Kalman Filtered Compressed Sensing

TL;DR

This paper proposes a real-time method combining compressed sensing and Kalman filtering to improve the reconstruction of time-varying sparse signals from limited measurements, leveraging temporal support dynamics.

Contribution

It introduces a novel approach that integrates Kalman filtering with compressed sensing to exploit temporal support changes in sparse signals.

Findings

Enhanced reconstruction accuracy over standard CS methods

Effective support estimation using Kalman filter innovations

Applicable to real-time processing of spatially sparse signals

Abstract

We consider the problem of reconstructing time sequences of spatially sparse signals (with unknown and time-varying sparsity patterns) from a limited number of linear "incoherent" measurements, in real-time. The signals are sparse in some transform domain referred to as the sparsity basis. For a single spatial signal, the solution is provided by Compressed Sensing (CS). The question that we address is, for a sequence of sparse signals, can we do better than CS, if (a) the sparsity pattern of the signal's transform coefficients' vector changes slowly over time, and (b) a simple prior model on the temporal dynamics of its current non-zero elements is available. The overall idea of our solution is to use CS to estimate the support set of the initial signal's transform vector. At future times, run a reduced order Kalman filter with the currently estimated support and estimate new additions to the support set by applying CS to the Kalman innovations or filtering error (whenever it is "large").