SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Invited Talk

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Second-order Riemannian Active Contours for Image Segmentation

G. Gallego, A. Valdés and J.I. Ronda

Abstract
This work aims at providing algorithms for geodesic active contours optimization based on second-order approximations of functionals. In contrast to the standard approach based on gradient schemes, that define a time-continuous curve evolution that must be discretized, we propose algorithms that are intrinsically time-discrete. To this purpose, we obtain the second-order approximation of the Riemannian curve energy and length and show how the extrema of these approximations can be characterized by means of linear second order differential equations. Moreover, we prove that the solution of the equations for the length functional is given by a simple closed-form expression, while the damped case requires the solution of a second-order ordinary differential equation that can be considered as a generalization of the classical equation defining the Jacobi fields along the curve.

Bibliography
[2] L. Bar and G. Sapiro, Generalized newton-type methods for energy formulations in image processing, SIAM J. Imaging Sciences, vol. 2(2) (2009), pp. 508-531.
[2] M. Hintermüller and W. Ring, A Second Order Shape Optimization Approach for Image Segmentation, SIAM J. Applied Mathematics, vol. 64(2) (2004), pp. 442-467.

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