Tracking dynamic sparse signals using Hierarchical Bayesian Kalman filters

In this work we are interested in the problem of reconstructing time-varying signals for which the support is assumed to be sparse. For a single time instance it is possible to reconstruct the original signal efficiently by employing a suitable algorithm for sparse signal recovery, given the sparsity level of the signal. In the case of time-varying sparse signals the sparsity level is not necessarily known a-priori. Furthermore conventional tracking by Kalman filtering fails to promote sparsity. Instead, a hierarchical Bayesian model is used in the tracking process which succeeds in modelling sparsity. One theorem is provided that extends previous work by providing some more general results. A second theorem gives the conditions under which all sparse signals are recovered exactly. It is demonstrated that the proposed method succeeds in recovering time-varying sparse signals with greater accuracy than the classic Kalman filter approach.