wavelet scattering transforms · problem wavelet scattering transformdigit classification: mnist...
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![Page 1: Wavelet Scattering Transforms · Problem Wavelet Scattering TransformDigit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks The Problem 79paintings](https://reader030.vdocuments.net/reader030/viewer/2022040716/5e20d3808a3b2e22921177b5/html5/thumbnails/1.jpg)
Problem Wavelet Scattering Transform Digit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks
Wavelet Scattering Transforms
Haixia Liu
Department of MathematicsThe Hong Kong University of Science and Technology
February 6, 2018
![Page 2: Wavelet Scattering Transforms · Problem Wavelet Scattering TransformDigit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks The Problem 79paintings](https://reader030.vdocuments.net/reader030/viewer/2022040716/5e20d3808a3b2e22921177b5/html5/thumbnails/2.jpg)
Problem Wavelet Scattering Transform Digit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks
Outline
1 ProblemDatasetProblemtwo subproblemsoutline of image classification problem
2 Wavelet Scattering TransformReview of Multiscale Wavelet TransformWhy Wavelets?Wavelet Convolutional Networks
3 Digit Classification: MNIST by Joan Bruna et al.
4 MATLAB code of Wavelet convolutional Networks
![Page 3: Wavelet Scattering Transforms · Problem Wavelet Scattering TransformDigit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks The Problem 79paintings](https://reader030.vdocuments.net/reader030/viewer/2022040716/5e20d3808a3b2e22921177b5/html5/thumbnails/3.jpg)
Problem Wavelet Scattering Transform Digit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks
Digit classification
Translation
Deformation
![Page 4: Wavelet Scattering Transforms · Problem Wavelet Scattering TransformDigit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks The Problem 79paintings](https://reader030.vdocuments.net/reader030/viewer/2022040716/5e20d3808a3b2e22921177b5/html5/thumbnails/4.jpg)
Problem Wavelet Scattering Transform Digit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks
Digit classification
Translation
Deformation
![Page 5: Wavelet Scattering Transforms · Problem Wavelet Scattering TransformDigit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks The Problem 79paintings](https://reader030.vdocuments.net/reader030/viewer/2022040716/5e20d3808a3b2e22921177b5/html5/thumbnails/5.jpg)
Problem Wavelet Scattering Transform Digit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks
Dataset
(a) f249 (b) f371 (c) f522 (d) f752
Figure: van Gogh’s paintings.
(a) f253a (b) f418 (c) f687 (d) s205 (e) s206v
Figure: Forgeries.
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Problem Wavelet Scattering Transform Digit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks
The Problem
79 paintings authenticated by experts
64 genuine paintings and 15 forgeries
Forgeries are ‘quite’ genuine with 6 historically wronglyattributed to van Gogh
High-resolution professional images provided by van GoghMuseum and Kroller-Muller Museum
Design an algorithm to determine if a painting is from van Goghor NOT
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Problem Wavelet Scattering Transform Digit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks
Image classification can be contributed to the following twosubproblems:
Feature extraction (image processing),
Fourier Transform,Wavelet,EMD,Tight frame...
Clustering or classification (data analysis).
SVM,HMM,...
![Page 8: Wavelet Scattering Transforms · Problem Wavelet Scattering TransformDigit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks The Problem 79paintings](https://reader030.vdocuments.net/reader030/viewer/2022040716/5e20d3808a3b2e22921177b5/html5/thumbnails/8.jpg)
Problem Wavelet Scattering Transform Digit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks
Image classification can be contributed to the following twosubproblems:
Feature extraction (image processing),
Fourier Transform,Wavelet,EMD,Tight frame...
Clustering or classification (data analysis).
SVM,HMM,...
![Page 9: Wavelet Scattering Transforms · Problem Wavelet Scattering TransformDigit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks The Problem 79paintings](https://reader030.vdocuments.net/reader030/viewer/2022040716/5e20d3808a3b2e22921177b5/html5/thumbnails/9.jpg)
Problem Wavelet Scattering Transform Digit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks
Image Classification
Feature Extraction
Classification (classifiers)
![Page 10: Wavelet Scattering Transforms · Problem Wavelet Scattering TransformDigit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks The Problem 79paintings](https://reader030.vdocuments.net/reader030/viewer/2022040716/5e20d3808a3b2e22921177b5/html5/thumbnails/10.jpg)
Problem Wavelet Scattering Transform Digit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks
Aims
AIM: Classify correctly although translation and deformation, i.e.,
Globally invariant to the translation group
Locally invariant to small deformation
Wavelet Scattering Transform
Some advantages of Wavelet Scattering Transform:
Share hierarchical structure of DNNs
replace data-driven filters by wavelets
have strong theoretical support
better performance for small-sample data
![Page 11: Wavelet Scattering Transforms · Problem Wavelet Scattering TransformDigit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks The Problem 79paintings](https://reader030.vdocuments.net/reader030/viewer/2022040716/5e20d3808a3b2e22921177b5/html5/thumbnails/11.jpg)
Problem Wavelet Scattering Transform Digit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks
Aims
AIM: Classify correctly although translation and deformation, i.e.,
Globally invariant to the translation group
Locally invariant to small deformation
Wavelet Scattering Transform
Some advantages of Wavelet Scattering Transform:
Share hierarchical structure of DNNs
replace data-driven filters by wavelets
have strong theoretical support
better performance for small-sample data
![Page 12: Wavelet Scattering Transforms · Problem Wavelet Scattering TransformDigit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks The Problem 79paintings](https://reader030.vdocuments.net/reader030/viewer/2022040716/5e20d3808a3b2e22921177b5/html5/thumbnails/12.jpg)
Problem Wavelet Scattering Transform Digit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks
Haar wavelet transform
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Problem Wavelet Scattering Transform Digit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks
Haar Filtering
Hx(u) = x ∗ h(2u) and Gx(u) = x ∗ g(2u)
where h is a low frequency and g is a high frequency.
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Problem Wavelet Scattering Transform Digit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks
Review of Multiscale Wavelet Transform
wavelet filters {ψλ}λ
Dilated Wavelets: ψλ(t) = 2jψ(2jt) with λ = 2j.
Multiscale and oritented wavelet filters
ψλ = 2jψ(2jθx)
where θ ∈ R(R2) be a rotation matrix and λ = (2j,θ).
x ∗ ψλ(ω) =∫
x(u)ψλ(ω− u)⇒ x ∗ ψλ(ω) = x · ψλ
Wavelet transform:
Wx =[
x ∗ φ2J(t)x ∗ ψλ(t)
]λ≤2J
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Problem Wavelet Scattering Transform Digit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks
Advantages of Wavelets
Wavelets separate multiscale information
Wavelets provide sparse representation
Wavelets are uniformly stable to deformations.If ψλ,τ = ψλ(t− τ(t)), then
‖ψλ − ψλ,τ‖ ≤ C supt|∇τ|
Modulus improves invariance
Fourier transform on translated function, modulus lead totranslation invariance
|W|x =[
x ∗ φ2J(t)|x ∗ ψλ(t)|
]λ≤2J
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Problem Wavelet Scattering Transform Digit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks
![Page 17: Wavelet Scattering Transforms · Problem Wavelet Scattering TransformDigit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks The Problem 79paintings](https://reader030.vdocuments.net/reader030/viewer/2022040716/5e20d3808a3b2e22921177b5/html5/thumbnails/17.jpg)
Problem Wavelet Scattering Transform Digit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks
Scattering Coefficients
first-layer scattering coefficients
S1,J((λ1),x) = |X ∗ ψλ1 | ∗ φJ(x)
second-layer scattering coefficients
S2,J((λ1,λ2),x) = ||X ∗ ψλ1 | ∗ ψλ2 | ∗ φJ(x)
m-th layer scattering coefficients
S2,J((λ1,λ2, · · · ,λm),x) = ||X ∗ ψλ1 | · · · ∗ ψλm | ∗ φJ(x)
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Problem Wavelet Scattering Transform Digit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks
![Page 19: Wavelet Scattering Transforms · Problem Wavelet Scattering TransformDigit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks The Problem 79paintings](https://reader030.vdocuments.net/reader030/viewer/2022040716/5e20d3808a3b2e22921177b5/html5/thumbnails/19.jpg)
Problem Wavelet Scattering Transform Digit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks
Renormalization
S1,J((λ1)) = S1,J((λ1))
and
S2,J((λ1,λ2)) =S2,J((λ1,λ2))
S1,J((λ1))
Paper Deep Scattering Spectrum points out second coefficients can bedecorrelated to increase their invariance through a renormalization.
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Problem Wavelet Scattering Transform Digit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks
Features based on Scattering Coefficients
One choice is to take spatial averages of scattering coefficients
Sm,J = ∑x
Sm,J((λ1, · · · ,λm),x).
dimension reduction
destroy the spatial information contained in scatteringcoefficients
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Problem Wavelet Scattering Transform Digit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks
Classifiers
There are a lot of classifiers can be used if features are extracted
Logistic regression
Random forest
SVM
LDA
Sparse SVM
Sparse LDA
and so on · · ·
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Problem Wavelet Scattering Transform Digit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks
Numerical results
Figure: Results from paper Invariant Scattering Convolution Networks
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Problem Wavelet Scattering Transform Digit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks
Software
Code can be downloaded fromhttp://www.di.ens.fr/data/software/.
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Problem Wavelet Scattering Transform Digit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks
Thank you!!!