Boykov, Yuri; Kolmogorov, VladimirISTA; Cremers, Daniel; Delong, Andrew
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 functional and it is flexible with respect to distance metrics on the space of contours/surfaces. Preliminary tests for standard L2 distance metric demonstrate numerical stability, topological changes and an absence of any oscillatory motion.
409 - 422
ECCV: European Conference on Computer Vision
Boykov Y, Kolmogorov V, Cremers D, Delong A. An integral solution to surface evolution PDEs via geo cuts. In: Vol 3953. Springer; 2006:409-422. doi:10.1007/11744078_32
Boykov, Y., Kolmogorov, V., Cremers, D., & Delong, A. (2006). An integral solution to surface evolution PDEs via geo cuts (Vol. 3953, pp. 409–422). Presented at the ECCV: European Conference on Computer Vision, Springer. https://doi.org/10.1007/11744078_32
Boykov, Yuri, Vladimir Kolmogorov, Daniel Cremers, and Andrew Delong. “An Integral Solution to Surface Evolution PDEs via Geo Cuts,” 3953:409–22. Springer, 2006. https://doi.org/10.1007/11744078_32.
Y. Boykov, V. Kolmogorov, D. Cremers, and A. Delong, “An integral solution to surface evolution PDEs via geo cuts,” presented at the ECCV: European Conference on Computer Vision, 2006, vol. 3953, pp. 409–422.
Boykov Y, Kolmogorov V, Cremers D, Delong A. 2006. An integral solution to surface evolution PDEs via geo cuts. ECCV: European Conference on Computer Vision, LNCS, vol. 3953, 409–422.
Boykov, Yuri, et al. An Integral Solution to Surface Evolution PDEs via Geo Cuts. Vol. 3953, Springer, 2006, pp. 409–22, doi:10.1007/11744078_32.