Abstract
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 [4] 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.
Chapter PDF
Similar content being viewed by others
References
Amini, A.A., Weymouth, T.E., Jain, R.C.: Using dynamic programming for solving variational problems in vision. IEEE Transactions on Pattern Analysis and Machine Intelligence 12(9), 855–867 (1990)
Appleton, B., Talbot, H.: Globally minimal surfaces by continuous maximal flows. IEEE transactions on Pattern Analysis and Pattern Recognition (PAMI) 28(1), 106–118 (2006)
Black, M.J., Anandan, P.: The robust estimation of multiple motions: Parametric and piecewise–smooth flow fields. cvgip-iu 63(1), 75–104 (1996)
Boykov, Y., Kolmogorov, V.: Computing geodesics and minimal surfaces via graph cuts. In: Int. Conf. on Computer Vision, vol. I, pp. 26–33 (2003)
Boykov, Y., Kolmogorov, V.: An experimental comparison of mincut/max-flow algorithms for energy minimization in vision. IEEE Transactions on Pattern Analysis and Machine Intelligence 26(9), 1124–1137 (2004)
Chan, T.F., Vese, L.A.: Active contours without edges. IEEE Trans. Image Processing 10(2), 266–277 (2001)
Charpiat, G., Faugeras, O., Keriven, R.: Approximations of shape metrics and application to shape warping and empirical shape statistics. Journal of Foundations of Computational Mathematics 5(1), 1–58 (2005)
Cremers, D., Tischhäuser, F., Weickert, J., Schnörr, C.: Diffusion Snakes: Introducing statistical shape knowledge into the Mumford–Shah functional. IJCV 50(3), 295–313 (2002)
Horn, B.K.P., Schunck, B.G.: Determining optical flow. Artificial Intelligence 17, 185–203 (1981)
Kass, M., Witkin, A., Terzolpoulos, D.: Snakes: Active contour models. International Journal of Computer Vision 1(4), 321–331 (1988)
Kirsanov, D., Gortler, S.-J.: A discrete global minimization algorithm for continuous variational problems. Harvard CS. Tech. Rep., TR-14-04 (July 2004)
Kolmogorov, V., Boykov, Y.: What metrics can be approximated by geo-cuts, or global optimization of length/area and flux. In: ICCV (October 2005)
Kornprobst, P., Deriche, R., Aubert, G.: Image sequence analysis via partial differential equations. Journal of Math. Imaging and Vision 11(1), 5–26 (1999)
Paragios, N.: Shape-based segmentation and tracking in cardiac image analysis. IEEE Transactions on Medical Image Analysis, 402–407 (2003)
Perona, P., Malik, J.: Scale-space and edge-detection. IEEE Trans. on Pattern Analysis and Machine Intelligence 12(7), 629–639 (1990)
Weickert, J.: Anisotropic diffusion in image processing. Teubner, Stuttgart (1998)
Weickert, J., Schnörr, C.: A theoretical framework for convex regularizers in PDE–based computation of image motion. IJCV 45(3), 245–264 (2001)
Xu, N., Bansal, R., Ahuja, N.: Object segmentation using graph cuts based active contours. In: CVPR, vol. II, pp. 46–53 (2003)
Yezzi, A., Mennucci, A.: Conformal metrics and true “gradient flows” for curves. IEEE Intl. Conf. on Comp. Vis. (2005)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Boykov, Y., Kolmogorov, V., Cremers, D., Delong, A. (2006). An Integral Solution to Surface Evolution PDEs Via Geo-cuts. In: Leonardis, A., Bischof, H., Pinz, A. (eds) Computer Vision – ECCV 2006. ECCV 2006. Lecture Notes in Computer Science, vol 3953. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11744078_32
Download citation
DOI: https://doi.org/10.1007/11744078_32
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-33836-9
Online ISBN: 978-3-540-33837-6
eBook Packages: Computer ScienceComputer Science (R0)Springer Nature Proceedings Computer Science