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

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

Problem Wavelet Scattering Transform Digit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks Outline Problem 1 Dataset Problem two subproblems outline of image classification problem Wavelet Scattering Transform 2 Review of Multiscale Wavelet Transform Why Wavelets? Wavelet Convolutional Networks Digit Classification: MNIST by Joan Bruna et al. 3 MATLAB code of Wavelet convolutional Networks 4

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

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

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.

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¨ oller-M¨ uller Museum Design an algorithm to determine if a painting is from van Gogh or NOT

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 ...

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, ...

Problem Wavelet Scattering Transform Digit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks Image Classification Feature Extraction Classification (classifiers)

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

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

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

Problem Wavelet Scattering Transform Digit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks Haar Filtering Hx ( u ) = x ∗ h ( 2 u ) and Gx ( u ) = x ∗ g ( 2 u ) where h is a low frequency and g is a high frequency.

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 { ψ λ } λ Dilated Wavelets: ψ λ ( t ) = 2 j ψ ( 2 j t ) with λ = 2 j . Multiscale and oritented wavelet filters ψ λ = 2 j ψ ( 2 j θ x ) where θ ∈ R ( R 2 ) be a rotation matrix and λ = ( 2 j , θ ) . � x ( u ) ψ λ ( ω − u ) ⇒ � x · � x ∗ ψ λ ( ω ) = x ∗ ψ λ ( ω ) = � ψ λ Wavelet transform: � x ∗ φ 2 J ( t ) � Wx = x ∗ ψ λ ( t ) λ ≤ 2 J

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 ψ λ , τ = ψ λ ( t − τ ( t )) , then � ψ λ − ψ λ , τ � ≤ C sup t |∇ τ | Modulus improves invariance Fourier transform on translated function, modulus lead to translation invariance � x ∗ φ 2 J ( t ) � | W | x = | x ∗ ψ λ ( t ) | λ ≤ 2 J

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

Problem Wavelet Scattering Transform Digit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks Scattering Coefficients first-layer scattering coefficients S 1 , J (( λ 1 ) , x ) = | X ∗ ψ λ 1 | ∗ φ J ( x ) second-layer scattering coefficients S 2 , J (( λ 1 , λ 2 ) , x ) = || X ∗ ψ λ 1 | ∗ ψ λ 2 | ∗ φ J ( x ) m -th layer scattering coefficients S 2 , J (( λ 1 , λ 2 , ··· , λ m ) , x ) = || X ∗ ψ λ 1 |··· ∗ ψ λ m | ∗ φ J ( x )

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

Problem Wavelet Scattering Transform Digit Classification: MNIST by Joan Bruna et al. MATLAB code of Wavelet convolutional Networks Renormalization S 1 , J (( λ 1 )) = S 1 , J (( λ 1 )) ˜ and S 2 , J (( λ 1 , λ 2 )) = S 2 , J (( λ 1 , λ 2 )) ˜ S 1 , J (( λ 1 )) Paper Deep Scattering Spectrum points out second coefficients can be decorrelated to increase their invariance through a renormalization.

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 S m , J = ∑ ¯ ˜ S m , J (( λ 1 , ··· , λ m ) , x ) . x dimension reduction destroy the spatial information contained in scattering coefficients

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 ···

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

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/ .

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