An Integral Solution to Surface Evolution PDEs Via Geo-cuts
We introduce a new approach to modelling gradient flows of contours and surfaces. While standard variational methods (e.g. level sets) compute local interface motion in a differential fashion by estimating local contour velocity via energy derivatives, we propose to solve surface evolution PDEs by explicitly estimating integral motion of the whole surface. We formulate an optimization problem directly based on an integral characterization of gradient flow as an infinitesimal move of the (whole) surface giving the largest energy decrease among all moves of equal size. We show that this problem can be efficiently solved using recent advances in algorithms for global hypersurface optimization [4,2,11]. In particular, we employ the geo-cuts method  that uses ideas from integral geometry to represent continuous surfaces as cuts on discrete graphs. The resulting interface evolution algorithm is validated on some 2D and 3D examples similar to typical demonstrations of level-set methods. Our method can compute gradient flows of hypersurfaces with respect to a fairly general class of continuous functionals and it is flexible with respect to distance metrics on the space of contours/surfaces. Preliminary tests for standard L 2 distance metric demonstrate numerical stability, topological changes and an absence of any oscillatory motion.
Unable to display preview. Download preview PDF.
- 3.Black, M.J., Anandan, P.: The robust estimation of multiple motions: Parametric and piecewise–smooth flow fields. cvgip-iu 63(1), 75–104 (1996)Google Scholar
- 4.Boykov, Y., Kolmogorov, V.: Computing geodesics and minimal surfaces via graph cuts. In: Int. Conf. on Computer Vision, vol. I, pp. 26–33 (2003)Google Scholar
- 11.Kirsanov, D., Gortler, S.-J.: A discrete global minimization algorithm for continuous variational problems. Harvard CS. Tech. Rep., TR-14-04 (July 2004)Google Scholar
- 12.Kolmogorov, V., Boykov, Y.: What metrics can be approximated by geo-cuts, or global optimization of length/area and flux. In: ICCV (October 2005)Google Scholar
- 14.Paragios, N.: Shape-based segmentation and tracking in cardiac image analysis. IEEE Transactions on Medical Image Analysis, 402–407 (2003)Google Scholar
- 16.Weickert, J.: Anisotropic diffusion in image processing. Teubner, Stuttgart (1998)Google Scholar
- 18.Xu, N., Bansal, R., Ahuja, N.: Object segmentation using graph cuts based active contours. In: CVPR, vol. II, pp. 46–53 (2003)Google Scholar
- 19.Yezzi, A., Mennucci, A.: Conformal metrics and true “gradient flows” for curves. IEEE Intl. Conf. on Comp. Vis. (2005)Google Scholar