Abstract：This paper presents an efficient algorithm to design M-channel oversampled graph filter banks with better overall performance. In the new algorithm, a two-step scheme is exploited to tackle the design task. Firstly, for controlling the spectral selectivity, the analysis filter is designed by solving a constraint optimization problem that minimizes the passband ripple and stopband energy subject to 3 dB constraint; secondly, by taking the Perfect Reconstruction (PR) condition into account, the design problem of synthesis filters is formulated into an optimization problem that minimizes the stopband energy subject to PR constraint. Both the optimization problems are Semi-Definite Programming (SDP), which can be efficiently solved. Since the proposed method fully considerate the spectral characteristic and PR condition, M-channel biorthogonal oversampled graph filter banks with better performance can be obtained. Numerical examples and comparison show that compared with the existing methods, the proposed method can lead to graph filter banks with smaller reconstruction error.
CROVELLA M and KOLACZYK E. Graph wavelets for spatial traffic analysis[C]. Joint Conference of the IEEE Computer and Communications, San Francisco, CA, USA, 2003: 1848-1857. doi: 10.1109/INFCOM.2003.1209207.
[2]
NARANG S K, CHAO Y H, and ORTEGA A. Graph-wavelet filterbanks for edge-aware image processing[C]. Statistical Signal Processing Workshop, IEEE, Ann Arbor, MI, 2012: 141-144. doi: 10.1109/SSP.2012.6319643.
[3]
GIRVAN M and NEWMAN M E. Community structure in social and biological networks[J]. Proceedings of the National Academy of Sciences of the United States of America, 2002, 99(12): 7821-7826. doi: 10.1073/pnas.122653799.
[4]
SHEN G and ORTEGA A. Optimized distributed 2D transforms for irregularly sampled sensor network grids using wavelet lifting[C]. IEEE International Conference on Acoustics, Speech and Signal Processing, Las Vegas, NV, 2008: 2513-2516. doi: 10.1109/ICASSP.2008.4518159.
[5]
WANG W and RAMCHANDRAN K. Random multiresolution representations for arbitrary sensor network graphs[C]. IEEE International Conference on Acoustics, Speech and Signal Processing, Toulouse, 2006: IV161-IV164. doi: 10.1109/ICASSP.2006.1660930.
[6]
TAY D B H, TANAKA Y, and SAKIYAMA A. Near orthogonal oversampled graph filter banks[J]. IEEE Signal Processing Letters, 2016, 23(2): 277-281. doi: 10.1109/LSP. 2016.2514490.
[7]
TANAKA Y and SAKIYAMA A. M-channel oversampled graph filter banks[J]. IEEE Transactions on Signal Processing, 2014, 62(14): 3578-3590. doi: 10.1109/TSP.2014.2328983.
[8]
SAKIYAMA A and TANAKA Y. Oversampled graph Laplacian matrix for graph filter banks[J]. IEEE Transactions on Signal Processing, 2014, 62(24): 6425-6437. doi: 10.1109/ TSP.2014.2365761.
[9]
TAY D B H and LIN Z. Design of near orthogonal graph filter banks[J]. IEEE Signal Processing Letters, 2015, 22(6): 701-704. doi: 10.1109/LSP.2014.2368128.
[10]
NARANG S K and ORTEGA A. Perfect reconstruction two-channel wavelet filter banks for graph structured data[J]. IEEE Transactions on Signal Processing, 2012, 60(6): 2786-2799. doi: 10.1109/TSP.2012.2188718.
[11]
NARANG S K and ORTEGA A. Compact support biorthogonal wavelet filterbanks for arbitrary undirected graphs[J]. IEEE Transactions on Signal Processing, 2013, 61(19): 4673-4685. doi: 10.1109/TSP.2013.2273197.
[12]
JIANG J Z, ZHOU F, and SHUI P L. Optimization design of two-channel biorthogonal graph filter banks[J]. Circuits, Systems, and Signal Processing, 2016, 35(2): 685-692. doi: 10.1007/s00034-015-0073-x.
[13]
SAKIYAMA A and TANAKA Y. Oversampled graph Laplacian matrix for graph signals[C]. IEEE, Signal Processing Conference, Lisbon, Portugal, 2014: 2225-2229.
[14]
SAKIYAMA A and TANAKA Y. Edge-aware image graph expansion methods for oversampled graph Laplacian matrix[C]. IEEE International Conference on Image Processing, Paris, France, 2014: 2958-2962. doi: 10.1109/ ICIP.2014.7025598.
[15]
CHEN S, SANDRYHAILA A, MOURA J M F, et al. Signal denoising on graphs via graph filtering[C]. IEEE, Global Conference on Signal and Information Processing (GlobalSIP). Atlanta, GA, 2015: 872-876. doi: 10.1109/ GlobalSIP.2014. 7032244.
[16]
SHI X, FENG H, ZHAI M, et al. Infinite impulse response graph filters in wireless sensor networks[J]. IEEE Signal Processing Letters, 2015, 22(8): 1113-1117. doi: 10.1109/LSP. 2014.2387204.
