Calculus of Variations and Geometric Measure Theory

A. C. G. Mennucci - G. Sapiro - G. Sundaramoorthi - A. Yezzi

New possibilities with Sobolev Active Contours

created by mennucci on 06 Nov 2007
modified on 14 Nov 2012


Published Paper

Inserted: 6 nov 2007
Last Updated: 14 nov 2012

Journal: International Journal of Computer Vision
Volume: 84
Number: 2
Pages: 113-129
Year: 2008
Doi: 10.1007/s11263-008-0133-9


Recently, the Sobolev metric was introduced to define gradient flows of various geometric active contour energies. It was shown that the Sobolev metric outperforms the traditional metric for the same energy in many cases such as for tracking where the coarse scale changes of the contour are important. Some interesting properties of Sobolev gradient flows include that they stabilize certain unstable traditional flows, and the order of the evolution PDEs are reduced when compared with traditional gradient flows of the same energies. In this paper, we explore new possibilities for active contours made possible by Sobolev metrics. The Sobolev method allows one to implement new energy-based active contour models that were not otherwise considered because the traditional minimizing method render them ill-posed or numerically infeasible. In particular, we exploit the stabilizing and the order reducing properties of Sobolev gradients to implement the gradient descent of these new energies. We give examples of this class of energies, which include some simple geometric priors and new edge-based energies. We also show that these energies can be quite useful for segmentation and tracking. We also show that the gradient flows using the traditional metric are either ill-posed or numerically difficult to implement, and then show that the flows can be implemented in a stable and numerically feasible manner using the Sobolev gradient.

Keywords: Gradient Flow, space of curves, sobolev active contours, shape optimization, shape priors, ill-posed flows