2021/12/09 by Hsi-Wei Hsieh, Hsieh, Hsi-Wei, Nicolas Charon +1
Engineering · Mathematics · #3D Shape Modeling and Analysis #FOS: Mathematics #Geometric Analysis and Curvature Flows #Morphological variations and asymmetry #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2112.04644
openalex publication_date 2021/12/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.