My Theses

This books gives a detailed overview on the mathematical background of the Laplace Beltrami operator (LBO) for a Riemannian manifold. A few analytical computations are presented for special domains in 2D and 3D. Furthermore the numerical computation of the spectra of the LBO are described using a special method for planar domains and the FEM method with up to cubic form functions for the general setup in 2D and 3D. Since the spectrum of the LBO is an isometry invariant, it is invariant under translation, rotation, and invariant under a change of parametrization. It is therefore possible to construct complex objects by gluing several parameter spaces to each other. Some examples are presented, also employing the medial axis as a tool to guide the parametrization of any 2D domain in a way that ensures good meshes without any error at the boundary.

Furthermore this book describes the extraction of geometric data from the spectrum. For this purpose the rapid convergence of the Heat trace is shown. Finally several applications and examples are presented to employ the spectrum in the field of shape recognition. Examples of isospectral 3D solids are shown, that can be distinguished by the spectrum of their boundary surface. The robustness is demonstrated when objects on different mesh resolutions are successfully compared. It is shown how the Neumann boundary condition can be used instead of the Dirichlet condition in cases where the boundary is not supposed to play an important role. finally several complex objects are matched successfully employing their Laplace-Beltrami spectrum.

Subject of this thesis is the numerical computation of the eigenvalues of the Laplace-Beltrami operator employing the Finite-Element-Method (FEM) with quadratic and cubic form functions on triangular elements. The Laplace-Beltrami operator is the natural extension of the Laplacian for Riemannian manifolds (such as curved surfaces). First an overview on the necessary background on differential equations, eigenvalue problems, FEM (variational formulation, Galerkin method) and the form functions of higher degree is given. The numbering of the vertices using the Cuthill algorithm, a mesh refinement method and meshing techniques are described. A similarity invariant for triangles is introduced and used as a measure for the quality of a triangle. The structure of the "Laplace" software project as a powerful tool to compute Laplace-Beltrami spectra on parametrized surfaces is explained. Numerical computations of the spectra of several surfaces in R² and R³ are presented and the dependency of the eigenvalues on deformations of the surface and on the triangulation is analyzed.

Nature is rich in highly irregular structures such as trees, clouds, flashes or coast lines. They can absolutely not be described by simple geometric objects but rather embody a new level of geometry. With the help of a family of scale invariant "fractals" these irregular structures can be described. The fractal dimension measures the degree of irregularity that stays constant on any resolution of the object. The fractal dimension presents a good measure of the roughness of objects or textures. This concept can be transfered to gray scale images to get a measure of the roughness of the depicted texture. This work presents and compares different techniques to compute the fractal dimension for discrete data to classify gray scale images concerning their roughness. For this purpose the program "FDim" is introduced and described.

This paper gives an overview on one of the most successful methods to detect the dissimilarity of knots. A polynomial is assigned to every knot. Since these polynomials can be computed from any knot projection and since they always stay the same for the same knot they are called knot invariants. If two such polynomials differ, it is sure that they cannot describe the same knot. First the Jones polynomial is derived together with the prove that it is a knot invariant (using the Reidemeister moves). Afterwards a closer look is taken at alternating knots and their invariants. The final section describes the Alexander and the HOMFLY polynomials and their properties.