Proximal iterative hard thresholding methods for wavelet frame based image restoration

The iterative thresholding algorithms started in [1] (both soft and hard) and in [2, 3, 4] (soft) for wavelet based linear inverse problems restoration with sparsity constraint. The analysis of iterate soft thresholding algorithms has been well studied under the framework of foward-backward splitting method [5, 6] and inspired many works for different applications and related minimization problems. However, iterative hard thresholding methods are less understood due to its non-convexity, although there are some studies related to sparse signal recovery [7, 8] and image restorations [9, 10, 11]. In this paper, we consider a proximal iterative hard thresholding algorithms for `0-norm regularized wavelet frame balanced approach for image restoration, based on KurdykaLojasiewicz property recently studied in [12, 13, 14]. In particular, we study the convergence of two algorithms, namely proximal iterative hard thresholding (PIHT) algorithm and extrapolated proximal iterative hard thresholding (EPIHT) algorithm for solving this class of problems. We first demonstrate that, given an initial point, the sequence generated by PIHT will converge to a local minimizer of the objective function and the sequential error rate is at o(1/k). Then, we show the convergence of EPIHT by proving that the sequence generated by this algorithm is bounded, and any accumulation point of the sequence is a local minimizer of the objective function. Furthermore, we conduct numerical experiments on compressive sensing sparse signal reconstruction and wavelet frame based image restoration, such as CT reconstruction, image deblurring and parallel MRI image reconstruction, to demonstrate the improvement of `0-norm based regularization models as well as the effectiveness of the proposed algorithms compared some prevailing `1-norm based models and algorithms. We also show in some numerical experiments that the iteration complexity of the proposed EPIHT is lower than that of PIHT.