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A continuum mechanical approach to geodesics in shape space

Publication ,  Journal Article
Wirth, B; Bar, L; Rumpf, M; Sapiro, G
Published in: International Journal of Computer Vision
July 1, 2011

In this paper concepts from continuum mechanics are used to define geodesic paths in the space of shapes, where shapes are implicitly described as boundary contours of objects. The proposed shape metric is derived from a continuum mechanical notion of viscous dissipation. A geodesic path is defined as the family of shapes such that the total amount of viscous dissipation caused by an optimal material transport along the path is minimized. The approach can easily be generalized to shapes given as segment contours of multi-labeled images and to geodesic paths between partially occluded objects. The proposed computational framework for finding such a minimizer is based on the time discretization of a geodesic path as a sequence of pairwise matching problems, which is strictly invariant with respect to rigid body motions and ensures a 1-1 correspondence along the induced flow in shape space. When decreasing the time step size, the proposed model leads to the minimization of the actual geodesic length, where the Hessian of the pairwise matching energy reflects the chosen Riemannian metric on the underlying shape space. If the constraint of pairwise shape correspondence is replaced by the volume of the shape mismatch as a penalty functional, one obtains for decreasing time step size an optical flow term controlling the transport of the shape by the underlying motion field. The method is implemented via a level set representation of shapes, and a finite element approximation is employed as spatial discretization both for the pairwise matching deformations and for the level set representations. The numerical relaxation of the energy is performed via an efficient multi-scale procedure in space and time. Various examples for 2D and 3D shapes underline the effectiveness and robustness of the proposed approach. © 2010 Springer Science+Business Media, LLC.

Duke Scholars

Published In

International Journal of Computer Vision

DOI

EISSN

1573-1405

ISSN

0920-5691

Publication Date

July 1, 2011

Volume

93

Issue

3

Start / End Page

293 / 318

Related Subject Headings

  • Artificial Intelligence & Image Processing
  • 4611 Machine learning
  • 4607 Graphics, augmented reality and games
  • 4603 Computer vision and multimedia computation
  • 0801 Artificial Intelligence and Image Processing
 

Citation

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Wirth, B., Bar, L., Rumpf, M., & Sapiro, G. (2011). A continuum mechanical approach to geodesics in shape space. International Journal of Computer Vision, 93(3), 293–318. https://doi.org/10.1007/s11263-010-0416-9
Wirth, B., L. Bar, M. Rumpf, and G. Sapiro. “A continuum mechanical approach to geodesics in shape space.” International Journal of Computer Vision 93, no. 3 (July 1, 2011): 293–318. https://doi.org/10.1007/s11263-010-0416-9.
Wirth B, Bar L, Rumpf M, Sapiro G. A continuum mechanical approach to geodesics in shape space. International Journal of Computer Vision. 2011 Jul 1;93(3):293–318.
Wirth, B., et al. “A continuum mechanical approach to geodesics in shape space.” International Journal of Computer Vision, vol. 93, no. 3, July 2011, pp. 293–318. Scopus, doi:10.1007/s11263-010-0416-9.
Wirth B, Bar L, Rumpf M, Sapiro G. A continuum mechanical approach to geodesics in shape space. International Journal of Computer Vision. 2011 Jul 1;93(3):293–318.
Journal cover image

Published In

International Journal of Computer Vision

DOI

EISSN

1573-1405

ISSN

0920-5691

Publication Date

July 1, 2011

Volume

93

Issue

3

Start / End Page

293 / 318

Related Subject Headings

  • Artificial Intelligence & Image Processing
  • 4611 Machine learning
  • 4607 Graphics, augmented reality and games
  • 4603 Computer vision and multimedia computation
  • 0801 Artificial Intelligence and Image Processing