Compressed Sensing of Block Sparse Signals with Known and Unknown Block Borders

In many practical applications, a large number of sensors may be deployed to collect data from an area of interest. However, it may be the case that only a small number of sensors will be active at any given time. In such cases, the data collected by the sensors is sparse. This sparsity enables the measured data to be estimated using a small number of observations, giving rise to the so-called compressed sensing paradigm. In this paradigm, sparsity is invoked to reconstruct the original data. In many practical applications of compressed sensing, the signal has a block sparse structure. In this structure, the non-zero entries appear in blocks, endowing the desired signal with an additional structure that can be exploited to reduce the number of samples necessary for reconstruction. In this work, we develop novel recovery algorithms for block sparse signals. Two cases are considered. In the first case, the block borders are known whereas in the second case, the block borders are unknown. To address the first problem, we propose two novel Bayesian algorithms which differ from existing ones in that in each iteration the optimal block covariances are obtained. However, unlike existing ones the algorithms proposed herein do not rely on prior assumptions. In the first algorithm, the decision as to whether a block is declared zero or non-zero is based on prescribed thresholds, whereas in the second algorithm, this decision is based on hypothesis testing, which ensures that the probability of erroneous detection of the non-zero blocks is minimized. To address the problem of recovering the block sparse signals with unknown borders, we introduce a new prior model which is characterized by elastic dependencies between neighboring signal elements. Using this model, we develop a novel Bayesian learning algorithm which iterates between estimating the dependencies between the signal elements and updating the Gaussian prior model. The effectiveness of the proposed Bayesian algorithm is demonstrated through experimental results. We will also show that the proposed algorithm can be implemented using the message passing techniques, thereby enabling it to be presented under a unified Bayesian framework.

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