400 500
0.2
0.4
0.6
0.8
1.0DataApproximation
0 50 100 150 200 2500.2
0.1
0.0
0.1
0.2
Haar mother wavelets in the frequency domain
200 150 100 50 0 50 100 150 200Frequency
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16 Width: 200Width: 100Width: 50
Time-frequency support of basis vectors
STFT Wavelets
2D Wavelets
Extension to 2D by using outer products of 1D basis vectors
To build a 2D basis vector at scale (m1,m2) and shift (s1, s2) we set
v2D[s1,s2,m1,m2]:= v1D[s1,m1]
(v1D[s2,m2]
)T,
where v1D can refer to 1D father or mother wavelets
Nonseparable designs: steerable pyramid, curvelets, bandlets...
2D Haar wavelet basis vectors
Image
2D Haar wavelet decomposition
Approximation Coefficients
300
310
320
330
340
350
2D Haar wavelet decomposition
Approximation Coefficients
0
20
40
60
80
100
0
20
40
60
80
100
0
20
40
60
80
100
2D Haar wavelet decomposition
Approximation Coefficients
5051015202530
5051015202530
5051015202530
2D Haar wavelet decomposition
Approximation Coefficients
5
0
5
10
15
5
0
5
10
15
5
0
5
10
15
2D Haar wavelet decomposition
Approximation Coefficients
642
0246
642
0246
642
0246
2D Haar wavelet decomposition
Approximation Coefficients
6
4
2
0
2
4
6
4
2
0
2
4
6
4
2
0
2
4
2D Haar wavelet decomposition
Approximation Coefficients
2
1
0
1
2
3
2
1
0
1
2
3
2
1
0
1
2
3
2D Haar wavelet decomposition
Approximation Coefficients
2.01.51.00.5
0.00.51.01.5
2.01.51.00.5
0.00.51.01.5
2.01.51.00.5
0.00.51.01.5
2D Haar wavelet decomposition
Approximation Coefficients
1.0
0.5
0.0
0.5
1.0
1.0
0.5
0.0
0.5
1.0
1.0
0.5
0.0
0.5
1.0
2D Haar wavelet decomposition
Approximation Coefficients
0.60.40.2
0.00.20.40.6
0.60.40.2
0.00.20.40.6
0.60.40.2
0.00.20.40.6
What have we learned
Framework for multiresolution analysis based on wavelets
Implementation based on Haar wavelets
Extension to two dimensions
