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6 25 DFT spectrum Frequency index 4 5

7 DFT spectrum Frequency index 4 5

9 .8 GFT spectrum Graph frequency (eigenvalue)

21 1 GFT spectrum Graph frequency (eigenvalue) DFT spectrum Frequency index

34 GFT Fold & add Warping spectrum

35 1.5 Downsampling by two, path graph (N = 1 to 5) 1 Spectrum Original GD1 GD2 GD3 Eigenvalues of the original graph Eigenvalues of the reduced-size graph λ

36 2 Downsampling by two, random regular graph (N = 1 to 5) Spectrum Original GD1 GD2 GD3 Eigenvalues of the original graph Eigenvalues of the downsampled graph λ

37 Analysis Synthesis

39 C. A. level: 1.5 C. A. level: C. A. level: C. A. level: C. A P. E. level: P. E. level: 1 P. E. level: 2 P. E GLP w/ vertex domain sampling C. A. C. A. level: C. A. level: C. A. level: C. A. level: P. E. P. E. level: P. E. level: P. E. level: GLP w/ spectral domain sampling Level 1 2 3

40 GLP w/ vertex domain sampling GLP w/ spectral domain sampling

44 GFT Spectrum Eigenvalue

45 GFT Spectrum 6 4 GFT Spectrum Eigenvalue Eigenvalue

46 GFT Spectrum 6 4 GFT Spectrum Eigenvalue Eigenvalue

49 2.5 Ideal Characteristic: SGWT 1.4 Ideal Characteristic: Tight Meyer H k (λ) 1 H k (λ) λ λ

50 2.5 (a) graph QMF 2.5 (b) graphbior H k (λ) 1 H k (λ) λ λ

52 λ λ

53 1.4 Ideal Characteristic: DCT GFB 1.4 Ideal Characteristic: LOT GFB H k (λ).6 H k (λ) λ λ

56 graph spectrum [.6,.8], [.3,.5], [3.2, 3.7], respectively, as in [3]. (c) Yale Coat of Arms. The signals of Figs. (c) and (d) were toolbox [13]. (d) Swiss Roll. 周波数領域グラフ周波数領域 TABLE V DENOISED RESULTS (AVERAGE OF TEN EXECUTIONS): SNR (DB) ノイズ除去 AMA et al.: SPECTRAL GRAPH WAVELETS AND FILTER BANKS WITH LOW APPROXIMATION ERROR W APPROXIMATION ERROR Original graph signals. (a) Minnesota Traffic Graph (Example 1). The signal was reproduced from the MATLAB code by Narang and Ortega [32]. nesota Traffic Graph (Example 2). The signal is obtained by dividing nodes into three clusters and calculating the signal so that each cluster contains the pectrum [.6,.8], [.3,.5], [3.2, 3.7], respectively, as in [3]. (c) Yale Coat of Arms. The signals of Figs. (c) and (d) were created by using SGWT [13]. (d) Swiss Roll. TABLE V DENOISED RESULTS (AVERAGE OF TEN EXECUTIONS): SNR (DB) he signal was reproduced from the MATLAB code by Narang and Ortega [32]. des into three clusters and calculating the signal so that each cluster contains the ale Coat of Arms. The signals of Figs. (c) and (d) were created by using SGWT LE V TEN EXECUTIONS): SNR (DB) OF Y. Tanaka (TUAT/JST PRESTO): Graph signal processing 53 transforms had a filter length of p = 1, fo of filter performance, and used edge-awar the Coins image [67]. Note that the CDF

57 Analysis bank Synthesis bank Analysis bank Synthesis bank

58 SNR [db] GraphBior Meyer-CSSGFB (normalized) Meyer-CSSGFB (combinatorial) Haar-CSSGFB (normalized) Haar-CSSGFB (combinatorial) Ideal-CSSGFB (normalized) Ideal-CSSGFB (combinatorial) Fraction of coeffs. Coeffs λ

Wavelet HSI / [1] JPEG2000 9/7Wavelet [2][6] 2:1 9/7Wavelet Wavelet 80 Wavelet i

17 Wavelet Image Enhancement by Wavelet Transform 1060326 2006 3 10 Wavelet HSI / [1] JPEG2000 9/7Wavelet [2][6] 2:1 9/7Wavelet Wavelet 80 Wavelet i Abstract Image Enhancement by Wavelet Transform Yuichi