Abstract. The nonconvex optimization method has attracted increasing attention due to its excellent ability of promoting sparsity in signal processing, image restoration, and machine learning. In this paper, we consider a new minimization method [Formula: see text] [Formula: see text] and its applications in signal recovery and image reconstruction because [Formula: see text] minimization provides an effective way to solve the [Formula: see text]-ratio sparsity minimization model. Our main contributions are to establish a convex hull decomposition for [Formula: see text] and investigate RIP-based conditions for stable signal recovery and image reconstruction by [Formula: see text] minimization. For one-dimensional signal recovery, our derived RIP condition extends existing results. For two-dimensional image recovery under [Formula: see text] minimization of image gradients, we provide the error estimate of the resulting optimal solutions in terms of sparsity and noise level, which is missing in the literature. Numerical results of the limited angle problem in computed tomography imaging and image deblurring are presented to validate the efficiency and superiority of the proposed minimization method among the state-of-art image recovery methods.

Transformed $L_1$ (TL1) regularization has been shown to have comparable signal recovery capability with $L_1-L_2$ regularization and $L_1/L_2$ regularization, regardless of whether the measurement matrix satisfies the restricted isometry property (RIP). In the spirit of the TL1 method, we introduce a transformed total variation (TTV) minimization model to investigate robust image recovery from a certain number of noisy measurements by the proposed TTV minimization model in this paper. An optimal error bound, up to a logarithmic factor, of robust image recovery from compressed measurements via the TTV minimization model is established, and the RIP based condition is improved compared with total variation (TV) minimization. Numerical results of image reconstruction demonstrate our theoretical results and illustrate the efficiency of the TTV minimization model among state-of-the-art methods. Empirically, the error bound between the reconstructed image and the original image is shown to be better than that produced by TV minimization.

Many objects in real world have convex shapes. It is a difficult task to have representations for convex shapes with good and fast numerical solutions. This paper proposes a method to incorporate convex shape prior for multi-object segmentation using level set method. The relationship between the convexity of the segmented objects and the signed distance function corresponding to their union is analyzed theoretically. This result is combined with Gaussian mixture method for the multiple objects segmentation with convexity shape prior. Alternating direction method of multiplier (ADMM) is adopted to solve the proposed model. Special boundary conditions are also imposed to obtain efficient algorithms for 4th order partial differential equations in one step of ADMM algorithm. In addition, our method only needs one level set function regardless of the number of objects. So the increase in the number of objects does not result in the increase of model and algorithm complexity. Various numerical experiments are illustrated to show the performance and advantages of the proposed method.