Perfect Reconstruction Two-Channel Wavelet Filter Banks for Graph Structured Data
Sunil K. Narang,Antonio Ortega +1 more
TL;DR: This work proposes the construction of two-channel wavelet filter banks for analyzing functions defined on the vertices of any arbitrary finite weighted undirected graph, and proposes quadrature mirror filters for bipartite graph which cancel aliasing and lead to perfect reconstruction.
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Abstract: In this work, we propose the construction of two-channel wavelet filter banks for analyzing functions defined on the vertices of any arbitrary finite weighted undirected graph. These graph based functions are referred to as graph-signals as we build a framework in which many concepts from the classical signal processing domain, such as Fourier decomposition, signal filtering and downsampling can be extended to graph domain. Especially, we observe a spectral folding phenomenon in bipartite graphs which occurs during downsampling of these graphs and produces aliasing in graph signals. This property of bipartite graphs, allows us to design critically sampled two-channel filter banks, and we propose quadrature mirror filters (referred to as graph-QMF) for bipartite graph which cancel aliasing and lead to perfect reconstruction. For arbitrary graphs we present a bipartite subgraph decomposition which produces an edge-disjoint collection of bipartite subgraphs. Graph-QMFs are then constructed on each bipartite subgraph leading to “multi-dimensional” separable wavelet filter banks on graphs. Our proposed filter banks are critically sampled and we state necessary and sufficient conditions for orthogonality, aliasing cancellation and perfect reconstruction. The filter banks are realized by Chebychev polynomial approximations.
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Fig. 5: Some of the graph-formulation of a 2D image lattice: (a) shows an 8-connected image graph G formed by connecting each pixel with its 8 nearest neighbors. The graph is 4-colorable, and the nodes of different shapes (squares,circles,triangles and diamonds) represent different colors. (b) shows the image-graph Gr by connecting each pixel with its rectangular (NWSE) neighbors only, (c) the image graph Gv with vertical links only (d) the image-graph Gh with horizontal links only. and (e) shows image-graph Gd with each pixel linked to its 4 diagonal neighbors The graphs shown in (b), (c), (d) and (e) are bipartite graphs, with the partitions represented as nodes with different colors and shapes (red-circles vs. blue-squares). 
Fig. 6: Discrete Fourier frequency magnitude responses of ideal lowpass filters on some bipartite image-graphs. Fig. (a) ideal lowpass filter response on NWSE bipartite subgraph Gr shown in 5b, Fig. (b) ideal lowpass filter response on diagonally connected bipartite subgraph Gd shown in 5c, Fig. (c) ideal lowpass filter on vertical-links only bipartite subgraph shown in 5d, Fig. (d) ideal lowpass filter on horizontal-links only bipartite subgraph shown in 5e. 
Fig. 4: Example of 2-dimensional separable downsampling on a graph: (a) original graph G, (b) the first bipartite graph B1 = (L1, H1,E1 ), containing all the links in G between sets L1 and H1. (c) the second bipartite graph B2 = (L2, H2,E2 ), containing all the links in G− B1, between sets L2 and H2 
Fig. 11: The Delaunay triangulation plots of output wavelet coefficients of the proposed filterbanks with parameter m = 6. The edge-color reflects the value of the coefficients at that point. (a) original graph signal (b) LL channel wavelet coefficients (c) LH channel wavelet coefficients (d) HH channel wavelet coefficients 
Fig. 9: (a) The Minnesota traffic graph and (b) the scatter-plot of a graph-signal to be analyzed. The colors of the nodes represent the sample values. 
Fig. 10: Bipartite decomposition of Minnesota graph into two bipartite subgraphs using Harary’s decomposition.
Citations
Reduced dimension policy iteration for wireless network control via multiscale analysis
Marco Levorato,Sunil K. Narang,Urbashi Mitra,Antonio Ortega +3 more
- 01 Dec 2012
TL;DR: The proposed methodology is based on the intrinsic multi-dimensional/multi-scale structure of the state space of the FSM and enables the analysis and minimization of cost-to-go functions, i.e., functions measuring the expected long-term cost associated with a control strategy, on coarser versions of the original graph.
24
A fast algorithm for vertex-frequency representations of signals on graphs
TL;DR: The results showed that graphs can be reconstructed from the vertex-frequency representations obtained with the proposed algorithms and showed that noise has no effect on the results of the algorithm for the fast windowed graph Fourier transform or on the graph S-transform.
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Distributed recursive least squares strategies for adaptive reconstruction of graph signals
Paolo Di Lorenzo,Elvin Isufi,Paolo Banelli,Sergio Barbarossa,Geert Leus +4 more
- 01 Aug 2017
TL;DR: A sparse sensing method is proposed that selects the sampling probability at each node in the graph in order to guarantee adaptive signal reconstruction and a target steady-state performance and a distributed RLS strategy is derived and is shown to be convergent to its centralized counterpart.
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Multiscale Local Polynomial Smoothing in a Lifted Pyramid for Non-Equispaced Data
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