Research in geometry processing concerns the design of algorithms and mathematical models for the analysis and manipulation of geometric data. Examples of its applications are shape projection (e.g. smoothing and filtering), shape correspondence (e.g. functional maps), shape descriptors (e.g. heat and wave kernel signatures), segmentation, and surface parameterization. A set of tools that have proven to be useful for solving such tasks are spectral methods. In general, spectral methods solve geometry processing problems by taking the benefit of the spectra and the eigenfunctions of the Laplacian operator defined on a surface mesh. This allows us to extend the notion of Fourier analysis from signal and image processing to surface processing, a theoretically sound and well-researched concept. In practice, the decomposition of the Laplacian operator into a diagonal matrix of eigenvalues and a rectangular matrix of eigenvectors enables efficient treatment of a broad range of geometry processing problems.
A main adversity in spectral geometry processing is the expensive computational cost attached to the eigendecomposition of the Laplacian operator, before we can use the spectra and the eigenfunctions for the applications. Since analytical solutions are not known, one needs to opt for a numerical method to solve the eigenvalue problem. It is a numerically expensive computation, especially for a complex mesh. Another challenge comes from the storage requirement. Considering that the Laplace--Beltrami operator has global support, it takes a dense matrix to represent the eigenvectors. Therefore, the memory requirement for saving the eigenbasis can be high, particularly when a large number of eigenfunctions need to be stored. These challenges hinder the use of spectral methods for geometry processing applications.
In this thesis, we introduce new methods addressing the aforementioned challenges. In Chapter 2, we propose a fast algorithm that allows for approximating the smallest eigenvalues and the corresponding eigenvectors of the Laplace--Beltrami operator in just a fraction of the time needed to solve the original eigenvalue problem. We construct subspaces of the space of all functions that include low frequency functions and restrict the solution of the eigenproblem to the subspace. It enables the fast approximation of the eigenproblem, independent of the size of the original problem. Our novel scheme also enables significantly more efficient storage of the approximated eigenfunctions. We show that the approximated spectra are close to the reference spectra and that the fast approximation method benefits geometry processing applications, such as shape classification, geodesic distance computation, shape projection (e.g. filtering), and vibration modes of deformable objects.
We consider localized eigenfields of the Hodge--Laplacian, which serve as a sparse basis for the efficient design and processing of tangential fields, in Chapter 3. The basis spans subspaces of the spaces of tangential vector, $n$-vector, and tensor fields on a surface mesh. Restricting the design and processing of tangential fields to the subspace allows us to decouple the degrees of freedom we use for design and processing tasks from the complexity of the mesh representation. The construction is scalable, so we can efficiently compute and store subspaces for large meshes. We evaluate the performance of the novel method on various modeling and processing tasks in vector fields (fur design), n-vector fields (n-field design and hatching/line-art design), and tensor fields (curvature fields smoothing) and show that the computation time decreases up to two orders of magnitude compared to that of the original problem.
Chapter 4 introduces a novel multigrid method for numerically solving the Laplace--Beltrami eigenproblems on a surface mesh. Our new technique, the Hierarchical Subspace Iteration Method (HSIM), works on a hierarchy of nested vector spaces, in which the solution of the coarser level is used as an initial solution on the finer level. We construct the coarsest level such that the eigenproblems can be solved efficiently using a dense eigensolver. On every level, the prolongation operator maps the solution from the coarser to the finer level. The result then can be used as an initialization for subspace iterations to approximate the eigenpairs. This approach significantly reduces the number of iterations at the finest level, compared to the non-hierarchical subspace iteration method. We show that HSIM outperforms the Locally Optimal Block Preconditioned Conjugate Gradient method and the state-of-the-art Lanczos-based eigensolvers, such as Matlab's eigs, Manifold Harmonics, and SpectrA.
In summary, each of the chapters in this thesis proposes efficient algorithms for computing the eigendecompositions of Laplace--Beltrami and Hodge--Laplace operators, mainly using model order reduction and multigrid approaches. These methods reduce computational costs (Chapter 1-3) and storage requirements (Chapter 1-2) for the spectral processing of scalar functions and tangential fields on surface meshes.