Mathematical Imaging and Surface Processing

Within the last decade image and geometry processing have become increasingly rigorous with solid foundations in mathematics. Both areas are research fields at the intersection of different mathematical disciplines, ranging from geometry and calculus of variations to PDE analysis and numerical analysis. The workshop brought together scientists from all these areas and a fruitful interplay took place. There was a lively exchange of ideas between geometry and image processing applications areas, characterized in a number of ways in this workshop. For example, optimal transport, first applied in computer vision is now used to define a distance measure between 3d shapes, spectral analysis as a tool in image processing can be applied in surface classification and matching, and so on. We have also seen the use of Riemannian geometry as a powerful tool to improve the analysis of multivalued images. This volume collects the abstracts for all the presentations covering this wide spectrum of tools and application domains

Efficient Methods for Continuous and Discrete Shape Analysis

When interpreting an image of a given object, humans are able to abstract from the presented color information in order to really see the presented object. This abstraction is also known as shape. The concept of shape is not defined exactly in Computer Vision and in this work, we use three different forms of these definitions in order to acquire and analyze shapes. This work is devoted to improve the efficiency of methods that solve important applications of shape analysis. The most important problem in order to analyze shapes is the problem of shape acquisition. To simplify this very challenging problem, numerous researchers have incorporated prior knowledge into the acquisition of shapes. We will present the first approach to acquire shapes given a certain shape knowledge that computes always the global minimum of the involved functional which incorporates a Mumford-Shah like functional with a certain class of shape priors including statistic shape prior and dynamical shape prior. In order to analyze shapes, it is not only important to acquire shapes, but also to classify shapes. In this work, we follow the concept of defining a distance function that measures the dissimilarity of two given shapes. There are two different ways of obtaining such a distance function that we address in this work. Firstly, we model the set of all shapes as a metric space induced by the shortest path on an orbifold. The shortest path will provide us with a shape morphing, i.e., a continuous transformation from one shape into another. Secondly, we address the problem of shape matching that finds corresponding points on two shapes with respect to a preselected feature. Our main contribution for the problem of shape morphing lies in the immense acceleration of the morphing computation. Instead of solving partial resp. ordinary differential equations, we are able to solve this problem via a gradient descent approach that subsequently shortens the length of a path on the given manifold. During our runtime test, we observed a run-time acceleration of up to a factor of 1000. Shape matching is a classical discrete problem. If each shape is discretized by N shape points, most Computer Vision methods needed a cubic run-time. We will provide two approaches how to reduce this worst-case complexity to O(N2 log(N)). One approach exploits the planarity of the involved graph in order to efficiently compute N shortest path in a graph of O(N2) vertices. The other approach computes a minimal cut in a planar graph in O(N log(N)). In order to make this approach applicable to shape matching, we improved the run-time of a recently developed graph cut approach by an empirical factor of 2–4

Differential invariant signatures and flows in computer vision : a symmetry group approach

Includes bibliographical references (p. 40-44).Supported by the National Science Foundation. DMS-9204192 DMS-8811084 ECS-9122106 Supported by the Air Force Office of Scientific Research. AFOSR-90-0024 Supported by the Rothschild Foundation-Yad Hanadiv and by Image Evolutions, Ltd.Peter J. Olver, Guillermo Sapiro, Allen Tannenbaum

The Role of Symmetry and Separation in Surface Evolution and Curve Shortening

With few exceptions, known explicit solutions of the curve shortening flow (CSE) of a plane curve, can be constructed by classical Lie point symmetry reductions or by functional separation of variables. One of the functionally separated solutions is the exact curve shortening flow of a closed, convex ''oval''-shaped curve and another is the smoothing of an initial periodic curve that is close to a square wave. The types of anisotropic evaporation coefficient are found for which the evaporation-condensation evolution does or does not have solutions that are analogous to the basic solutions of the CSE, namely the grim reaper travelling wave, the homothetic shrinking closed curve and the homothetically expanding grain boundary groove. Using equivalence classes of anisotropic diffusion equations, it is shown that physical models of evaporation-condensation must have a diffusivity function that decreases as the inverse square of large slope. Some exact separated solutions are constructed for physically consistent anisotropic diffusion equations

Processing Elastic Surfaces and Related Gradient Flows

Surface processing tools and techniques have a long history in the fields of computer graphics, computer aided geometric design and engineering. In this thesis we consider variational methods and geometric evolution problems for various surface processing applications including surface fairing, surface restoration and surface matching. Geometric evolution problems are often based on the gradient flow of geometric energies. The Willmore functional, defined as the integral of the squared mean curvature over the surface, is a geometric energy that measures the deviation of a surface from a sphere. Therefore, it is a suitable functional for surface restoration, where a destroyed surface patch is replaced by a smooth patch defined as the minimizer of the Willmore functional with boundary conditions for the position and the normal at the patch boundary. However, using the Willmore functional does not lead to satisfying results if an edge or a corner of the surface is destroyed. The anisotropic Willmore energy is a natural generalization of the Willmore energy which has crystal-shaped surfaces like cubes or octahedra as minimizers. The corresponding L2-gradient flow, the anisotropic Willmore flow, leads to a fourth-order partial differential equation that can be written as a system of two coupled second second order equations. Using linear Finite Elements, we develop a semi-implicit scheme for the anisotropic Willmore flow with boundary conditions. This approach suffer from significant restrictions on the time step size. Effectively, one usually has to enforce time steps smaller than the squared spatial grid size. Based on a natural approach for the time discretization of gradient flows we present a new scheme for the time and space discretization of the isotropic and anisotropic Willmore flow. The approach is variational and takes into account an approximation of the L2-distance between the surface at the current time step and the unknown surface at the new time step as well as a fully implicity approximation of the anisotropic Willmore functional at the new time step. To evaluate the anisotropic Willmore energy on the unknown surface of the next time step, we first ask for the solution of an inner, secondary variational problem describing a time step of anisotropic mean curvature motion. The time discrete velocity deduced from the solution of the latter problem is regarded as an approximation of the anisotropic mean curvature vector and enters the approximation of the actual anisotropic Willmore functional. The resulting two step time discretization of the Willmore flow is applied to polygonal curves and triangular surfaces and is independent of the co-dimension. Various numerical examples underline the stability of the new scheme, which enables time steps of the order of the spatial grid size. The Willmore functional of a surface is referred to as the elastic surface energy. Another interesting application of modeling elastic surfaces as minimizers of elastic energies is surface matching, where a correspondence between two surfaces is subject of investigation. There, we seek a mapping between two surfaces respecting certain properties of the surfaces. The approach is variational and based on well-established matching methods from image processing in the parameter domains of the surfaces instead of finding a correspondence between the two surfaces directly in 3D. Besides the appropriate modeling we analyze the derived model theoretically. The resulting deformations are globally smooth, one-to-one mappings. A physically proper morphing of characters in computer graphic is capable with the resulting computational approach