An Efficient Convex Approach For Computing Minimal Partitions

Image segmentation,as a basic and important research topic in image technology,has been developed for decades.Image segmentation refers to the use of certain features of an image to divide the image into disjoint subregions,so that in different subregions,certain properties of the image are consistent.In recent years,image segmentation based on variational energy functional has been widely used.The basic idea is to establish a variational model for image segmentation in the sense of continuous space.Since variational energy functionals are often nonconvex,numerical methods are used to solve the convex approximate solution of the original problem,the solution of the convex approximation problem is the expected image segmentation result.In this paper,based on the shortcomings of the continuous Potts model of the multi-region image segmentation problem,the model is improved,and the quality of image segmentation is improved from the construction of the original variational model.In the process of solving for the nonconvex variational energy functional,the accuracy of the convex approximation problem of the variational model and the simplicity and efficiency of algorithm are actually two parts that are difficult to achieve at the same time.Therefore,this paper proposes an efficient convex relaxation method,the quality of image segmentation is well balanced with the computational cost.For a class of problems(lower spatial dimensions and fewer partitions)that are numerically easy to solve,the proposed convex relaxation method is optimal.Regarding the solution to the convex approximation problem,first of all,the discrete form of the continuous convex approximation problem is given by the idea of the finite difference method,and the P-PD(Preconditional Primal-Dual)algorithm is used to solve it.Although the solution of the convex relaxation method proposed in this paper is not always the global optimal solution of the original image segmentation problem,in many convex approximation problems with similar calculation costs,the convex relaxation method in this paper has obvious advantages and can estimate the approximate solution.This paper shows the superiority of the improved continuous Potts model(minimum partition problem)through numerical experiments,and shows the ideal convergence of the P-PD algorithm used by the convergence curve.