In this work, we present a new method for convex shape representation, which is regardless of the dimension of the concerned objects, using
level-set approaches. To the best of our knowledge, the proposed prior is the
first one which can work for high dimensional objects. Convexity prior is very
useful for object completion in computer vision. It is a very challenging task
to represent high dimensional convex objects. In this paper, we first prove that
the convexity of the considered object is equivalent to the convexity of the associated signed distance function. Then, the second order condition of convex
functions is used to characterize the shape convexity equivalently. We apply
this new method to two applications: object segmentation with convexity prior
and convex hull problem (especially with outliers). For both applications, the
involved problems can be written as a general optimization problem with three
constraints. An algorithm based on the alternating direction method of multipliers is presented for the optimization problem. Numerical experiments are
conducted to verify the effectiveness of the proposed representation method
and algorithm.
