A level set representation method for N-dimensional convex shape and applications
This work addresses a domain-specific problem in computer vision for object completion, offering an incremental improvement in convex shape representation.
The paper tackles the challenge of representing high-dimensional convex shapes efficiently by proving that convexity is equivalent to the convexity of the signed distance function, using second-order conditions for characterization. It applies this method to object segmentation and convex hull problems, with numerical experiments verifying effectiveness and efficiency.
In this work, we present a new efficient method for convex shape representation, which is regardless of the dimension of the concerned objects, using level-set approaches. Convexity prior is very useful for object completion in computer vision. It is a very challenging task to design an efficient method for high dimensional convex objects representation. In this paper, we 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. Efficient algorithm based on alternating direction method of multipliers is presented for the optimization problem. Numerical experiments are conducted to verify the effectiveness and efficiency of the proposed representation method and algorithm.