Graph Wavelets and Filter Banks Designs in Graph Spectral Domain

Transcription

1 at CMU Graph Wavelets and Filter Banks Designs in Graph Spectral Domain Yuichi Tanaka Tokyo University of Agriculture and Technology PRESTO, Japan Science and Technology Agency This work was supported in part by JST PRESTO under grant JPMJPR1656.

2 Outline Review Desired specifications and properties List of spectral graph wavelets and filter banks Challenges 2

3 Review: Graph Graph Vertex set, edge set, weight set # of vertices Adjacency matrix For undirected graph, Degree matrix 3

4 Review: Graph (cont d) Graph Laplacian Combinatorial graph Laplacian: Symmetric normalized graph Laplacian: Random walk graph Laplacian: Eigendecomposition of L = U U Adjacency matrix If the graph is undirected, can be diagonalized by 4

5 Graph Fourier Transform Graph Fourier transform [Hammond+ 2011] Inner product of graph signal and eigenvector Matrix notation: Inverse GFT 5

6 Filtering in Spectral Domain Review: Filtering = Convolution Requires signal shifts Equivalent expression on DFT domain 6

7 Filtering in Spectral Domain (cont d) Convolution of graph signals: Defined on GFT domain : Projection matrix : Filter kernel 7

8 Filtering Example GFT IGFT GFT Spectrum LPF GFT Spectrum Attenuated Eigenvalue Eigenvalue 8

9 Spectral Domain Filtering Naive filtering Eigendecomposition is required High computational cost for large graphs Workaround 1: Design polynomial filters If a spectral domain filter is a polynomial of λ, we do not need GFT basis. Order-K polynomial in the spectral domain K- hop localized in the vertex domain 9

10 Polynomial Filters Ex. Graph lowpass filter Equivalent to: h(λ) Eigenvalue Product of the graph Laplacian and input signal: Eigendecomposition is not necessary. 10

11 Polynomial Approximation Workaround 2: Polynomial approximation of spectral response Minimax, least squares, Chebyshev etc. Chebyshev polynomial approximation [Hammond+ 2011, Shuman+ 2011] Widely used in GSP due to: Less errors in stopband Asymptotic property to minimax approx. 11

12 Polynomial Approximation (cont d) Ex. Meyer kernel Large order approximation: non-localized filter Response Ideal response 15-th order approx. 30-th order approx λ 12

13 Graph Wavelets/Filter Banks Analysis of graph signals w/ a set of graph filters Expansion of signals with basis/frame Decomposing signals into different graph frequency bands Subband-adaptive processing Compression, restoration, Analysis filters Downsampling Upsampling Synthesis filters Analysis transform Synthesis transform 13

14 Graph Wavelets/Filter Banks (cont d) Differences from classical SP Variable maximum frequency Classical SP: Maximum freq. = Graph SP: Maximum freq. = Downsampling Reducing signal length & vertices Aliasing (spectral folding phenomenon) Depending on the original and/or reduced-size graphs 14

15 What should we consider to design? Specifications Variation operator Redundancy Tolerable approximation error Trade-offs! Desired properties Perfect reconstruction Energy preservation property Fast computation Flexibility on eigenvalue distributions Ease of imposing constraints 15

16 Specs & Properties Specifications Variation operator Redundancy Tolerable approximation error Desired properties Perfect reconstruction Energy preservation property Fast computation Flexibility on eigenvalue distributions Ease of imposing constraints 16

17 Variation Operators Matrix used for graph Fourier basis Graph Laplacian Undirected graphs, always diagonalizable Typical graph Laplacians Combinatorial graph Laplacian Symmetric normalized graph Laplacian Random walk graph Laplacian Adjacency matrix (and its variations) Directed/undirected, not always diagonalizable Complex eigenvalues should be handled 17

18 Redundancy : # coefs. in the i-th subband Undecimated (R. = M) Pros Flexible filter design Simple structure Better performance for restoration Analysis filters Synthesis filters Cons Memory/storage requirement 18

19 Redundancy (cont d) Critically sampled (R. = 1) Pros Less memory requirement Better performance for compression Cons Constraints on graphs and/or filter characteristics Non-uniqueness of downsampling strategies Analysis filters Downsampling Upsampling Synthesis filters N N N 19

20 Redundancy (cont d) Oversampled (1 < R. < M) Pros Not-so-large memory requirements Good performance compromise of restoration and compression Cons Constraints on graphs and/or filter characteristics Non-uniqueness of downsampling strategies 20

21 Redundancy (cont d) Undersampled (R. < 1) Pros Much less memory requirement Cons Reconstruction error Analysis filters Downsampling Upsampling Synthesis filters 21

22 Approximation Error Polynomial filters Pros Low computational complexity Vertex localization Cons Approximation error Setting of polynomial degree Complex eigenvalues 22

23 Approximation Error (cont d) Non-polynomial filters Pros Ideal characteristics (no error) Cons Computational complexity! Globalized transform in vertex domain 23

24 Specs & Properties Specifications Variation operator Redundancy Tolerable approximation error Desired properties Perfect reconstruction Energy preservation property Fast computation Flexibility on eigenvalue distributions Ease of imposing constraints 24

25 Perfect Reconstruction Exact perfect reconstruction if subband coefs. are losslessly transmitted Undecimated case: Always PR as long as a family of vectors is a frame It is equivalent to Nonzero response for all graph frequencies 25

26 Perfect Reconstruction (cont d) Exact PR: Decimated case Case 1: Symmetric structure Synthesis part is a flipped version of the analysis part Possibility of PR depends on graph partitions and/or filters used 26

27 Perfect Reconstruction (cont d) Exact PR: Decimated case Case 2: Asymmetric structure The output is somehow interpolated from the filtered and sampled subband signals Easy to reconstruct (relatively) Subband-by-subband interpolation / Global interpolation Interpolation 27

28 Perfect Reconstruction (cont d) Near perfect reconstruction Condition ( depends on applications) Advantage compared to exact PR Flexibilities on filter design (for decimated graph filter banks) ex. Good stopband attenuation Would be better for compression w/ low bitrates 28

29 Energy Preservation Signal power in graph frequency domain Frame: Energy of the subband signals are upper and lower bounded by A and B Tight frame: A = B (Parseval frame: A = B = 1) In application: Easy to set thresholds for restoration or compression Synthesis transform = analysis transform 29

30 Fast Computation Eigendecomposition is costly Complexity ex.) ED for >10k vertices takes minutes Polynomial filters Complexity (p: Degree of the polynomial, E : # edges) Reducing samples before filtering Move downsampling before filtering Noble identities / polyphase structure 30

31 Flexibility on EV Distributions Eigenvalues are not equally spaced Frequency spacing of DFT: Examples of graph freq. spacing Comet graph (N = 32) Random regular graph (N = 100) 31

32 Flexibility on EV Distributions (cont d) Fixed set of passbands may cause a problem Only a few subbands keep most of the energy of the graph signal Uneven number of eigenvalues in each subbands Solution: Spectrum-adapted filters Warping the passbands according to the (estimated) eigenvalue distribution : Monotonically increasing warping function 32

33 Imposing Constraints Structurally satisfied constraints Avoiding DC leakage: Zeros at : Possible solutions Spectral factorization of polynomial filters Similar to design problems of wavelets in classical signal processing Customized polynomial approximation For non-polynomial filters 33

34 List of SGWTs & FBs Abbreviations Variation operator L (graph Laplacian) / A (Adjacency matrix) Redundancy UD (undecimated) / OS (oversampled) / CS (critically-sampled) / US (undersampled) Filter types P (polynomial) / NP (non-polynomial) 34

35 Cubic Spline aka SGWT [Hammond+ 2011] L UD NP: Not tight Wavelet function at scale t and vertex n Image courtesy of [Sakiyama+2016] 35

36 Tight Graph Wavelets Tight version of SGWT [Leonardi+ 2013] L UD NP: Tight Various specs from classical SP Meyer, Simoncelli, Image courtesy of [Sakiyama+2016] 36

37 Spectrum-Adapted Spectrum-adapted tight graph wavelets [Shuman+ 2015] L UD NP: Tight Based on Hann window Spectrum-adapted filter kernel Image courtesy of [Sakiyama+2016] 37

38 Frequency Conversion Converted FIR filter banks [Sakiyama+ 2016] L UD/OS/CS NP: Tight/not tight Impulse response (time domain) Discrete-time Fourier trans. Freq. response (freq. domain) Graph filter (Graph spectral domain) 1. Modulation 2. Changing of variables 38

39 Frequency Conversion (cont d) Design examples UD wavelets based on 9/7 DWT UD GFB based on LOT OS GFB based on OSLPPUFB CS wavelets based on 9/7 DWT Image courtesy of [Sakiyama+2016] 39

40 Frequency Conversion (cont d) Filter characteristics after polynomial approximation [Leonardi+ 2013] [Shuman+ 2015] DCT-based GFB Image courtesy of [Sakiyama+2016] 40 LOT-based GFB

41 2-ch Graph Wavelets First approach on CS spectral graph wavelets [Narang+ 2012, Narang+ 2013] L CS NP/P graphqmf: Orthogonal / NP graphbior: Biorthogonal/ P Symmetric normalized L. / bipartite graphs Spectral folding phenomenon Coloring-based downsampling 41

42 PR Polynomial Approx. PR-ensured polynomial approximations [Tay+ 2015, Tay+ 2017] L UD/OS/CS P Desired filter responses are approximated while ensuring PR: with (slight) sacrifices on filter shapes Imposing constraints on DC and highest graph freq. Original length 8 QMF Spectral Filters using Z 1 (µ) Image courtesy of [Tay+2015] Image courtesy of [Tay+2017]

43 M-ch Multirate GFBs Multirate GSP framework [Teke part papers] A CS P M-block cyclic graph (and its generalization) Eigenvectors have DFT-like property PR and aliasing cancellation For general graph: Find an invertible matrix which transforms to satisfy the condition ( ) Noble identities Image courtesy of [Teke+ 2017] 43

44 Subgraph-based GFBs Downsampling based on graph partition [Trembray+ 2016] L (partitioned subgraphs) CS P The original graph is partitioned by several subgraphs, then Haar-like filtering in each partition is performed PR is guaranteed Image courtesy of [Trembray+2016] 44

45 M-ch Critically Sampled GFBs Asymmetric structure w/ critically sampling [Jin+ 2017] L CS NP A subset of vertices is selected based on uniqueness set Subband-by-subband interpolation Images courtesy of [Jin+ 2017] 45

46 Bandlimiting filters Splitting signals into bandlimited ones [Chen+ 2015] A CS NP Sampling theory-based approach Bandlimiting operators, sampling operators and interpolators Images courtesy of [Chen+2015] 46

47 M-ch Oversampled GFBs Spectral factorization-based method [Tanaka+ 2014] L OS P Extension of graphbior to the oversampling system Symmetric normalized L. / bipartite graphs Image courtesy of [Tanaka+2014] 47

48 Undersampled GFBs Undersampled graph FB w/ row subset selection [Sakiyama+ 2016] L US NP Subset of rows in OS/CS GFB is extracted Asymmetry and global reconstruction Images courtesy of [Sakiyama+2016b] 48

49 Challenges Frequency analysis of signals on dynamic networks Like 3-D wavelet transform for spatiotemporal data Directionality of the data Faster computations Polyphase structure, fast graph Fourier transform Multiscale graph filter banks Partition of graphs, easier computations, symmetric structures 49

50 Challenges Cost functions to design Measure filter performances in some sense Easy to compute and optimize Adaptive graph wavelets Spectrum-adapted, signal-adapted Fine-tuning of initial filters Vertex/spectrum localization Tune for time-series graph signals (adaptive graph filter) 50

51 References [Hammond+ 2011] D.K.Hammond,P.Vandergheynst,andR.Gribonval, Wavelets on graphs via spectral graph theory, Appl. Comput. Harmonic Anal., vol. 30, no. 2, pp , [Shuman+ 2011] D. I. Shuman, P. Vandergheynst, and P. Frossard, Chebyshev polynomial approximation for distributed signal processing, in Proc. Int. Conf. Distrib. Comput. Sensor Syst. Workshops, 2011, pp [Leonardi+ 2013] N. Leonardi and D. VanDe Ville, Tight wavelet frames on multislice graphs, IEEE Trans. Signal Process., vol. 16, no. 13, pp , Jul [Shuman+ 2015] D. I. Shuman, C. Wiesmeyr, N. Holighaus, and P. Vandergheynst, Spectrum-adapted tight graph wavelet and vertex-frequency frames, IEEE Trans. Signal Process., vol. 63, no. 16, pp , Aug [Sakiyama+ 2016a] A. Sakiyama, K. Watanabe, and Y. Tanaka, Spectral graph wavelets and filter banks with low approximation error, IEEE Trans. Signal Inf. Process. Netw., vol. 2, no. 3, pp , [Narang+ 2012] S. K. Narang and A. Ortega, Perfect reconstruction two-channel wavelet filter banks for graph structured data, IEEE Trans. Signal Process., vol. 60, no. 6, pp , [Narang+ 2013] S. K. Narang and A. Ortega, Compact support biorthogonal wavelet filterbanks for arbitrary undirected graphs, IEEE Trans. Signal Process., vol. 61, no. 19, pp , [Tay+ 2015] D. B. H. Tay and Z. Lin, Design of near orthogonal graph filter banks, IEEE Signal Process. Lett., vol. 22, no. 6, pp , Jun [Tay+ 2017] D. B. H. Tay, Y. Tanaka, and A. Sakiyama, Almost tight spectral graph wavelets with polynomial filters, IEEE Journal of Selected Topics in Signal Processing, accepted. 51

52 References (cont d) [Teke+ 2017a] O. Teke and P. P. Vaidyanathan, Extending classical multirate signal processing theory to graphs-part I: Fundamentals, IEEE Trans. Signal Process., vol. 65, no. 2, pp , [Teke+ 2017b] O. Teke and P. P. Vaidyanathan, Extending classical multirate signal processing theory to graphs Part II: M-Channel filter banks, IEEE Trans. Signal Process., vol. 65, no. 2, pp , [Trembray+ 2016] N. Tremblay and P. Borgnat, Subgraph-based filterbanks for graph signals, IEEE Trans. Signal Process., vol. 64, no. 15, pp , [Jin+ 2017] Y. Jin and D. I. Shuman, An M-channel critically sampled filter bank for graph signals, in Proc. ICASSP 17, [Chen+ 2015] S. Chen, R. Varma, A. Sandryhaila, and J. Kovacevic, Discrete signal processing on graphs: Sampling theory, IEEE Trans. Signal Process., vol. 63, no. 24, pp , Dec [Tanaka+ 2014] Y. Tanaka and A. Sakiyama, M-channel oversampled graph filter banks, IEEE Trans. Signal Process., vol. 62, no. 14, pp , Jul [Sakiyama+ 2016b] A. Sakiyama and Y. Tanaka, Construction of undersampled graph filter banks via row subset selection, GlobalSIP 2016, Washington, D.C., Dec