There have been proposed several compressed imaging reconstruction algorithms for natural and MR images. In essence, however, most of them aim at the good reconstruction of edges in the images. In this paper, a nonconvex compressed sampling approach is proposed for structure-preserving image reconstruction, through imposing sparseness regularization on strong edges and also oscillating textures in images. The proposed approach can yield high-quality reconstruction as images are sampled at sampling ratios far below the Nyquist rate, due to the exploitation of a kind of approximate ？ 0 seminorms. Numerous experiments are performed on the natural images and MR images. Compared with several existing algorithms, the proposed approach is more efficient and robust, not only yielding higher signal to noise ratios but also reconstructing images of better visual effects. 1. Introduction In the past several decades, image compression [1, 2] and superresolution [3, 4] have been the primary techniques to alleviate the storage/transmission burden in image acquisition. As for image compression, it is known that, however, the compression-and-then-decompression scheme is not economical [5]. Though superresolution is capable of economically reconstructing high-resolution images to subpixel precision from multiple low-resolution images of the similar view, subpixel shifts have to be estimated in advance. It is a pity that accurate motion estimation is not an easy job for superresolution, thus resulting in a possible compromise of image quality (e.g., spatial resolution, signal to noise ratio (SNR)). Recently, a novel sampling theory, called compressed sensing or compressive sampling (CS) [5–9], asserts that one can reconstruct signals from far fewer samples or measurements than traditional sampling methods use. The emergence of CS has offered a great opportunity to economically acquire signals or images even as the sampling ratio is significantly below the Nyquist rate. In fact, CS has become one of the hottest research topics in the field of signal processing. Though there have been many relevant results on encoding and decoding of sparse signals, our focus in this paper is mainly on the compressed sampling of natural images and its applications to magnetic resonance imaging (MRI) reconstruction. Recently, there have been proposed several algorithms for compressed imaging reconstruction (e.g., [6, 11–17]). In essence, each of them solves a minimization problem of single ？ 1 norm or total variation (TV) or their combination either directly or asymptotically, and the
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