Gabor and Wavelet transforms for signals defined on graphs An invitation to signal processing on graphs
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1 Gabor and Wavelet transforms for signals defined on graphs An invitation to signal processing on graphs Benjamin Ricaud Signal Processing Lab 2, EPFL Lausanne, Switzerland SPLab, Brno, 10/2013 B. Ricaud (LTS2, EPFL) Signal processing on graphs SPLab, Brno, 10/ / 20
2 Lab Presentation Swiss federal institute of technology 9,306 students of over 125 nationalities 316 laboratories, 319 faculty Main focus: engineering, computer science, life science, biomedical engineering. B. Ricaud (LTS2, EPFL) Signal processing on graphs SPLab, Brno, 10/ / 20
3 Lab Presentation Signal Processing Lab. 2 Prof. Pierre Vandergheynst 2 postdocs, 7 Phd Students, 1 engineer (software) Signal and image processing 3D reconstruction, video tracking Sparsity, compressive sensing compressive sensing for MRI data Optimization, inverse problems Graphs and signal processing on graphs Analysis of brain data (fmri /dmri), graph of music, transportation networks From theory to applications and to start-ups. B. Ricaud (LTS2, EPFL) Signal processing on graphs SPLab, Brno, 10/ / 20
4 Motivation Graphs: models for many applications B. Ricaud (LTS2, EPFL) Signal processing on graphs SPLab, Brno, 10/ / 20
5 Motivation Graphs: models for many applications Nodes V, edges E, weight matrix w. Data on graphs A signal: a value or vector on each node f : V RN or CN. Here N = 1. B. Ricaud (LTS2, EPFL) Signal processing on graphs SPLab, Brno, 10/ / 20
6 Motivation Examples of applications B. Ricaud (LTS2, EPFL) Signal processing on graphs SPLab, Brno, 10/ / 20
7 Motivation Focus: denoising and sparsity in wavelets (1) signal, (2) noisy signal y, (3) smoothing: argmin f y f γ f, Lf Wavelet denoising. Wavelet transform + thresholding Decay of wavelet coefficients argmin a y W a γ a 1 B. Ricaud (LTS2, EPFL) Signal processing on graphs SPLab, Brno, 10/ / 20
8 Motivation Analysis of functions, key concepts Smoothness, regularity of functions on graphs The Laplacian Locality Wavelet transform Gabor transform B. Ricaud (LTS2, EPFL) Signal processing on graphs SPLab, Brno, 10/ / 20
9 The Laplacian The graph Laplacian L Regularity of a function on the graph Smooth function: function with small variations from node to node. Measures of the variations: Gradient: a value for each edge l 2 (V) l 2 (E) f(m, n) = w(m, n)[f(n) f(m)] Laplacian: a value for each node l 2 (V) l 2 (V) Lf(n) = f(n) = m w(m, n)[f(n) f(m)] Choice + graph Laplacian well studied in math + used in the wavelet definition L. B. Ricaud (LTS2, EPFL) Signal processing on graphs SPLab, Brno, 10/ / 20
10 Global regularity Global regularity Regularity based on the Laplacian. The graph Laplacian quadratic form (used for Tikhonov reg.) f L := f 2 = f, Lf. Sobolev regularity (p N): L p f 2 + f 2 f, L 1 f = 0.14 f, L 2 f = 1.31 f, L 3 f = 1.81 B. Ricaud (LTS2, EPFL) Signal processing on graphs SPLab, Brno, 10/ / 20
11 Wavelets Wavelets (Hammond et al. ACHA, 2011) Main challenges: define translation, define dilation. Classical case: translation: convolution with a delta, multiplication in the Fourier domain, dilation: inverse dilation in the Fourier domain. Fourier on graphs Fourier domain = spectral domain of the Laplacian. Classical case: Le ikx = d2 dx 2 e ikx = k 2 e ikx. Graph case: eigenvectors {u k } of L = Fourier modes Lu k = λ k u k. Graph Fourier transform: f(k) = f, u k. B. Ricaud (LTS2, EPFL) Signal processing on graphs SPLab, Brno, 10/ / 20
12 Wavelets Translations on the graph Generalization of the convolution Localization around point a: convolution f δ a. Multiplication in Fourier Classical: T a f(x) = f(x a) = f(k)e 2πika e 2πikx dk Graph: T a f(j) = ˆf(n)u n(a)u n (j) n R We prove that it stays localized. but the shape changes as well as the l 2 -norm. B. Ricaud (LTS2, EPFL) Signal processing on graphs SPLab, Brno, 10/ / 20
13 Wavelets Dilations on the graph s 1 =17.8 s 2 =6.7 s 3 =2.0 s 4 =0.5 s 5 = Fourier domain Dilation in the Fourier domain. Fourier domain = a line 1 define a continuous function ĝ on R +, Wavelet kernel, 2 dilate it at different scales ĝ s (λ) = ĝ(sλ), 3 take the discretized version of ĝ s on the spectrum, 4 compute the inverse Fourier transform of each ĝ s. ψ s,a (j) = n (ĝ s (n).u n(a)) u n (j) B. Ricaud (LTS2, EPFL) Signal processing on graphs SPLab, Brno, 10/ / 20
14 Wavelets Dilations on the graph (2) B. Ricaud (LTS2, EPFL) Signal processing on graphs SPLab, Brno, 10/ / 20
15 Wavelets Wavelets (Hammond et al. ACHA, 2011) Eigendecomposition of the Laplacian. Kernel ĝ defined on the spectral domain. ĝ : R + R continuous. Wavelet: ψ s,n (j) = l ĝ(sλ l)u l (n)u l (j). Wavelet transform: W f(s, n) = f, ψ s,n. Invertible transform: add scaling function h (low-pass filter) Admissibility cond. ĝ(x) 2 dx/x < 0 s 1 =17.8 s 2 =6.7 s 3 =2.0 s 4 =0.5 s 5 = B. Ricaud (LTS2, EPFL) Signal processing on graphs SPLab, Brno, 10/ / 20
16 Wavelets A Gabor transform on graphs Ingredients a window, a translation, a modulation. B. Ricaud (LTS2, EPFL) Signal processing on graphs SPLab, Brno, 10/ / 20
17 Wavelets Modulations on the graph Modulation: multiplication by a Fourier mode (eigenmode of the Laplacian). M k f(i) = Nu k (i)f(i). B. Ricaud (LTS2, EPFL) Signal processing on graphs SPLab, Brno, 10/ / 20
18 Wavelets Gabor frame The set M b T a g(i) = Nu b (i) n ĝ(n)u n(a)u n (i) is a frame: A f 2 2 N N 1 a=1 b=0 f, M b T a g 2 B f < N ĝ(0) 2 A = min a={1,2,...,n} N T ag 2 2 B max N T a g 2 2 a Shuman, Ricaud, Vandergheynst, Vertex-frequency analysis on graphs, B. Ricaud (LTS2, EPFL) Signal processing on graphs SPLab, Brno, 10/ / 20
19 Wavelets Visualization B. Ricaud (LTS2, EPFL) Signal processing on graphs SPLab, Brno, 10/ / 20
20 Wavelets Conclusion Function behavior on the graph Globally smooth, slow variations of f, piecewise smooth (communities), spikes, fast variations in small regions, oscillations. Representation and sparsity Piecewise smooth functions on graphs: Sparse representation in wavelets (localized) oscillations: sparsity in Gabor Applications compression, compressive sensing, denoising, inpainting, label propagation... B. Ricaud (LTS2, EPFL) Signal processing on graphs SPLab, Brno, 10/ / 20
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