A Combinatorial Solution for Model-Based Image Segmentation and Real-Time Tracking

Abstract

We propose a combinatorial solution to determine the optimal elastic matching of a deformable template to an image. The central idea is to cast the optimal matching of each template point to a corresponding image pixel as a problem of finding a minimum cost cyclic path in the three-dimensional product space spanned by the template and the input image. We introduce a cost functional associated with each cycle, which consists of three terms: a data fidelity term favoring strong intensity gradients, a shape consistency term favoring similarity of tangent angles of corresponding points, and an elastic penalty for stretching or shrinking. The functional is normalized with respect to the total length to avoid a bias toward shorter curves. Optimization is performed by Lawler's Minimum Ratio Cycle algorithm parallelized on state-of-the-art graphics cards. The algorithm provides the optimal segmentation and point correspondence between template and segmented curve in computation times that are essentially linear in the number of pixels. To the best of our knowledge, this is the only existing globally optimal algorithm for real-time tracking of deformable shapes.

Moment Constraints in Convex Optimization for Segmentation and Tracking

“…• Apart from a few exceptions such as [23] -computable solutions are only locally optimal. As a consequence, one typically needs appropriate initializations and solutions may be arbitrarily far from the globally optimal ones.…”

A convex framework for image segmentation with moment constraints

“…• Apart from a few exceptions such as [20] -computable solutions are only locally optimal. As a consequence, one typically needs appropriate initializations and solutions may be arbitrarily far from the globally minimal ones.…”

Image Segmentation with Shape Priors: Explicit Versus Implicit Representations

“…[84,74,1,87,89] hybrid repres. & LP relaxation [90] Implicit level set methods [39,72], convex relaxation [11,31] graph cut methods [49,6] Figure 1: Shapes can be represented explicitly or implicitly, in a spatially continuous or a spatially discrete setting.…”

“…Instead, the extension of explicit shape representations to higher dimensions is by no means straight forward: The notion of arc-length parameterization of curves does not extend to surfaces. Moreover, the discrete polynomial-time shortest-path algorithms [1,85,89] which allow to optimally identify pairwise correspondence of points on either shape do not directly extend to minimal-surface algorithms.…”

“…Instead, for spatially continuous parametric curves, modeling the transition from a single closed curve to a multiply connected object boundary requires sophisticated splitting and merging techniques [61,65,60,38]. Similarly, discrete polynomial-time algorithms are typically constrained to finding open [1,23,20] or closed curves [86,89].…”