2019

Permanent URI for this collection

https://diglib.eg.org/handle/10.2312/2632668

Browse

Browsing 2019 by Subject "Geometry Processing"

Now showing 1 - 2 of 2
  • No Thumbnail Available
    Item
    Locally Solving Linear Systems for Geometry Processing
    (2019) Herholz, Philipp
    Geometry processing algorithms commonly need to solve linear systems involving discrete Laplacians. In many cases this constitutes a central building block of the algorithm and dominates runtime. Usually highly optimized libraries are employed to solve these systems, however, they are built to solve very general linear systems. I argue that it is possible to create more efficient algorithms by exploiting domain knowledge and modifying the structure of these solvers accordingly. In this thesis I take a first step in that direction. The focus lies on Cholesky factorizations that are commonly employed in the context of geometry processing. More specifically I am interested in the solution of linear systems where variables are associated with vertices of a mesh. The central question is: Given the Cholesky factorization of a linear system defined on the full mesh, how can we efficiently obtain solutions for a local set of vertices, possibly with new boundary conditions? I present methods to achieve this without computing the value at all vertices or refactoring the system from scratch. Developing these algorithms requires a detailed understanding of sparse Cholesky factorizations and modifications of their implementation. The methods are analyzed and validated in concrete applications. Ideally this thesis will stimulates research in geometry processing and related fields to jointly develop algorithms and numerical methods rather than treating them as distinct blocks.
  • No Thumbnail Available
    Item
    Model Reduction for Interactive Geometry Processing
    (n/a, 2019-04-01) Brandt, Christopher
    The research field of geometry processing is concerned with the representation, analysis, modeling, simulation and optimization of geometric data. In this thesis, we introduce novel techniques and efficient algorithms for problems in geometry processing, such as the modeling and simulation of elastic deformable objects, the design of tangential vector fields or the automatic generation of spline curves. The complexity of the geometric data determines the computation time of algorithms within these applications. The high resolution of modern meshes, for example, poses a big challenge when geometric processing tools are expected to perform at interactive rates. To this end the goal of this thesis is to introduce fast approximation techniques for problems in geometry processing. One line of research to achieve this goal will be to introduce novel model order reduction techniques to problems in geometry processing. Model order reduction is a concept to reduce the computational complexity of models in numerical simulations, energy optimizations and modeling problems. New specialized model order reduction approaches are introduced and existing techniques are applied to enhance tools within the field of geometry processing. In addition to introducing model reduction techniques, we make several other contributions to the field. We present novel discrete differential operators and higher order smoothness energies for the modeling of tangential n-vector fields. These are used, to develop novel tools for the modeling of fur, stroke based renderings or anisotropic reflection properties on meshes. We propose a geometric flow for curves in shape space that allows for the processing and creation of animations of elastic deformable objects. A new optimization scheme for sparsity regularized functionals is introduced and used to compute natural, localized deformations of geometrical objects. Lastly, we reformulate the classical problem of spline optimization as a sparsity regularized optimization problem.