Geometry-aware bases for shape approximation

Olga Sorkine, Daniel Cohen-Or, Dror Irony, Sivan Toledo

Research output: Contribution to journalArticlepeer-review

Abstract

We introduce a new class of shape approximation techniques for irregular triangular meshes. Our method approximates the geometry of the mesh using a linear combination of a small number of basis vectors. The basis vectors are functions of the mesh connectivity and of the mesh indices of a number of anchor vertices. There is a fundamental difference between the bases generated by our method and those generated by geometry-oblivious methods, such as Laplacian-based spectral methods. In the latter methods, the basis vectors are functions of the connectivity alone. The basis vectors of our method, in contrast, are geometry-aware since they depend on both the connectivity and on a binary tagging of vertices that are "geometrically important" in the given mesh (e.g., extrema). We show that, by defining the basis vectors to be the solutions of certain least-squares problems, the reconstruction problem reduces to solving a single sparse linear least-squares problem. We also show that this problem can be solved quickly using a state-of-the-art sparse-matrix factorization algorithm. We show how to select the anchor vertices to define a compact effective basis from which an approximated shape can be reconstructed. Furthermore, we develop an incremental update of the factorization of the least-squares system. This allows a progressive scheme where an initial approximation is incrementally refined by a stream of anchor points. We show that the incremental update and solving the factored system are fast enough to allow an online refinement of the mesh geometry.

Original languageEnglish
Pages (from-to)171-179
Number of pages9
JournalIEEE Transactions on Visualization and Computer Graphics
Volume11
Issue number2
DOIs
StatePublished - Mar 2005

Keywords

  • Basis
  • Linear least-squares
  • Mesh Laplacian
  • Shape approximation

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