We extend the scope of analysis for linesearch optimization algorithms on (possibly infinitedimensional) Riemannian manifolds to the convergence analysis of the BFGS quasi-Newton scheme and the Fletcher-Reeves conjugate gradient iteration. Numerical implementations for exemplary problems in shape spaces show the practical applicability of these methods.

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Abstract.In dimension one it is proved that the solution to a total variation-regularized least-squares problem is always a function which is "constant almost everywhere", provided that the data are in a certain sense outside the range of the operator to be inverted. A similar, but weaker result is derived in dimension two.Mathematics Subject Classification. 26B30, 46N10, 47A52, 49J52, 49N45, 65K10.

The problem of segmentation of a given image using the active contour technique is considered. An abstract calculus to find appropriate speed functions for active contour models in image segmentation or related problems based on variational principles is presented. The speed method from shape sensitivity analysis is used to derive speed functions which correspond to gradient or Newton-type directions for the underlying optimization problem. The Newton-type speed function is found by solving an elliptic problem on the current active contour in every time step. Numerical experiments comparing the classical gradient method with Newton's method are presented.