Globally Optimal Surfaces By Continuous Maximal Flows

In this paper we consider the problem of computing globally minimal continuous curves and surfaces for image segmentation and 3D reconstruction. This is solved using a maximal flow approach expressed as a PDE model. Previously proposed techniques yield either grid-biased solutions (graph based approaches) or sub-optimal solutions (active contours and surfaces). The proposed algorithm simulates the flow of an ideal fluid with a spatially varying velocity constraint derived from image data. A proof is given that the algorithm gives the globally maximal flow at convergence, along with an implementation scheme. The globally minimal surface may be obtained trivially from its output. The new algorithm is applied to segmentation in 2D and 3D medical images and to 3D reconstruction from a stereo image pair. The results in 2D agree remarkably well with an existing planar minimal contour algorithm and the results in 3D segmentation and reconstruction demonstrate that the new algorithm is free from grid bias

Globally minimal surfaces by continuous maximal flows

In this paper we address the computation of globally minimal curves and surfaces for image segmentation and stereo reconstruction. We present a solution, simulating a continuous maximal flow by a novel system of partial differential equations. Existing methods are either grid-biased (graph-based methods) or sub-optimal (active contours and surfaces). The solution simulates the flow of an ideal fluid with isotropic velocity constraints. Velocity constraints are defined by a metric derived from image data. An auxiliary potential function is introduced to create a system of partial differential equations. It is proven that the algorithm produces a globally maximal continuous flow at convergence, and that the globally minimal surface may be obtained trivially from the auxiliary potential. The bias of minimal surface methods toward small objects is also addressed. An efficient implementation is given for the flow simulation. The globally minimal surface algorithm is applied to segmentation in 2D and 3D as well as to stereo matching. Results in 2D agree with an existing minimal contour algorithm for planar images. Results in 3D segmentation and stereo matching demonstrate that the new algorithm is robust and free from grid bias

On the flow of non-axisymmetric perturbations of cylinders via surface diffusion

We study the surface diffusion flow acting on a class of general (non--axisymmetric) perturbations of cylinders $\mathcal{C}_r$ in ${\rm I \! R}^3$. Using tools from parabolic theory on uniformly regular manifolds, and maximal regularity, we establish existence and uniqueness of solutions to surface diffusion flow starting from (spatially--unbounded) surfaces defined over $\mathcal{C}_r$ via scalar height functions which are uniformly bounded away from the central cylindrical axis. Additionally, we show that $\mathcal{C}_r$ is normally stable with respect to $2 \pi$--axially--periodic perturbations if the radius $r > 1$,and unstable if $0 < r < 1$. Stability is also shown to hold in settings with axial Neumann boundary conditions.Comment: 20 pages, 1 figure, submitted for publicatio

Curve diffusion and straightening flows on parallel lines

In this paper, we study families of immersed curves $\gamma:(-1,1)\times[0,T)\rightarrow\mathbb{R}^2$ with free boundary supported on parallel lines $\{\eta_1, \eta_2\}:\mathbb{R}\rightarrow\mathbb{R}^2$ evolving by the curve diffusion flow and the curve straightening flow. The evolving curves are orthogonal to the boundary and satisfy a no-flux condition. We give estimates and monotonicity on the normalised oscillation of curvature, yielding global results for the flows.Comment: 35 pages, 3 figure

The surface diffusion and the Willmore flow for uniformly regular hypersurfaces

We consider the surface diffusion and Willmore flows acting on a general class of (possibly non-compact) hypersurfaces parameterized over a uniformly regular reference manifold possessing a tubular neighborhood with uniform radius. The surface diffusion and Willmore flows each give rise to a fourth-order quasilinear parabolic equation with nonlinear terms satisfying a specific singular structure. We establish well-posedness of both flows for initial surfaces that are $C^{1+\alpha}$-regular and parameterized over a uniformly regular hypersurface. For the Willmore flow, we also show long-term existence for initial surfaces which are $C^{1+\alpha}$-close to a sphere, and we prove that these solutions become spherical as time goes to infinity.Comment: 22 page

Combinatorial Continuous Maximal Flows

Maximum flow (and minimum cut) algorithms have had a strong impact on computer vision. In particular, graph cuts algorithms provide a mechanism for the discrete optimization of an energy functional which has been used in a variety of applications such as image segmentation, stereo, image stitching and texture synthesis. Algorithms based on the classical formulation of max-flow defined on a graph are known to exhibit metrication artefacts in the solution. Therefore, a recent trend has been to instead employ a spatially continuous maximum flow (or the dual min-cut problem) in these same applications to produce solutions with no metrication errors. However, known fast continuous max-flow algorithms have no stopping criteria or have not been proved to converge. In this work, we revisit the continuous max-flow problem and show that the analogous discrete formulation is different from the classical max-flow problem. We then apply an appropriate combinatorial optimization technique to this combinatorial continuous max-flow CCMF problem to find a null-divergence solution that exhibits no metrication artefacts and may be solved exactly by a fast, efficient algorithm with provable convergence. Finally, by exhibiting the dual problem of our CCMF formulation, we clarify the fact, already proved by Nozawa in the continuous setting, that the max-flow and the total variation problems are not always equivalent.Comment: 26 page

Capacitive flows on a 2D random net

This paper concerns maximal flows on $\mathbb{Z}^2$ traveling from a convex set to infinity, the flows being restricted by a random capacity. For every compact convex set $A$, we prove that the maximal flow $\Phi(nA)$ between $nA$ and infinity is such that $\Phi(nA)/n$ almost surely converges to the integral of a deterministic function over the boundary of $A$. The limit can also be interpreted as the optimum of a deterministic continuous max-flow problem. We derive some properties of the infinite cluster in supercritical Bernoulli percolation.Comment: 20 pages, 1 figure published in The Annals of Applied Probability http://www.imstat.org/aap