Bilevel optimization of block compressive sensing with perceptually nonlocal similarity

Abstract

Dictionary learning (DL) based block compressive sensing (BCS) aims to obtain both good sparse representation and reconstructed image with high precision. Traditional methods always combines these two objectives together into one single-level optimization problem by Lagrangian multiplier or optimize one objective by fixing the other one as a constraint, which makes the problem much easier to solve. However, when independent measurement noise exists, the recovered sub-block and the sparse coefficients are no longer simply bridged by linear function but have a more complex relationship with each other. In addition, the major task in BCS focuses on optimizing the recovered sub-block. To accurately address the intrinsically mutual influences between the two tasks and stress the importance of major task, DL based BCS is formulated as a bi-level optimization problem in which the upper level is to approximate the reconstructed sub-block by minimizing the CS measurement discrepancy and the lower level is to optimize the sparse coefficients represented by locally learned dictionary by minimizing the sparsity of the image sub-block. In this bilevel problem, the perceptual nonlocal similarity (PNLS) is proposed as the constraint for the upper-level optimization, which can reduce the block artifact among the sub-blocks. In order to solve this problem, a combination of l1 and l2 norm minimization method is used. Experimental results demonstrate that the proposed bilevel optimization method is effective and achieves higher performance on numerical and visual results than some state-of-the-art single-level optimization methods in BCS.