A convex framework for image segmentation with moment constraints

Abstract

Convex relaxation techniques have become a popular approach to image segmentation as they allow to compute solutions independent of initialization to a variety of image segmentation problems. In this paper, 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 lowerorder moments correspond to the overall volume, 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 easily impose such constraints with a few mouse clicks, giving rise 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.

Moment Constraints in Convex Optimization for Segmentation and Tracking

“…This constraint is particularly meaningful if one additionally constrains the centroid to be µ, i.e. considers the intersection of the set (19) with a set of the form (14).…”

“…In the following, we will describe the moment constraints presented in [14]. We will successively constrain the moments of the segmentation and show how all of these constraints give rise to nested convex sets.…”

Global Binary Optimization on Graphs for Classification of High Dimensional Data

“…where (34) is the flow capacity constraint on edges between paiwise nodes, (35) and (36) are flow capacities on the source and sink edges, and (37) is the flow conservation constraint. The objective function (33) measures the total amount of flow on the graph.…”

Fast Trust Region for Segmentation

“…Our energy is defined as Figure 6 shows an example of liver segmentation with the shape prior constraint. The target shape moments as well as the foreground and background appearance models are computed from an input ellipse (top-left) provided by user as in [11]. We used moments of up to order d = 2 (including the center of mass and shape covariance but excluding the volume).…”

“…In the recent years there is a general trend in computer vision towards using complex non-linear energies with higher-order regional terms for the task of image segmentation, co-segmentation and stereo [10,7,14,2,1,11,8].…”