Journal Publications

This work introduces a method to hierarchically segment articulated shapes into meaningful parts and to register these parts across populations of near-isometric shapes (e.g. head, arms and legs of humans in different body postures). The method exploits the isometry invariance of eigenfunctions of the Laplace-Beltrami operator and uses topological features (level sets at important saddles) for the segmentation. Concepts from persistent homology are employed for a hierarchical representation, for the elimination of topological noise and for the comparison of eigenfunctions. The obtained parts can be registered via their spectral projection across a population of near isometric shapes. This work also presents the highly accurate computation of eigenfunctions and eigenvalues with cubic FEM on triangle meshes and discusses the construction of persistence diagrams from the Morse-Smale complex as well as the relation to size functions.

This paper proposes the use of the surface based Laplace-Beltrami and the volumetric Laplace eigenvalues and -functions as shape descriptors for the comparison and analysis of shapes. These spectral measures are isometry invariant and therefore allow for shape comparisons with minimal shape pre-processing. In particular, no registration, mapping, or remeshing is necessary. The discriminatory power of the 2D surface and 3D solid methods is demonstrated at a population of female caudate nuclei (a subcortical gray matter structure of the brain, involved in memory function, emotion processing, and learning) of normal control subjects and of subjects with schizotypal personality disorder. The behavior and properties of the Laplace-Beltrami eigenvalues and -functions are discussed extensively for both the Dirichlet and Neumann boundary condition showing advantages of the Neumann vs. the Dirichlet spectra in 3D. Furthermore, topological analyses employing the Morse-Smale complex (on the surfaces) and the Reeb graph (in the solids) are performed on selected eigenfunctions, yielding shape descriptors, that are capable of localizing geometric properties and detecting shape differences by indirectly registering topological features such as critical points, level sets and integral lines of the gradient field across subjects. The use of these topological features of the Laplace-Beltrami eigenfunctions in 2D and 3D for statistical shape analysis is novel.

We present a method for solving arbitrary systems of N nonlinear polynomials in n variables over an n-dimensional simplicial domain based on polynomial representation in the barycentric Bernstein basis and subdivision. The roots are approximated to arbitrary precision by iteratively constructing a series of smaller bounding simplices. We use geometric subdivision to isolate multiple roots within a simplex. An algorithm implementing this method in rounded interval arithmetic is described and analyzed. We find that when the total order of polynomials is close to the maximum order of each variable, an iteration of this solver algorithm is asymptotically more efficient than the corresponding step in a similar algorithm which relies on polynomial representation in the tensor product Bernstein basis. We also discuss various implementation issues and identify topics for further study.

In the area of image retrieval from data bases and for copyright protection of large image collections there is a growing demand for unique but easily computable fingerprints for images. These fingerprints can be used to quickly identify every image within a larger set of possibly similar images. This paper introduces a novel method to automatically obtain such fingerprints from an image. It is based on a re-interpretation of an image as a Riemannian manifold. This representation is feasible for gray value images and color images. We discuss the use of the spectrum of eigenvalues of different variants of the Laplace operator as a fingerprint and show the usability of this approach in several use cases. Contrary to existing works in this area we do not only use the discrete Laplacian, but also with a particular emphasis the underlying continuous operator. This allows better results in comparing the resulting spectra and deeper insights in the problems arising. We show how the well known discrete Laplacian is related to the continuous Laplace-Beltrami operator. Furthermore we introduce the new concept of solid height functions to overcome some potential limitations of the method.

This paper describes in detail a method to extract fingerprints of any surface or solid object by taking the eigenvalues of its respective Laplace-Beltrami operator. Since the spectrum is an isometry invariant it is independent of the objects representation including parametrization, spatial position and of its representation (e.g. NURBS or any parametrized or implicitly represented surface or even for polyhedra). Therefore checking if two objects are isometric needs no prior alignment (registration / localization) of the objects, but only a comparison of their (normalized) spectra, the Shape-DNA.
In this paper we describe the computation of the Shape-DNA and their comparison. We also give an overview on implementation issues like meshing, numerics and the used "Atlas" data structure allowing principally to compute eigenvalues of topologically complex objects by gluing surface patches. Exploiting the isometry invariance of the Laplace-Beltrami operator we succeed in computing eigenvalues for smoothly bounded objects without discretization errors caused by approximation of the boundary.
Furthermore we show the rapid convergence of the heat trace series and demonstrate that it is computationally feasible to extract geometrical data such as the volume, the boundary length and even the Euler characteristic from the numerically calculated fingerprint data. This fact not only confirms the exactness of our computed eigenvalues, but also underlines the geometrical importance of the spectrum.
With the help of the here described Shape-DNA it is possible to support copyright protection, database retrieval and quality assessment of digital data representing surfaces and solids.