Surface fairing, generating free-form surfaces satisfying aesthetic requirements, is important for many computer graphics and geometric modeling applications. A common approach for fair surface design consists of minimization of fairness measures penalizing large curvature values and curvature oscillations. The paper develops a numerical approach for fair surface modeling via curvature-driven evolutions of triangle meshes. Consider a smooth surface each point of which moves in the normal direction with speed equal to a function of curvature and curvature derivatives. Chosen the speed function properly, the evolving surface converges to a desired shape minimizing a given fairness measure. Smooth surface evolutions are approximated by evolutions of triangle meshes. A tangent speed component is used to improve the quality of the evolving mesh and to increase computational stability. Contributions of the paper include also an improved method for estimating the mean curvature.

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. 1.

Two challenges in computer vision are to accommodate noisy data and missing data. Many problems in computer vision, such as segmentation, filtering, stereo, reconstruction, inpainting and optical flow seek solutions that match the data while satisfying an additional regularization, such as total variation or boundary length. A regularization which has received less attention is to minimize the curvature of the solution. One reason why this regularization has received less attention is due to the difficulty in finding an optimal solution to this image model, since many existing methods are complicated, slow and/or provide a suboptimal solution. Following the recent progress of Schoenemann et al. [28], we provide a simple formulation of curvature regularization which admits a fast optimization which gives globally optimal solutions in practice. We demonstrate the effectiveness of this method by applying this curvature regularization to image segmentation. 1.

Noname manuscript No. (will be inserted by the editor) A linear framework for region-based image segmentation

Abstract. Length and area regularization are commonplace for inverse problems today. It has however turned out to be much more difficult to incorporate a curvature prior. In this paper we propose several improvements to a recently proposed framework based on global optimization. We identify and solve an issue with extraneous arcs in the original formulation by introducing region consistency constraints. The mesh geometry is analyzed both from a theoretical and experimental viewpoint and hexagonal meshes are shown to be superior. We demonstrate that adaptively generated meshes significantly improve the performance. Our final contribution is that we generalize the framework to handle mean curvature regularization for 3D surface completion and segmentation. 1

The bending energy of a thin, naturally straight, homogeneous and isotropic elastic rod of length L is given by ∫ L F (γ) = |κ(s) | 2 ds, 0 where γ: [0, L] → Rm is the arclength parametrisation and κ = γ ′′(s) the curvature vector. Consider the following boundary value problem: Given points P, Q ∈ Rm and unit vectors v, w ∈ Sm−1 find the shapes of static elastic curves with clamped ends and fixed length. Defining the space C = γ ∈ L 2 ([0, L]; R m) ∣ γ ′ ∈ L2 ([0, L]; Sm−1), γ(0) = P, γ(L) = Q, γ ′ ′ ∈ L2 ([0, L]; Rm), γ ′(0) = v, γ ′(L) = w this can be reformulated to find the minimizers of F: C → R., A widely used discrete bending energy for a polygonal line p = (p0, p1, · · · , pn) with pi ∈ Rm is given by n−1 ( ) 2 ϕi Fn(p) = ℓi, i=1 where ϕi is the turning angle and ℓi is given by ℓi = 1 2 (|pi+1 − pi | + |pi − pi−1|). We restrict ourselves to evenly segmented polygons, i. e. |pi − pi−1 | = L n for all i = 1,..., n. It is straightforward to formulate a discrete analogon of the boundary value problem above: Find the minimizers of Fn: Cn → R with the discrete ansatz space Cn = (p0,..., pn) ∈ (R m) n ∣ |pi − pi−1 | = L n, p0 = P, pn = Q, p1 − p0 = L n v, pn − pn−1 = L n w There has been an attempt by Bruckstein et al. [1] to relate argmin(Fn) and argmin(F) via techniques from the theory of epi-convergence. However, epiconvergence of Fn to F only guarantees that some minimizers of F can be approximated by those of Fn. We are able to improve this result in various ways: • The metric on configuration space is strenghtened from Fréchet-distance to W 1,2-distance. • We settle some subtleties concerning the length constraint. • If certain growth conditions of F, Fn can be established, the method yields convergence rates for Hausdorff distance of argmin(F) and argmin(Fn). As metric space we choose