R-letter of August 2013 - IEEE Communications Society

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IEEE COMSOC MMTC R-LetterExtending Signal Processing Techniques to Graph DomainA short review for “Perfect Reconstruction Two-Channel Wavelet Filter Banks for Graph Structured Data”Edited by Jun ZhouSunil K. Narang and Antonio Ortega, “Perfect Reconstruction Two-Channel Wavelet FilterBanks for Graph Structured Data”, IEEE Transactions on Signal Processing, Vol. 60, No. 6,pages 2786-2799, 2012.Graph theory has been successfully adopted inmany computer vision and pattern recognitionapplications. When dealing with large scale data,one of the problems that hinders the wideadoption of graphical models is the very highcomputational complexity caused by largenumber of nodes and vertices in graph. Toaddress this challenge, one would expect thatonly a few nodes in the graph be used to form acompacted representation of the original graph.Then data processing can be performed only on asmall neighborhood of each node. Some recentefforts in this direction have explored traditionalsignal processing techniques, such as wavelettransform, as possible solutions.The paper published by Narang and Ortega inIEEE TSP is a seminal work on graph samplingand design of critically sampled wavelet filterbanks on graphs. It not only provides acomprehensive review of the spatial/spectralrepresentation of graph signals and existing workon two-channel filter banks, but also proposesthe important characteristics of the samplingstrategy and filter banks for perfectreconstruction of bipartite graphs.The key idea behind this method is applying atwo-channel filter banks that decompose a graphinto high-pass and low-pass channels, eachcontaining only part of the nodes in the graphafter downsampling and following upsamplingoperations. When these two channels arecombined, they form a perfect reconstruction ofthe original graph representation. In order toachieve such distortion-free reconstruction, analiasing component, which is composed of filterbanks and downsampling functions, shall be setto zero. Therefore, the goal of this research is tofigure out what are the proper filter banks anddownsampling functions to meet the aboverequirement.To develop the downsampleing strategy, theauthors proposed that the decomposed high-passthe low-pass channels shall contain complementnode sets of the original graph. This leads to thebuilding of a bipartition of the graph nodes [1].Based on the graph spectral theory, this strategygenerates spectral coefficients at symmetricgraph frequencies around a central frequency,which is equivalent to the aliasing component ofthe reconstruction function.To design the filter banks, the authors pointedout that they shall meet three conditions, i.e.,aliasing cancellation, perfect reconstruction, andorthogonality. Therefore, a quadrature mirrorfilter bank method [2] (wavelet is one of suchmethod) was chosen and extended to bipartitegraph. This method allows a single basis spectralkernel be created, while all other kernels are builton top of the basis kernel.Whilst it is straightforward to adopt the waveletfilter banks on bipartite graph, the application ofthis framework to arbitrary graph requiresgenerating a series of bipartite subgraphs fromthe original graph. Then each subgraph can beprocessed independently with a cascadedtransform being implemented at the end. In thispaper, the authors proposed to use the biparticitymethod from Harary et al [3] for subgraphgeneration.Two experiments have been performed todemonstrate how the proposed method can beapplied to image processing (as an example ofregular graph) and traffic graph analysis (as anexample of irregular graph). These examplesshow that the two-channel wavelet filter banksand the sampling method form a practicalsolution for graph decomposition andreconstruction. It enables efficient graphcomputation, which has been expected by theresearch community. I believe this work willgenerate long-term impact to the development ofgraph theory because it provides an elegant wayhttp://committees.comsoc.org/mmc 8/22 Vol.4, No.4, August 2013
IEEE COMSOC MMTC R-Letterof applying signal processing techniques to solvestructured pattern recognition problems.References:[2] S. Narang and A. Ortega, “Downsamplinggraphs using spectral theory,” Proceedingsof the International Conference onAcoustics, Speech and Signal Processing,pages 4208-4211, 2011.[3] J. Johnston. “A filter family designed foruse in quadrature mirror filter banks,”Proceedings of the IEEE InternationalConference on Acoustics, Speech andSignal, pages 291-294, 1980.[4] F. Harary, D. Hsu, and Z.Miller, “Thebiparticity of a graph,” Journal of GraphTheory, vol. 1, no. 2, pp. 131–133, 1977.Jun Zhou received the B.S.degree in computer scienceand the B.E. degree ininternational business fromNanjing University ofScience and Technology,China, in 1996 and 1998,respectively. He receivedthe M.S. degree in computerscience from ConcordiaUniversity, Canada, in 2002,and the Ph.D. degree incomputing science from University of Alberta, Canada,in 2006.He joined the School of Information andCommunication Technology in Griffith University asa lecturer in June 2012. Prior to this appointment, hehad been a research fellow in the Australian NationalUniversity, and a researcher at NICTA. His researchinterests are in statistical pattern recognition,interactive computer vision, and their applications tohyperspectral imaging and environmental informatics.http://committees.comsoc.org/mmc 9/22 Vol.4, No.4, August 2013
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