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Fidelity metrics between curves and surfaces: currents, varifolds, and normal cycles. (English) Zbl 1453.92156

Pennec, Xavier (ed.) et al., Riemannian geometric statistics in medical image analysis. Amsterdam: Elsevier/Academic Press. Elsevier Miccai Soc. Book Ser., 441-477 (2020).
Summary: Following the classical setting proposed by Grenander, metrics on shape spaces were defined through the action of diffeomorphism groups equipped with right-invariant metrics. In particular, the LDDMM framework introduced earlier provides a convenient way to generate diffeomorphic transformations and such right-invariant metrics. In that case the resulting distance between two given shapes is given through the solution of an exact registration problem obtained by optimizing the deformation cost over all possible deformation fields that match the source shape on the target.
This approach, however, only applies if both shapes belong to the same orbit; in other words, if there exists a deformation in the group that can exactly deform one shape on the other. Such an assumption is routinely violated in practical scenarios involving shapes extracted from biomedical imaging data. Indeed, those shapes are typically affected by many other variations including noise, potential topological variations, or segmentation artifacts, all of which are poorly modeled within a pure diffeomorphic setting. From a statistical perspective it is in fact more reasonable to make the computation of shape distances rather as insensitive as possible to those types of perturbations that are not morphologically relevant.
For the entire collection see [Zbl 1428.92004].

MSC:

92C55 Biomedical imaging and signal processing
58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
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