### An Integral Solution to Surface Evolution PDEs via Geo-Cuts

In *European Conference on Computer Vision* (ECCV), LNCS 3953, vol.III, pp.409-422, Graz, Austria, May 2006.

### 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 dierential 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 L2 distance metric demonstrate numerical
stability, topological changes and an absence of any oscillatory motion.

##### WHOLE PAPER: pdf file (699 Kb)