Moment Constraints in Convex Optimization for Segmentation and Tracking

Abstract

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.

Chan–Vese Reformulation for Selective Image Segmentation

“…The minimisation problem is given asThe model consists of weighted TV regularisation with a geodesic distance constraint as in [35]. However, alternative constraints are possible, such as Euclidean [39], or moments [24]. It is important to note that we have defined the model in a similar framework to the related approaches discussed previously.…”

“…A common approach in selective segmentation is to discriminate between objects of a similar intensity [34, 35, 39]. However, the fitting terms in previous formulations [24, 34, 35, 39] aren’t applicable in many cases as there are contradictions in the formulation in this context. We will address this in detail in the following section.…”

Entropy Minimization for Convex Relaxation Approaches

“…We consider a fixed volume constraint on the solution of the segmentation problem. Volume constraints have been used with convex relaxation methods for image segmentation [22,14], imagebased modeling [27,20] and they have also been generalized to higher order moment constraints [8]. In [14] they also addressed the problem of the non-tight relaxation, but their suggested algorithm is less general as user-provided seed points are required.…”

“…The total variation weight is defined as g = exp(−|∇f |). Note that one easily adapts the approach to bound the volume with inequality constraints [8,14], but the problems of the relaxation can be better demonstrated with an equality constraint. Because of the volume constraint there is no free choice of thresholds to obtain a binary solution and the thresholding theorem [17], which directly relates solutions of the relaxed problem to binary one, does not apply anymore.…”

“…For more general multi-label problems the computed solutions are often far from binary and rounding may drastically increase the energy and the corresponding a-posteriori optimality bound (the energetic difference between rounded and relaxed solution). Moreover, a number of works incorporate additional constraints on the solution in order to constraint the volume/area [22,27], or higher order moments [8] of the resulting segmentation, or to enforce size proportions of respective segments [15]. While these constraints are meaningful for the binary problem (1.2), they often change their physical meaning in the relaxed setting (1.3).…”