Signal Processing on Graphs: Recent Results, Challenges and Applications
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1 Signal Processing on Graphs: Recent Results, Challenges and Applications Oversampled Graph Transforms and Image Processing Yuichi Tanaka Tokyo University of Agriculture and Technology Nov. 12, 214 Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
2 Outline Oversampled Graph Transforms Introduction Graph Oversampling Oversampled Graph Filter Banks Image Processing Image Denoising with Collaborative Graph Wavelet Shrinkage Trilateral Filter on Graph Spectral Domain Conclusions Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
3 Introduction Collaborators & Acknowledgments Collaborators S. Ono & M. Yamagishi (Tokyo Institute of Technology) Y. Iizuka, M. Onuki & A. Sakiyama (Graduate Students) Grants MEXT/JST Tenure Track Promotion Program MIC SCOPE Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
4 Introduction Outline Oversampled Graph Transforms Introduction Graph Oversampling Oversampled Graph Filter Banks Image Processing Image Denoising with Collaborative Graph Wavelet Shrinkage Trilateral Filter on Graph Spectral Domain Conclusions Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
5 Introduction Conventional Graph Filter Banks (GFBs) Undecimated GFBs (redundancy: M) Spectral graph wavelets (SGWT) [Hammond et al. 211] Tight graph wavelets [Leonardi et al. 213], [Shuman et al. 214] Critically sampled GFBs (redundancy: 1) Two-channel orthogonal GFBs [Narang and Ortega 212] Two-channel biorthogonal GFBs [Narang and Ortega 213] Figure: Critically sampled graph filter bank. Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
6 Introduction Oversampled Graph Transforms Graph transforms with redundancy R (1 < R < M) Topics: Perfect reconstruction condition Redundancy Design method graph oversampling graph undersampling Figure: Graph oversampling followed by M-channel oversampled graph filter bank. Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
7 Introduction Oversampling Graph Signals Two possibilities to oversample graph signals: 1. Oversampling-then-filtering by oversampled graph Laplacian matrix [Sakiyama and Tanaka 214] Simultaneous oversampling of graph signal and graph Laplacian matrix 2. Transformation by M (> 2) filters by oversampled GFBs [Tanaka and Sakiyama 214] IGFT IGFT Graph oversampling GFT IGFT Figure: Signal processing flow of oversampled graph filter bank. Symbols just below the skewed lines indicate the number of signals at typical positions. Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
8 Graph Oversampling Outline Oversampled Graph Transforms Introduction Graph Oversampling Oversampled Graph Filter Banks Image Processing Image Denoising with Collaborative Graph Wavelet Shrinkage Trilateral Filter on Graph Spectral Domain Conclusions Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
9 Graph Oversampling Oversampled Graph Laplacian Matrix Original GLM of size N N : L = D A Oversampled GLM of size N 1 N 1 L = D 1/2 L D 1/2 where L = D [ Ã ] A A Ã = 1 A T 1 N1 N Ã, D: oversampled adjacency/degree matrix A 1 : edge information between original nodes and appended ones Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
10 Graph Oversampling Yuichi Tanaka (TUAT) PCSJ/IMPS Figure: 214 Nov. 12, / 49 Effective Graph Expansion Methods How to choose a good A 1? Oversampled bipartite graphs with all edges of the original graph If oversampled graph is bipartite: GFBs with downsampling can be used. We can control the overall redundancy. Our work (partially) answers the following questions: Can any graphs be converted into an oversampled bipartite graph? Systematic construction method? Relationship to graph theory?
11 Graph Oversampling Conversion of K-colorable graph One K-colorable graph log 2 K bipartite subgraphs [Harary et al. 1977], [Narang and Ortega 212] Bipartite subgraphs can be merged into one OS bipartite graph. Step-by-step example: Conversion of 3-colorable graph Definition F 1 to F 3 : Colored node sets B 1 : Subgraph 1 (having edges linking F 1 F 2 and F 3 ) B 2 : Subgraph 2 (having edges linking F 1 and F 2 ) Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
12 Graph Oversampling Oversampled Graph Construction Step 1: Decide foundation bipartite graph Foundation bipartite graph: 1st (or ground) floor of OS graph In this example, we chose B 1 as the foundation bipartite graph. (a) Original graph (b) Foundation bpt. graph Figure: OS bpt. graph construction Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
13 Graph Oversampling Oversampled Graph Construction Step 2: Append and connect oversampled nodes 1. Additional nodes F 1 and F 2 are placed just above F 1 and F Connect F 1 and F 2 and F 2 and F 1. (a) Original graph (b) OS bpt. graph (c) L and H Figure: OS bpt. graph construction Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
14 Graph Oversampling K-Colorable Case Similar construction to the 3-colorable case Freedom for choosing the foundation bipartite graph (a) Original graph (b) Foundation bpt. graph (c) Remaining edges (d) OS bpt. graph Figure: OS bpt. graph construction for a five-colorable graph Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
15 Graph Oversampling Example: Graph Oversampling of Ring Graph Ring graph with odd # of nodes: 3-colorable Critically sampled bpt. graphs # of nodes are heavily biased. A few relationships btw nodes becomes very weak. Oversampled bpt. graphs # of nodes are (almost) even. Small redundancy: (2n + 3)/(2n + 1) Strong connection (a) Original graph (c) Bpt. subgraph 1 (d) Bpt. subgraph 2 (b) OS bpt. graph (e) L and H Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
16 Graph Oversampling Relationship w/ Bipartite Double Cover (BDC) BDC of G: GBDC = G K 2 (K 2 : complete graph of two vertices) [ ] A Adjacency matrix: ÃBDC = A [ ] Normalized GLM: L I D 1 2 AD 1 2 BDC = D 1 2 AD 1 2 I 2N nodes and 2 E edges in G BDC (redundancy: 2) (f) BDC (g) L and H Figure: Bipartite graph cover for a 3-colorable graph Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
17 Graph Oversampling Graph Fourier Spectrum of BDC Eigenvalues and eigenvectors uλi : eigenvector of L (original graph) Eigenvectors of L BDC ũ λi = 1 2 [u T λ i u T λ i ] T ( λ i = λ i ), 1 2 [u T λ i u T λ i ] T ( λ i = 2 λ i ) Graph Fourier coefficients of f = [f T f T ]T [ ] [ ] ũλ T i f = 2 1 u T λi uλ T f i = 2u f λ T i f = 2f (λ i ) [ ] [ ] ũ2 λ T i f = 2 1 u T λi uλ T f i = # of nonzero coefficients: N Spectrum of BDC = Spectrum of the original graph f Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
18 Graph Oversampling Graph Fourier Spectrum of OS Bipartite Graph [ ] [ ] Adjacency and degree matrix: Ã = Af A r, A r D D = D r Af : Adjacency matrix of the foundation bpt. graph Ar : Adjacency matrix of the remaining graph A = A f + A r, D = D f + D r [ Normalized GLM: L I D 1 2 A = f D 1 2 D 1 2 A r D ] 1 2 r D 1 2 r A r D 1 2 I L has the eigenvector ũλi = 1 2 [uλ T i λ i = λ i D r = D (= BDC) u T λ i ] T with BDC is a special case of the oversampled GLM. Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
19 Graph Oversampling Experimental Results Nonlinear approximation of images All lowpass + some fractions of highpass coefficients Compared with CDF 9/7 DWT Laplacian pyramid with 9/7 DWT GraphBior [Narang and Ortega 212] Graph Laplacian pyramid [Shuman et al. 213] Figure: Left: Cameraman ( ). Right: Ballet ( ). Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
20 Graph Oversampling Experimental Results NLA: PSNR comparison 38 PSNR Comparison: Cameraman 65 PSNR Comparison: Ballet PSNR (db) /7 filter Laplacian pyramid graphbior graph Laplacian pyramid proposed method Number of highpass coefficients PSNR (db) Figure: Left: Cameraman. Right: Ballet /7 filter Laplacian pyramid graphbior graph Laplacian pyramid proposed method Number of highpass coefficients Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
21 Graph Oversampling Experimental Results Side-by-side quality comparison: All lowpass + 3% highpass (a) Original (b) 9/7 DWT (c) LP (d) GraphBior (e) Graph LP (f) OSGLM Figure: Reconstructed Coins image Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
22 Graph Oversampling Experimental Results Graph signal denoising: SNR comparison Table: Denoised Results of Minnesota Traffic Graph: SNR (db) σ noisy sym8 sym8 graphbior GLP (1 level) (5 levels) (CSGLM) 1/ / / / / Redundancy σ SGWT OSGFB graphbior graphbior OSGFB (CSGLM) (BDC) (OSGLM) (OSGLM) 1/ / / / / Redundancy Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
23 Graph Oversampling Experimental Results Graph signal denoising: Side-by-side comparison (a) Minnesota Traffic Graph (b) Graph signal 1.5 (c) Bpt. subgraph 1 (d) Bpt. subgraph 2 Figure: Minnesota Traffic Graph Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
24 Graph Oversampling (a) Noisy observation (5.99 db) (b) sym8 (5 level): R = 1. (5.76 db) (c) graphbior: R = 1. (8.5 db) (d) SGWT: R = 4. (1.33 db) (e) graphbior with OSGLM: R = 1.37 (9. db) (f) OSGFB with OSGLM: R = 2.74 (12.31 db) Figure: Denoising results of Minnesota Traffic Graph. Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
25 Oversampled Graph Filter Banks Outline Oversampled Graph Transforms Introduction Graph Oversampling Oversampled Graph Filter Banks Image Processing Image Denoising with Collaborative Graph Wavelet Shrinkage Trilateral Filter on Graph Spectral Domain Conclusions Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
26 Oversampled Graph Filter Banks Notations Eigenspace projection matrix: P λi := λ=λ i ũ λ ũ T λ k-th analysis filter: H k = λ i σ( L) h k(λ i ) P λi (synthesis: G k ) h k (λ), g k (λ): filter kernel (real function for λ λ max = 2) f out = Udiag{h k (λ i )}U T f in Figure: Oversampled graph filter bank. Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
27 Oversampled Graph Filter Banks Perfect Reconstruction Condition Transfer function T = 1 2 λ i M 1 k= g k (λ i )h k (λ i ) }{{} Amplitude term P λi + 1 { g k (λ i )h k (2 λ i ) + g 2 k+ M (λ i )h 2 k+ M (2 λ i )} P λi J 2 }{{} λ i M 2 1 k= Spectral folding term leads to the following condition similar to the critically sampled case: M 1 k= M 2 1 k= g k (λ)h k (λ) = 2 g k (λ i )h k (2 λ i ) + g k+ M 2 (λ i )h k+ M (2 λ i ) = 2 Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
28 Oversampled Graph Filter Banks Perfect Reconstruction Condition Spectral folding effect can be cancelled with the following filter selection: g k (λ) = h k+m/2 (2 λ), g k+m/2 (λ) = h k (2 λ) Product filters p k (λ) = g k (λ)h k (λ) must satisfy M/2 1 k= p k (λ) + p k (2 λ) = 2 Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
29 Oversampled Graph Filter Banks Design of Oversampled Graph Filter Banks Design methodology 1. Design two-channel halfband filters. 2. Design arbitrary M 2 filters. 3. Subtract M 2 product filters from the two-channel halfband filter and factorize it to obtain remaining 2 filters. Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
30 Oversampled Graph Filter Banks Design Examples OSGFB designs for different # of zeros of halfband filters Filter length (polynomial order): 1 for lowpass 11 for highpass 2 Oversampled Graph Filter Bank 2 Oversampled Graph Filter Bank 2 Oversampled Graph Filter Bank Magnitude Response 1.5 h h 1 h 2 h 3 NO h NO h 1 Magnitude Response 1.5 h h 1 h 2 h 3 NO h NO h 1 Magnitude Response 1.5 h h 1 h 2 h 3 NO h NO h λ (a) (2, 2) zeros λ (b) (4, 4) zeros λ (c) (8, 8) zeros Figure: Four-channel oversampled graph filter banks (black lines indicate graphbior(6,6) [Narang and Ortega 213]) Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
31 Oversampled Graph Filter Banks Graph Signal Decomposition Coins image Edge-aware image graph 2. Minnesota Traffic Graph Three-colorable graph Figure: Graph signals used. Left: Coins. Right: Minnesota Traffic Graph Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
32 Oversampled Graph Filter Banks original Figure: Multiresolution Coins image after three-level decomposition using the oversampled graph filter bank. The original image on the same scale is shown at the top right. The values of the transformed coefficients are scaled to be in the range [, 1] for the sake of visualization. Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
33 Oversampled Graph Filter Banks Channel # 1 Channel #1.4 LL Channel #4.3 Channel # Channel #2.5 Channel #3.4 LH Channel #6.3 Channel # Channel #1 5 x 1 Channel # HH Channel #14 4 x 1 Channel # Figure: Graphs decomposed by the proposed oversampled graph filter bank. We use a two-dimensional four-channel filter bank leading to 4 2 = 16 channels. Note that the graph is three-colorable: therefore, channels 8, 9, 12, and 13 (corresponding to the HL channel for the critically sampled filter banks) are empty. Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
34 Image Processing Image Denoising with Collaborative Graph Wavelet Shrinkage Outline Oversampled Graph Transforms Introduction Graph Oversampling Oversampled Graph Filter Banks Image Processing Image Denoising with Collaborative Graph Wavelet Shrinkage Trilateral Filter on Graph Spectral Domain Conclusions Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
35 Image Processing Image Denoising with Collaborative Graph Wavelet Shrinkage BM3D: Image Denoising State-of-the-Art Two-step algorithm in BM3D [Dabov et al. 27] 1. Basic estimate 1.1 Grouping 1.2 Collaborative separable filtering + hard-thresholding 1.3 Aggregation 2. Final estimate 2.1 Grouping 2.2 Collaborative Wiener filtering 2.3 Aggregation Block matching Collaborative filtering 3-D transform Input image Weight Hard thresholding Aggregation Inv. 3-D transform Basic-estimated image Figure: Basic estimate of BM3D Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
36 Image Processing Image Denoising with Collaborative Graph Wavelet Shrinkage Problem on BM3D BEST objective performance Problems due to aggregation and Wiener filtering 1. Unnatural artifacts 2. Oversmoothing (a) (b) Figure: (a) original, (b) denoised image by BM3D Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
37 Image Processing Image Denoising with Collaborative Graph Wavelet Shrinkage Collaborative Graph Wavelet Shrinkage Improved basic estimate step based on graph signal processing Algorithm 1. Grouping 2. Inter-/Intra-patch graph construction 3. Collaborative filtering with graph wavelets + hard-thresholding 4. Aggregation Figure: CGWS Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
38 Image Processing Image Denoising with Collaborative Graph Wavelet Shrinkage Experiments on Depth Map Denoising Depth map: Piecewise smooth (constant) images Depth maps used (all from Middlebury Stereo Datasets) art ( ), ballet ( ), Ryerson ( ), cones (45 375), teddy (45 375) Corrupted by AWGN with σ = 1, 3, 5, 1 Comparison with Gaussian filtering + L Smoothing [Xu et al. 211] K-SVD [Aharon and Elad 26] BM3D Figure: Left: cones. Right: teddy. Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
39 Image Processing Image Denoising with Collaborative Graph Wavelet Shrinkage Experimental Results: PSNR Comparisons σ/psnr method art ballet Ryerson cones teddy 1 / / / / 8.14 GF+L-Sm K-SVD BM3D prop GF+L-Sm K-SVD BM3D prop GF+L-Sm K-SVD BM3D prop GF+L-Sm K-SVD BM3D prop Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
40 Image Processing Image Denoising with Collaborative Graph Wavelet Shrinkage Experimental Results: Side-by-Side Comparisons teddy, σ = 1 (a) Original (b) Noisy (c) GF+L (d) K-SVD (e) BM3D (f) CGWS Figure: Denoising results. Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
41 Image Processing Trilateral Filter on Graph Spectral Domain Outline Oversampled Graph Transforms Introduction Graph Oversampling Oversampled Graph Filter Banks Image Processing Image Denoising with Collaborative Graph Wavelet Shrinkage Trilateral Filter on Graph Spectral Domain Conclusions Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
42 Image Processing Trilateral Filter on Graph Spectral Domain Image Processing with Nonlocal Filters Many nonlocal filters and processing so far Bilateral filter : Two Gaussian weights for distance and pixel values Weights between i and j-th pixels: w ij = exp ( p i p j 2 ) exp ( (x i x j ) 2 ) 2σ 2 c p i : Coordinate of i-th pixel, x i : i-th pixel value, σ c, σ s : Std. dev. for Gaussian function Trilateral filter : Extension of bilateral filter Gradient and pixel smoothing with BF High smoothing performance (compared to bilateral filter) Image filtering, tone-mapping of high dynamic range imaging Problem of nonlocal filters Pixel-dependent filtering: 2σ 2 s No expression in frequency domain Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
43 Image Processing Trilateral Filter on Graph Spectral Domain Bilateral Filter in Graph Spectral Domain Graph signal processing enables to represent nonlocal filters on graph spectral domain [Gadde et al. 213]. x, x: Vectorized input and output pixels W = [w ij ], D jj = j w ij Graph bilateral filter: x = D 1 Wx = U }{{} (I Λ) }{{} Inv. GFT Graph LPF h(λ) = 1 λ U T }{{} GFT Graph filter kernels can be arbitrary changed according to applications. x Original Kernel Proposed Kernel Figure: Kernels of graph bilateral filter Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
44 Image Processing Trilateral Filter on Graph Spectral Domain Graph Trilateral Filter Gradient smoothing can also be represented as graph spectral filters: Double lowpass filters in graph spectral domain Algorithm 1. Calculate image gradient and construct gradient graph 2. Gradient smoothing by graph bilateral filter 3. construct image graph with smoothed gradient 4. Pixel value smoothing by graph bilateral filter Both smoothing filters can be chosen arbitrary Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
45 Image Processing Trilateral Filter on Graph Spectral Domain Experimental Results AWGN denoising experiment Test images ( ): Lena, Watch, Boat, and Monarch White gaussian noise : σ = 2, 3, 4, 5 Comparison with regular/graph bilateral filter and regular trilateral filter Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
46 Image Processing Trilateral Filter on Graph Spectral Domain Experimental Results: PSNR Comparisons Table: Denoising Results: PSNR (db) Images σ Noisy Lena Watch Boat Monarch BF TF SGBF Proposed BF TF SGBF Proposed BF TF SGBF Proposed BF TF SGBF Proposed Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
47 Image Processing Trilateral Filter on Graph Spectral Domain Experimental results: Side-by-Side Comparison Lena, σ = 3 Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
48 Conclusions Conclusions Oversampled graph transforms Graph oversampling with oversampled graph Laplacian matrix Oversampled graph filter banks Good for graph signal analysis and reasonable redundancy Image processing Modification of BM3D with collaborative graph wavelet shrinkage Image smoothing by graph-based trilateral filter Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
49 Conclusions Further Reading Papers and Proceedings 1. A. Sakiyama and Y. Tanaka, Oversampled graph Laplacian matrix for graph filter banks, IEEE Trans. Signal Process., 214, in press. (Open Access) 2. Y. Tanaka and A. Sakiyama, M-channel oversampled graph filter banks, IEEE Trans. Signal Process., vol. 62, no. 14, pp , 214. (Open Access) 3. Y. Iizuka and Y. Tanaka, Depth map denoising using collaborative graph wavelet shrinkage on connected image patches, ICIP 214, Paris, France, Oct M. Onuki and Y. Tanaka, Trilateral filter on graph spectral domain, ICIP 214, Paris, France, Oct Review Article ( in Japanese! ) 1. Y. Tanaka, Recent Advances in Signal Processing on Graphs, IEICE Fundamentals Review, vol. 8, no. 1, pp , 214. MATLAB codes and slides Yuichi Tanaka (TUAT) PCSJ/IMPS 214 Nov. 12, / 49
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