Browsing by Author "Brandt, Christopher"

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    Locally Supported Tangential Vector, n-Vector, and Tensor Fields
    (The Eurographics Association and John Wiley & Sons Ltd., 2020) Nasikun, Ahmad; Brandt, Christopher; Hildebrandt, Klaus; Panozzo, Daniele and Assarsson, Ulf
    We introduce a construction of subspaces of the spaces of tangential vector, n-vector, and tensor fields on surfaces. The resulting subspaces can be used as the basis of fast approximation algorithms for design and processing problems that involve tangential fields. Important features of our construction are that it is based on a general principle, from which constructions for different types of tangential fields can be derived, and that it is scalable, making it possible to efficiently compute and store large subspace bases for large meshes. Moreover, the construction is adaptive, which allows for controlling the distribution of the degrees of freedom of the subspaces over the surface. We evaluate our construction in several experiments addressing approximation quality, scalability, adaptivity, computation times and memory requirements. Our design choices are justified by comparing our construction to possible alternatives. Finally, we discuss examples of how subspace methods can be used to build interactive tools for tangential field design and processing tasks.
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    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.
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    Spectral Processing of Tangential Vector Fields
    (© 2017 The Eurographics Association and John Wiley & Sons Ltd., 2017) Brandt, Christopher; Scandolo, Leonardo; Eisemann, Elmar; Hildebrandt, Klaus; Chen, Min and Zhang, Hao (Richard)
    We propose a framework for the spectral processing of tangential vector fields on surfaces. The basis is a Fourier‐type representation of tangential vector fields that associates frequencies with tangential vector fields. To implement the representation for piecewise constant tangential vector fields on triangle meshes, we introduce a discrete Hodge–Laplace operator that fits conceptually to the prominent discretization of the Laplace–Beltrami operator. Based on the Fourier representation, we introduce schemes for spectral analysis, filtering and compression of tangential vector fields. Moreover, we introduce a spline‐type editor for modelling of tangential vector fields with interpolation constraints for the field itself and its divergence and curl. Using the spectral representation, we propose a numerical scheme that allows for real‐time modelling of tangential vector fields.We propose a framework for the spectral processing of tangential vector fields on surfaces. The basis is a Fourier‐type representation of tangential vector fields that associates frequencies with tangential vector fields. To implement the representation for piecewise constant tangential vector fields on triangle meshes, we introduce a discrete Hodge–Laplace operator that fits conceptually to the prominent discretization of the Laplace–Beltrami operator. Based on the Fourier representation, we introduce schemes for spectral analysis, filtering and compression of tangential vector fields.