A Binary Characterization Method For Shape Convexity And Applications

Convex prior is one of the main cues for human vision and shape completion,and has received a lot of attention in many fields such as computer vision and image processing.This thesis provides a computable mathematical characterization method for shape convexity,and illustrates its applications in image segmentation and convex hull computation,in which,the convex objects are characterized by a series of quadratic inequality constraints on their corresponding indicator functions.Based on this characterization method,we propose some models for image segmentation with convexity prior,and for the convex hull computation of a given set with or without noise.We show that,these models can be summarized to a unified optimization problem on binary function(s)consisting of the quadratic inequality constraint because of similar structures.To deal with the difficulty caused by the quadratic function and the inequality constraint,we skillfully employ a linearization technique.Subsequently,a linearized minimization problem is formed so that the efficient alternating direction method of multipliers can be used.Besides,an interactive strategy is employed to improve the segmentation accuracy gradually.To demonstrate the effectiveness of the proposed convexity characterization method and the efficiency of the proximal alternating direction method of multipliers,we conduct a series of experiments on the single object,multiple objects and convex ring object.The experiments show that,compared with the 1-0-1 method and level set method,the image segmentation model proposed in this thesis has significant advantages in terms of computational time,shape-distance and Dice coefficient.Additionally,it is also verified that the performance of our method in computing the convex hull of a given data is evidently obvious.It should be noted that,compared to the popular level set function method,our proposed method has significant advantages in terms of shape-distance.At last,we also demonstrated that,when mutually independent objects are sufficiently far apart,using one indicator function can do the task of image segmentation and convex hull computation.