Sparse graph signal reconstruction and image processing on circulant graphs

In this work, we present extensions of the framework of sampling and reconstructing signals with a finite rate of innovation (FRI) to the graph domain, by tackling the problem of _R"-sparse graph signal reconstruction on perturbed circulant graphs, simulating network clusters within a large network. Given a dimensionality-reduced approximation of the GFT of the original graph signal, we develop a reconstruction approach, whereby, we operate on each subgraph individually using a set of approximation and denoising schemes. In particular, we employ a variation of Prony\´s method with Cadzow\´s algorithm, and further iterative denoising, which can lead to perfect reconstruction. In addition, we extend the application of recently developed circulant graph-wavelet fllterbanks to images featuring patterns, in a novel model inspired by image segmentation, which involves a localized operation of the graph wavelet transform on individual segments of homogeneous intensity content, employing the nearest circulant matrix approximation scheme. The proposed method outperforms traditional methods in the classical domain in nonlinear approximation performance. We give preliminary results and discuss generalizations to arbitrary graphs.