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arXiv · 2112.04644

Weight metamorphosis of varifolds and the LDDMM-Fisher-Rao metric

Hsi-Wei Hsieh·Nicolas Charon
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Abstract

This paper introduces and studies a metamorphosis framework for geometric measures known as varifolds, which extends the diffeomorphic registration model for objects such as curves, surfaces and measures by complementing diffeomorphic deformations with a transformation process on the varifold weights. We consider two classes of cost functionals to penalize those combined transformations, in particular the LDDMM-Fisher-Rao energy which, as we show, leads to a well-defined Riemannian metric on the space of varifolds with existence of corresponding geodesics. We further introduce relaxed formulations of the respective optimal control problems, study their well-posedness and derive optimality conditions for the solutions. From these, we propose a numerical approach to compute optimal metamorphoses between discrete varifolds and illustrate the interest of this model in the situation of partially missing data.

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Hsi-Wei Hsieh, Nicolas Charon. 2021-12-09. Weight metamorphosis of varifolds and the LDDMM-Fisher-Rao metric. https://arxiv.org/abs/2112.04644

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