Nonconvex Lp Nuclear Norm based ADMM Framework for Compressed Sensing

Chen Zhao, Jian Zhang, Siwei Ma, Wen Gao

Abstract: Compressed Sensing (CS) has drawn quite an amount of attention as a joint sampling and compression methodology. Recent studies further show that image prior models play an important role in image CS recovery. By exploiting the non-local self-similarity of natural images and clustering similar patches, low-rank prior model is adopted in this paper. Different from traditional nuclear norm, we extend the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> (0 <; <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> <; 1) penalty function on singular values of a matrix to characterize low-rank prior model, and propose a new non-nonvex <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> nuclear norm prior model for image CS recovery, which is able to more accurately enforce image structural sparsity and self-similarity at the same time. The proposed optimization problem is efficiently solved within the alternative direction multiplier method (ADMM) framework. Experimental results demonstrate that the proposed <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> nuclear norm based ADMM framework for image CS recovery framework exhibits good convergence and achieves significant performance improvements over the current state-of-the-art methods.