Block Compressed Sensing of Images Using Directional Transforms

Recent years have seen significant interest in the paradigm of compressed sensing (CS) which permits, under certain conditions, signals to be sampled at sub-Nyquist rates via linear projection onto a random basis while still enabling exact reconstruction of the original signal. As applied to 2D images, however, CS faces several challenges including a computationally expensive reconstruction process and huge memory required to store the random sampling operator. Recently, several fast algorithms have been developed for CS reconstruction, while the latter challenge was addressed by Gan using a block-based sampling operation as well as projection-based Landweber iterations to accomplish fast CS reconstruction while simultaneously imposing smoothing with the goal of improving the reconstructed-image quality by eliminating blocking artifacts. In this technique, smoothing is achieved by interleaving Wiener filtering with the Landweber iterations, a process facilitated by the relative simple implementation of the Landweber algorithm. In this work, we adopt Gan's basic framework of block-based CS sampling of images coupled with iterative projection-based reconstruction with smoothing. Our contribution lies in that we cast the reconstruction in the domain of recent transforms that feature a highly directional decomposition. These transforms---specifically, contourlets and complex-valued dual-tree wavelets---have shown promise to overcome deficiencies of widely-used wavelet transforms in several application areas. In their application to iterative projection-based CS recovery, we adapt bivariate shrinkage to their directional decomposition structure to provide sparsity-enforcing thresholding, while a Wiener-filter step encourages smoothness of the result. In experimental simulations, we find that the proposed CS reconstruction based on directional transforms outperforms equivalent reconstruction using common wavelet and cosine transforms. Additionally, the proposed technique usually matches or exceeds the quality of total-variation (TV) reconstruction, a popular approach to CS recovery for images whose gradient-based operation also promotes smoothing but runs several orders of magnitude slower than our proposed algorithm.