Fast Rotationally Symmetric Direction Fields on 3D Surfaces

Christian Clemenz
Fast Rotationally Symmetric Direction Fields on 3D Surfaces
[thesis]

Information

Abstract

We demonstrate the implementation of the Globally Optimal Direction Field algorithm by Knöppel et al. as a plugin for a geometry processing software. The plugin constructs N-RoSy fields of arbitrary degree by solving a smallest eigenvalue problem. For that, we use a sparse Cholesky solver and the Inverse Power Method. The field can optionally be aligned to the principal curvature induced by the geometry. We also added the option to use the improvements proposed by Pellenard et al. These improvements contain constraints imposed on certain areas of the mesh. A linear least squares approach is then used for solving the over-constrained system. Our main contribution is to clarify ambiguities we found in these papers, especially regarding the constraints.

We tested the algorithm using meshes of different common sizes used in 3D modeling for the computation time and ease of usage. Although the algorithm is very fast the responsiveness starts to decline at about 6 * 10^4 polygons. We recommend not to use it on huge meshes or detailed 3D scans if fast results are important. The degree of curvature alignment can be difficult to adjust. However, together with fast results, different parameter settings can be tested relatively easy.

The results look very smooth and singularities are often located at geometric features. Using constraints helps to align the field to mesh boundaries, sharp edges or, if it is warped, to the principal curvature directions. Their use is very easy because the results are predictable. Only curvature constraints can sometimes be hard to predict and are best used in conjunction with other constraints.

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BibTeX

@bachelorsthesis{clemenz_2019_rosy_fields,
  title =      "Fast Rotationally Symmetric Direction Fields on 3D Surfaces",
  author =     "Christian Clemenz",
  year =       "2019",
  abstract =   "We demonstrate the implementation of the Globally Optimal
               Direction Field algorithm by Kn\"{o}ppel et al. as a plugin
               for a geometry processing software. The plugin constructs
               N-RoSy fields of arbitrary degree by solving a smallest
               eigenvalue problem. For that, we use a sparse Cholesky
               solver and the Inverse Power Method. The field can
               optionally be aligned to the principal curvature induced by
               the geometry. We also added the option to use the
               improvements proposed by Pellenard et al. These improvements
               contain constraints imposed on certain areas of the mesh. A
               linear least squares approach is then used for solving the
               over-constrained system. Our main contribution is to clarify
               ambiguities we found in these papers, especially regarding
               the constraints.  We tested the algorithm using meshes of
               different common sizes used in 3D modeling for the
               computation time and ease of usage. Although the algorithm
               is very fast the responsiveness starts to decline at about 6
               * 10^4 polygons. We recommend not to use it on huge meshes
               or detailed 3D scans if fast results are important. The
               degree of curvature alignment can be difficult to adjust.
               However, together with fast results, different parameter
               settings can be tested relatively easy.  The results look
               very smooth and singularities are often located at geometric
               features. Using constraints helps to align the field to mesh
               boundaries, sharp edges or, if it is warped, to the
               principal curvature directions. Their use is very easy
               because the results are predictable. Only curvature
               constraints can sometimes be hard to predict and are best
               used in conjunction with other constraints.",
  month =      jun,
  address =    "Favoritenstrasse 9-11/E193-02, A-1040 Vienna, Austria",
  school =     "Research Unit of Computer Graphics, Institute of Visual
               Computing and Human-Centered Technology, Faculty of
               Informatics, TU Wien ",
  keywords =   "Direction Fields, RoSy Fields, Global Approach",
  URL =        "https://www.cg.tuwien.ac.at/research/publications/2019/clemenz_2019_rosy_fields/",
}