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• A computational binary characterization method for convex object(s). • A unified model for convex object(s) segmentation and convex hull computation. • Linearization technique employed to reduce the difficulty of the model solving. • A convergent proximal alternating direction method of multipliers for the linearization problem. • An interactive procedure to improve the segmentation accurac y in terms of shape distance and D ice coefficient. Convexity prior is one of the main cues for human vision and shape completion with important applications in image processing and computer vision. This paper provides a computable characterization method for shape convexity, and illustrates its applications in image segmentation and convex hull computation. We prove that the convexity of a region is equivalent to a series of quadratic inequality constraints on its indicator function. By incorporating this result, models are proposed for image segmentation with convexity prior and convex hull computation of clean and noisy data sets, respectively. Then, these models are summarized into a unified optimization problem on binary function(s) with the quadratic inequality constraints. Numerical method is proposed by linearizing the quadratic constraints. The linearization problem is solved by a proximal alternating direction method of multipliers, the convergence of which is guaranteed under some proper conditions. Numerical experiments demonstrate the efficiency and effectiveness of the proposed methods for image segmentation and convex hull computation. [ABSTRACT FROM AUTHOR]

• This paper propose a new image denoising model by coupling the rotation matrix and the weighted matrix into the TVp quasinorm regularization. • Since the proposed model is nonconvex and non-Lipschitz, we employe the ADMM to solve it with the help of the half-quadratic scheme. • The convergence of the proposed numerical methods is also discussed. • Numerical comparisons show that the proposed model leaded to considerable performance gains when testing on several denoising tasks. Image denoising problem still remains an active research field in the image processing. To improve the denoising quality, it is very important to describe the local structure of the image in the proposed model. This fact motivates us to introduce an adaptive weighted TVp regularization-based denoising model, where the rotation matrix and the weighted matrix depend on the local structure of the image. Specially, these two matrices can enhance the diffusion of the responding Euler-Lagrangian equation along with the tangential direction of the edge. This procedure offers more control over the regularization and then allows more denoising in smooth regions and less denoising when processing edge regions. In addition, since the proposed model is nonsmooth and non-Lipschitz, we employ the alternating direction method of multipliers (ADMM) to solve it with the help of using the half-quadratic scheme to solve the related ℓ 2 − ℓ p subproblem. In particular, we also provide the convergence analysis of the used numerical methods. Some numerical comparisons show that the proposed model leads to considerable performance gains when tested on several denoising tasks. [ABSTRACT FROM AUTHOR]