Reconstruction of compressively sampled images using a nonlinear Bayesian prior

This paper presents a procedure for reconstruction of spatially localized images from compressively sampled measurements making use of Bayesian priors. The contribution of this paper is twofold: firstly, we analytically derive the expected value of wavelet domain signal structures conditional to a suitably defined noisy estimate; secondly, we exploit such conditional expectation within a nonlinear estimation stage that is added to an iterative reconstruction algorithm at a very low computational cost. We present numerical results focusing on spatially localized images and assessing the accuracy of the resulting algorithm, which definitely outperforms state-of-the-art competitors in very ill-posed conditions characterized by a low number of measurements. This contribution highlights the strong analogy between compressive sampling reconstruction and blind deconvolution, and paves the way to further work on joint design of image deconvolution/reconstruction from compressively sampled measurements.