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We consider a class of region-based energies for image segmentation and inpainting which combine region integrals with curvature regularity of the region boundary. To minimize such energies, we formulate an integer linear program which jointly estimates regions and their boundaries. Curvature regularity is imposed by respective costs on pairs of adjacent boundary segments.By solving the associated linear programming relaxation and thresholding the solution one obtains an approximate solution to the original integer problem. To our knowledge this is the first approach to impose curvature regularity in region-based formulations in a manner that is independent of initialization and allows to compute a bound on the optimal energy.In a variety of experiments on segmentation and inpainting, we demonstrate the advantages of higher-order regularity. Moreover, we demonstrate that for most experiments the optimality gap is smaller than 2% of the global optimum. For many instances we are even able to compute the global optimum.

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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.

Abstract. Shape optimization is a problem which arises in numerous computer vision problems such as image segmentation and multiview reconstruction. In this paper, we focus on a certain class of binary labeling problems which can be globally optimized both in a spatially discrete setting and in a spatially continuous setting. The main contribution of this paper is to present a quantitative comparison of the reconstruction accuracy and computation times which allows to assess some of the strengths and limitations of both approaches. We also present a novel method to approximate length regularity in a graph cut based framework: Instead of using pairwise terms we introduce higher order terms. These allow to represent a more accurate discretization of the L2-norm in the length term.

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.

While the majority of competitive image segmentation methods are based on energy minimization, only few allow to efficiently determine globally optimal solutions. A graphtheoretic algorithm for finding globally optimal segmentations is given by the Minimum Ratio Cycles, first applied to segmentation in [8]. In this paper we show that the class of image segmentation problems solvable by Minimum Ratio Cycles is significantly larger than previously considered. In particular, they allow for the introduction of higher-order regularity of the region boundary.The key idea is to introduce an extended graph representation, where each node of the graph represents an image pixel as well as the orientation of the incoming line segment. With each graph edge representing a pair of adjacent line segments, edge weights can depend on the curvature. This way arbitrary positive functions of curvature can be introduced into globally optimal segmentation by Minimum Ratio Cycles. In numerous experiments we demonstrate that compared to length-regularity the integration of curvatureregularity will drastically improve segmentation results.Moreover, we show an interesting relation to the Snakes functional: Minimum Ratio Cycles provide a way to find one of the few cases where the Snakes functional has a meaningful global minimum.

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