A Linear Framework for Region-Based Image Segmentation and Inpainting Involving Curvature Penalization

Abstract

We present the first method to handle curvature regularity in region-based image segmentation and inpainting that is independent of initialization.To this end we start from a new formulation of length-based optimization schemes, based on surface continuation constraints, and discuss the connections to existing schemes. The formulation is based on a cell complex and considers basic regions and boundary elements. The corresponding optimization problem is cast as an integer linear program.We then show how the method can be extended to include curvature regularity, again cast as an integer linear program. Here, we are considering pairs of boundary elements to reflect curvature. Moreover, a constraint set is derived to ensure that the boundary variables indeed reflect the boundary of the regions described by the region variables.We show that by solving the linear programming relaxation one gets quite close to the global optimum, and that curvature regularity is indeed much better suited in the presence of long and thin objects compared to standard length regularity.

Convexity Shape Prior for Segmentation

“…. , x n ), see (3). In this case, it is easy to verify that Taylor expansion φ k of the potential φ around configuration x k is a linear function satisfying Prop.…”

“…These problems motivate active research on optimization of higher-order regularization energies, e.g. curvature [3,4,5,6], which can alleviate the shrinking bias and other issues. We propose a new higher-order regularization model: convexity shape constraint, see Fig.1.…”

Curvature Regularity for Multi-label Problems - Standard and Customized Linear Programming

“…To ensure the consistency between boundary and region variables, constraints are needed, and in [26] there are three sets. The first ensures that wherever the foreground region ends there is an appropriate boundary variable.…”

“…As shown in Figure 5 there is no clear winner: both methods can produce tighter lower bounds and better integral solutions than the respective other. [26]. In all cases the lower bounds and the energies of the thresholded solutions are given.…”

“…In a recent work, Schoenemann et al [25,26] proposed to solve region-based problems with curvature regularity as linear programs. This approach is independent of initialization and has the advantage over the boundary-based methods that region-terms can be included, self-intersections be excluded and that the computed regions need not be (singly-) connected.…”