Bayesian Signal Recovery Under Measurement Matrix Uncertainty: Performance Analysis

Compressive sensing (CS) has gained a lot of attention in recent years due to its benefits in saving measurement time and storage cost in many applications including biomedical imaging, wireless communications, image reconstruction, remote sensing, and so on. The CS framework enables signal recovery from a small number of linear measurements with an acceptable fidelity taking advantage of signal sparsity in some potentially unknown domain. The core idea of different variants of CS methods is incorporating prior knowledge about the input signal (e.g., prior distribution or sparsity of signals) into the recovery algorithm to restrict the search space and enhance the signal recovery performance. However, the accuracy of signal reconstruction can be significantly compromised if the designed and implemented measurement matrices do not fully match. Often times, the measurement matrix mismatch is treated as an additional noise term in the recovery algorithm ignoring the fact that this mismatch is a learnable quantity which includes random but constant or slow-varying terms during the lifetime of the measurement system. In this paper, we consider this problem for a simple case of Gaussian prior with a sparsity-driven diagonal covariance matrix and find strict bounds on the deviation of the reconstructed signal from the optimal case of fully known measurement matrix based on the properties of the mismatch matrix. The obtained bounds are general, and hence can be used to assess the performance of learning algorithms designed for learning measurement matrix uncertainty and eliminating its effect from the signal recovery. We provide numerical results to illustrate this concept in real-world applications.