title: Geometry-aware bases for shape approximation.
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.
authors.affiliation: The authors are with the School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. sorkine@tau.ac.il
authors.forename: Olga
authors.initials: O
authors.lastname: Sorkine
authors.forename: Daniel
authors.initials: D
authors.lastname: Cohen-Or
authors.forename: Dror
authors.initials: D
authors.lastname: Irony
authors.forename: Sivan
authors.initials: S
authors.lastname: Toledo
mesh_terms: D000465:Algorithms; D003196:Computer Graphics; D003198:Computer Simulation; D007090:Image Interpretation, Computer-Assisted / Q000379:methods*; D021621:Imaging, Three-Dimensional / Q000379:methods*; D016247:Information Storage and Retrieval / Q000379:methods*; D015233:Models, Statistical; D009716:Numerical Analysis, Computer-Assisted; D010363:Pattern Recognition, Automated / Q000379:methods*; D014584:User-Computer Interface