Signal inpainting on graphs via total variation minimization

Abstract

We propose a novel recovery algorithm for signals with complex, irregular structure that is commonly represented by graphs. Our approach is a generalization of the signal inpainting technique from classical signal processing. We formulate corresponding minimization problems and demonstrate that in many cases they have closed-form solutions. We discuss a relation of the proposed approach to regression, provide an upper bound on the error for our algorithm and compare the proposed technique with other existing algorithms on realworld datasets.

Signal denoising on graphs via graph filtering

“…However, the inversion of an N ×N matrix in (10) requires O(N 3 ) arithmetic operations, and for large values of N this operation is prohibitively expensive and numerically unstable. Furthermore, in many real-world applications the graph shift matrix A is sparse and has only a few nonzero entries, its inverse, as well as the inverse matrix in (10), are dense matrices that require O(N 2 ) arithmetic operations, again an expensive operation for large values of N . To avoid these issues, in the following section we derive an alternative solution that implements or approximates (10) with a graph filter (2).…”

“…This is a theoretical framework that generalizes fundamental concepts of the classical signal processing from regular domains, such as lines and rectangular lattices, to general graphs. Signal processing on graphs has found multiple applications, including approximation [5], sampling [6], [7], classification [8], [9], inpainting [10] and clustering [11] of signals on graphs.…”

Distributed algorithm for graph signal inpainting

“…For more details on graph signal inpainting, see [7,8,9]. We consider a corrupted graph signal measurement…”

“…Since the quadratic form of graph total variation is a regularization term, (5) is called the regularization approach. The closed-form solution is as follows [7]:…”

“…The approach via minimizing graph total variation provide a closed-form solution, which involves matrix inversion [7,8]. To solve this, we propose a distributed and decentralized algortithm for graph signal inpainting that can be computed locally in the vertex domain without transmitting signal coefficients to a computing center.…”