Wavelet Scattering Transforms Haixia Liu Department of Mathematics - - PowerPoint PPT Presentation

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wavelet scattering transforms

Wavelet Scattering Transforms Haixia Liu Department of Mathematics - - PowerPoint PPT Presentation

Jan 20, 2024 205 likes •461 views

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 Mathematics The Hong Kong University of Science and

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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 Mathematics The Hong Kong University of Science and Technology

February 6, 2018

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Problem Wavelet Scattering Transform Digit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks

Outline

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Problem Dataset Problem two subproblems

utline of image classification problem

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Wavelet Scattering Transform Review of Multiscale Wavelet Transform Why Wavelets? Wavelet Convolutional Networks

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Digit Classification: MNIST by Joan Bruna et al.

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MATLAB code of Wavelet convolutional Networks

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Problem Wavelet Scattering Transform Digit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks

Digit classification

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Problem Wavelet Scattering Transform Digit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks

Digit classification

Translation Deformation

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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 wrongly attributed to van Gogh High-resolution professional images provided by van Gogh Museum and Kr¨

ller-M¨

uller Museum Design an algorithm to determine if a painting is from van Gogh

r 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 two subproblems: Feature extraction (image processing),

Fourier Transform, Wavelet, EMD, Tight frame ...

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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 two subproblems: Feature extraction (image processing),

Fourier Transform, Wavelet, EMD, Tight frame ...

Clustering or classification (data analysis).

SVM, HMM, ...

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Problem Wavelet Scattering Transform Digit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks

Image Classification

Feature Extraction Classification (classifiers)

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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

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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

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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 ≤ Csup

t |∇τ|

Modulus improves invariance Fourier transform on translated function, modulus lead to translation 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

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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

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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 be decorrelated 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 scattering coefficients

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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 from http://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!!!