Constrained LMMSE‐based object‐specific reconstruction in compressive sensing

In many applications like surveillance, it is essential to detect the presence of specific objects by seeing an image or video without being concerned about details of the scene. The reconstruction algorithms proposed in the compressive sensing literature try to iteratively reconstruct the full image. Hence, those are computationally expensive. If some prior knowledge of the object is available, then a closed‐form reconstruction algorithm can be formulated. The goal is to reconstruct the object of interest efficiently without being bothered about the quality of reconstruction of the scene. To address this situation, a constraint is formulated and incorporated into linear minimum mean square error (LMMSE) estimator to form a closed‐form solution. This compact solution is capable of meaningfully reconstructing the object relative to the scene. In the proposed method, an ill‐conditioned matrix inversion problem has been faced and overcome by regularization method. To boost the speed of the algorithm, a modified Euler method is proposed for finding the regularization parameter. To further speedup the reconstruction process, larger images are divided into several pieces and each piece is reconstructed separately using constrained LMMSE. For a given number of measurements, the simulation‐based results demonstrate acceptable quality of reconstruction with minimal computational effort.