A Gauss-Newton method for the integration of spatial normal fields in shape Space

Jonathan Balzer

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

Abstract

We address the task of adjusting a surface to a vector field of desired surface normals in space. The described method is entirely geometric in the sense, that it does not depend on a particular parametrization of the surface in question. It amounts to solving a nonlinear least-squares problem in shape space. Previously, the corresponding minimization has been performed by gradient descent, which suffers from slow convergence and susceptibility to local minima. Newton-type methods, although significantly more robust and efficient, have not been attempted as they require second-order Hadamard differentials. These are difficult to compute for the problem of interest and in general fail to be positive-definite symmetric. We propose a novel approximation of the shape Hessian, which is not only rigorously justified but also leads to excellent numerical performance of the actual optimization. Moreover, a remarkable connection to Sobolev flows is exposed. Three other established algorithms from image and geometry processing turn out to be special cases of ours. Our numerical implementation founds on a fast finite-elements formulation on the minimizing sequence of triangulated shapes. A series of examples from a wide range of different applications is discussed to underline flexibility and efficiency of the approach. © 2011 Springer Science+Business Media, LLC.
Original languageEnglish (US)
Pages (from-to)65-79
Number of pages15
JournalJournal of Mathematical Imaging and Vision
Volume44
Issue number1
DOIs
StatePublished - Aug 9 2011

ASJC Scopus subject areas

  • Modeling and Simulation
  • Geometry and Topology
  • Applied Mathematics
  • Computer Vision and Pattern Recognition
  • Statistics and Probability
  • Condensed Matter Physics

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