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

Handle URI:
http://hdl.handle.net/10754/561839
Title:
A Gauss-Newton method for the integration of spatial normal fields in shape Space
Authors:
Balzer, Jonathan
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.
KAUST Department:
Visual Computing Center (VCC)
Publisher:
Springer Nature
Journal:
Journal of Mathematical Imaging and Vision
Issue Date:
9-Aug-2011
DOI:
10.1007/s10851-011-0311-1
Type:
Article
ISSN:
09249907
Appears in Collections:
Articles; Visual Computing Center (VCC)

Full metadata record

DC FieldValue Language
dc.contributor.authorBalzer, Jonathanen
dc.date.accessioned2015-08-03T09:32:09Zen
dc.date.available2015-08-03T09:32:09Zen
dc.date.issued2011-08-09en
dc.identifier.issn09249907en
dc.identifier.doi10.1007/s10851-011-0311-1en
dc.identifier.urihttp://hdl.handle.net/10754/561839en
dc.description.abstractWe 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.en
dc.publisherSpringer Natureen
dc.subjectGauss-Newton methoden
dc.subjectIntegrationen
dc.subjectMinimal surfaceen
dc.subjectNormal fielden
dc.subjectNormal mapen
dc.subjectShape Hessianen
dc.subjectShape spaceen
dc.subjectSobolev flowen
dc.titleA Gauss-Newton method for the integration of spatial normal fields in shape Spaceen
dc.typeArticleen
dc.contributor.departmentVisual Computing Center (VCC)en
dc.identifier.journalJournal of Mathematical Imaging and Visionen
kaust.authorBalzer, Jonathanen
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