A Level Set Representation Method for $N$-Dimensional Convex Shape and Applications (Communications in Mathematical Research, 2021)
In this work, we present a new method for convex shape representation, which is regardless of the dimension of the concerned objects, usinglevel-set approaches. To the best of our knowledge, the proposed prior is thefirst one which can work for high dimensional objects. Convexity prior is veryuseful for object completion in computer vision. It is a very challenging taskto represent high dimensional convex objects. In this paper, we first prove thatthe convexity of the considered object is equivalent to the convexity of the associated signed distance function. Then, the second order condition of convexfunctions is used to characterize the shape convexity equivalently. We applythis new method to two applications: object segmentation with convexity priorand convex hull problem (especially with outliers). For both applications, theinvolved problems can be written as a general optimization problem with threeconstraints. An algorithm based on the alternating direction method of multipliers is presented for the optimization problem. Numerical experiments areconducted to verify the effectiveness of the proposed representation methodand algorithm.
Keywords: Convex shape prior; level-set method; image segmentation; convex hull; ADMM.