Computing metamorphoses between discrete measures
The paper under review is a continuation of the analysis developed in [\textit{D. D. Holm} et al., Q. Appl. Math. 67, No. 4, 661-685 (2009; Zbl 1186.68413)]. There the authors formulated metamorphosis, that is, diffeomorphic pattern matching using distances on a space of images or shapes, in an abstract Lagrangian setting and derived the general evolution equations. The key point is that the metamorphosis equations contain the equations for a perfect complex fluid. A main question was left open in the case of matching point sets. An important case because of the extended use of landmarks to match images, for example. The authors proposed in the cited paper that whenever the template and the target were sums of Dirac measures, the geodesic evolution would remain a measure at each time, and this measure could be expressed as a sum of Dirac measures plus an absolutely continuous part. All this is rigorously proved in the paper under review. Nevertheless, beside Dirac measures, the setting needs to be augmented with distributions of Calderón-Zygmund type to obtain the right results. These additional singularities complicate the numerical approximation of the solutions, but this drawback is solved thanks to a Eulerian numerical method where the solutions are approximated using smooth densities and adapting then the discretization used for metamorphosis of smooth \(L^2\) densities. Finally, numerical experiments illustrate the method.
scientific article; zbMATH DE number 6170795
scientific article