Moment Constraints in Convex Optimization for Segmentation and Tracking

Convex relaxation techniques have become a popular approach to shape optimization as they allow to compute solutions independent of initialization to a variety of problems. In this chapter, we will show that shape priors in terms of moment constraints can be imposed within the convex optimization framework, since they give rise to convex constraints. In particular, the lower-order moments correspond to the overall area, the centroid, and the variance or covariance of the shape and can be easily imposed in interactive segmentation methods. Respective constraints can be imposed as hard constraints or soft constraints. Quantitative segmentation studies on a variety of images demonstrate that the user can impose such constraints with a few mouse clicks, leading to substantial improvements of the resulting segmentation, and reducing the average segmentation error from 12% to 0.35%. GPU-based computation times of around 1 second allow for interactive segmentation.