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We analyze a remarkable class of centrally symmetric polytopes, the Hansen
polytopes of split graphs. We confirm Kalai's 3^d-conjecture for such polytopes
(they all have at least 3^d nonempty faces) and show that the Hanner polytopes
among them (which have exactly 3^d nonempty faces) correspond to threshold
graphs. Our study produces a new family of Hansen polytopes that have only
3^d+16 nonempty faces.
In recent years, substantial progress in shape analysis has been achieved through methods that use the spectra and eigenfunctions of discrete Laplace operators. In this work, we study spectra and eigenfunctions of discrete differential operators that can serve as an alternative to discrete Laplacians for applications in shape analysis. We construct such operators as the Hessians of surface energies or deformation energies. In particular, we design a quadratic energy such that, on the one hand, its Hessian equals the Laplace operator if the surface is a part of the Euclidean plane, and, on the other hand, the Hessian eigenfunctions are sensitive to the extrinsic curvature (e.g. sharp bends) on curved
surfaces. Furthermore, we consider eigenvibrations induced by deformation energies, and we derive a closed form representation for the Hessian (at the rest state of the energy) for a general class of deformation energies. Based on these spectra and eigenmodes, we derive two shape signatures. One that measures the similarity of points on a surface, and another that can be used to identify features of surfaces.
A capillary surface in a negative gravitational field describes the shape of the surface of a hanging drop in a capillary tube with wetting material on the bottom. Mathematical modeling leads to the volume- and obstacle-constrained minimization of a nonconvex nonlinear energy functional of mean curvature type which is unbounded from below. In 1984 Huisken proved the existence and regularity of local minimizers of this energy under the condition on gravitation being sufficiently weak. We prove convergence of a first order finite element approximation of these minimizers. Numerical results demonstrating the theoretic convergence order are given.
We present a unified computational framework for matching 3d geometric objects (points, lines, surfaces, volumes) of highly varying shape. Our approach is based on the Large Deformation Diffeomorphic Metric Mapping (LDDMM) method acting on $m$-currents. After stating an optimization algorithm in the function space of admissible morph generating velocity fields, two innovative aspects in this framework are presented: First, we spatially discretize the velocity field with conforming adaptive finite elements and discuss advantages of this new approach. Secondly, we directly compute the temporal evolution of discrete $m$-current attributes. Several numerical experiments demonstrate the effectiveness of this approach.
Deformable surface models are often represented as triangular meshes in image segmentation applications. For a fast and easily regularized deformation onto the target object boundary, the vertices of the mesh are commonly moved along line segments (typically surface normals). However, in case of high mesh curvature, these lines may intersect with the target boundary at “non-corresponding” positions, or may not intersect at all. Consequently, certain deformations cannot be achieved. We propose omnidirectional displacements for deformable surfaces (ODDS) to overcome this limitation. ODDS allow each vertex to move not only along a line segment but within a surrounding sphere, and achieve globally optimal deformations subject to local regularization con-
straints. However, allowing a ball-shaped instead of a linear range of motion per vertex significantly increases runtime and memory. To alleviate this drawback, we propose a hybrid approach, fastODDS, with improved runtime and reduced memory requirements. Furthermore, fastODDS can also cope with simultaneous segmentation of multiple objects. We show the theoretical benefits of ODDS with experiments on synthetic data, and evaluate ODDS and fastODDS quantitatively on clinical image data of the mandible and the hip bones. There, we assess both the global segmentation accuracy as well as local accuracy in high curvature regions, such as the tip-shaped mandibular coronoid processes and the ridge-shaped acetabular rims of
the hip bones.
We introduce an algorithm for
diffusion weighted magnetic resonance imaging data enhancement based on structural adaptive smoothing in both space and diffusion direction.
The method, called POAS, does not refer to a specific model for the data, like the diffusion tensor or higher order models.
It works by embedding the measurement space into a space with defined metric and group operations, in this case the Lie group of three-dimensional Euclidean motion SE(3).
Subsequently, pairwise comparisons of the values of the diffusion
weighted signal are used for adaptation.
The position-orientation adaptive smoothing preserves the edges of the observed fine and anisotropic structures.
The POAS-algorithm is designed to reduce noise directly in the diffusion weighted images and consequently also to reduce bias and
variability of quantities derived from the data for specific models.
We evaluate the algorithm on simulated and experimental data and demonstrate that it can be used to reduce the number of applied diffusion gradients and
hence acquisition time while achieving similar quality of data, or to improve the quality of data acquired in a clinically feasible scan time setting.