Sampling and Reconstruction of Sparse Signals on Circulant Graphs - An Introduction to Graph-FRI

With the objective of employing graphs toward a more generalized theory of signal processing, we present a novel sampling framework for (wavelet-)sparse signals defined on circulant graphs which extends basic properties of Finite Rate of Innovation (FRI) theory to the graph domain, and can be applied to arbitrary graphs via suitable approximation schemes. At its core, the introduced Graph-FRI-framework states that any K-sparse signal on the vertices of a circulant graph can be perfectly reconstructed from its dimensionality-reduced representation in the graph spectral domain, the Graph Fourier Transform (GFT), of minimum size 2K. By leveraging the recently developed theory of e-splines and e-spline wavelets on graphs, one can decompose this graph spectral transformation into the multiresolution low-pass filtering operation with a graph e-spline filter, and subsequent transformation to the spectral graph domain; this allows to infer a distinct sampling pattern, and, ultimately, the structure of an associated coarsened graph, which preserves essential properties of the original, including circularity and, where applicable, the graph generating set.Comment: To appear in Appl. Comput. Harmon. Anal. (2017

From spline wavelet to sampling theory on circulant graphs and beyond– conceiving sparsity in graph signal processing

Graph Signal Processing (GSP), as the field concerned with the extension of classical signal processing concepts to the graph domain, is still at the beginning on the path toward providing a generalized theory of signal processing. As such, this thesis aspires to conceive the theory of sparse representations on graphs by traversing the cornerstones of wavelet and sampling theory on graphs. Beginning with the novel topic of graph spline wavelet theory, we introduce families of spline and e-spline wavelets, and associated filterbanks on circulant graphs, which lever- age an inherent vanishing moment property of circulant graph Laplacian matrices (and their parameterized generalizations), for the reproduction and annihilation of (exponen- tial) polynomial signals. Further, these families are shown to provide a stepping stone to generalized graph wavelet designs with adaptive (annihilation) properties. Circulant graphs, which serve as building blocks, facilitate intuitively equivalent signal processing concepts and operations, such that insights can be leveraged for and extended to more complex scenarios, including arbitrary undirected graphs, time-varying graphs, as well as associated signals with space- and time-variant properties, all the while retaining the focus on inducing sparse representations. Further, we shift from sparsity-inducing to sparsity-leveraging theory and present a novel sampling and graph coarsening framework for (wavelet-)sparse graph signals, inspired by Finite Rate of Innovation (FRI) theory and directly building upon (graph) spline wavelet theory. At its core, the introduced Graph-FRI-framework states that any K-sparse signal residing on the vertices of a circulant graph can be sampled and perfectly reconstructed from its dimensionality-reduced graph spectral representation of minimum size 2K, while the structure of an associated coarsened graph is simultaneously inferred. Extensions to arbitrary graphs can be enforced via suitable approximation schemes. Eventually, gained insights are unified in a graph-based image approximation framework which further leverages graph partitioning and re-labelling techniques for a maximally sparse graph wavelet representation.Open Acces

Perfect Reconstruction Two-Channel Filter Banks on Arbitrary Graphs

This paper extends the existing theory of perfect reconstruction two-channel filter banks from bipartite graphs to non-bipartite graphs. By generalizing the concept of downsampling/upsampling we establish the frame of two-channel filter bank on arbitrary connected, undirected and weighted graphs. Then the equations for perfect reconstruction of the filter banks are presented and solved under proper conditions. Algorithms for designing orthogonal and biorthogonal banks are given and two typical orthogonal two-channel filter banks are calculated. The locality and approximation properties of such filter banks are discussed theoretically and experimentally.Comment: 33 pages,11 figures. This manuscript has been submitted to ScienceDirect Applied and Computational Harmonic Analysis (ACHA) on Jan 27,202

Graph Signal Processing: Overview, Challenges and Applications

Research in Graph Signal Processing (GSP) aims to develop tools for processing data defined on irregular graph domains. In this paper we first provide an overview of core ideas in GSP and their connection to conventional digital signal processing. We then summarize recent developments in developing basic GSP tools, including methods for sampling, filtering or graph learning. Next, we review progress in several application areas using GSP, including processing and analysis of sensor network data, biological data, and applications to image processing and machine learning. We finish by providing a brief historical perspective to highlight how concepts recently developed in GSP build on top of prior research in other areas.Comment: To appear, Proceedings of the IEE

Infrared and Visible Image Fusion Based on Oversampled Graph Filter Banks

The infrared image (RI) and visible image (VI) fusion method merges complementary information from the infrared and visible imaging sensors to provide an effective way for understanding the scene. The graph filter bank-based graph wavelet transform possesses the advantages of the classic wavelet filter bank and graph representation of a signal. Therefore, we propose an RI and VI fusion method based on oversampled graph filter banks. Specifically, we consider the source images as signals on the regular graph and decompose them into the multiscale representations with M-channel oversampled graph filter banks. Then, the fusion rule for the low-frequency subband is constructed using the modified local coefficient of variation and the bilateral filter. The fusion maps of detail subbands are formed using the standard deviation-based local properties. Finally, the fusion image is obtained by applying the inverse transform on the fusion subband coefficients. The experimental results on benchmark images show the potential of the proposed method in the image fusion applications

Spline-Like Wavelet Filterbanks with Perfect Reconstruction on Arbitrary Graphs

In this work, we propose a class of spline-like wavelet filterbanks for graph signals. These filterbanks possess the properties of critical sampling and perfect reconstruction. Besides, the analysis filters are localized in the graph domain because they are polynomials of the normalized adjacency matrix of the graph. We generalize the spline-like filters in the literature so that they have the ability to annihilate signals of some specified frequencies. Optimization problems are posed for the analysis filters to approximate desired responses. We conduct some experiments to demonstrate the good locality of the proposed filters and the good performance of the filterbank in the denoising task.Comment: 11 pages,12 figures. This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessibl