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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.