Metrics of curves in shape optimization and analysis - Andrea Carlo ...

10.1.1 Related worksA family of metrics similar to H j above (but for the length dependent scalefactors 15 ) was studied (independently) in Michor and Mumford [36]: the Sobolevtypeweak Riemannian metric on Imm(S 1 , R 2 )∫〈h, k〉 Gjc=j∑〈Dsh, i Dsk〉 i ds ;c i=0in that paper the geodesic equation, horizontality, conserved momenta, lower andupper bounds on the induced distance, and scalar curvatures are computed. Notethat this metric is locally equivalent to the above metrics defined in equation(10.2), (10.3).Charpiat et al in [10, 11] studied (again independently) some generalizedmetrics and relative gradient flows; in particular they defined the Sobolev-typemetric∫〈h 1 , h 2 〉 + 〈D s h 1 , D s h 2 〉 ds .10.1.2 Properties of H j metricscThis is a list of important properties that will be discussed in the followingsections.• Flow regularization: Sobolev gradient flows are smoother than H 0 flows.• PDE order reducing property: Sobolev gradient flows are lower order thanH 0 flows.• SAC does not require derivatives of the curve to be defined for manycommonly used region-based and edge-based energies.• Coarse-to-fine deformation property: a SAC automatically favors coarsescaledeformations before moving to fine-scale deformations; this is idealfor visual tracking.• Sobolev-type norms induce a well-defined distance on space of curves;• moreover the structure of the completion of the space of immersed curvesw.r.to H 1 and H 2 norm is fairly well understood. So they offer a consistenttheory of shape optimization and shape analysis.10.2 Mathematical propertiesWe start summarizing the main mathematical properties, that were presented in[35] mostly. We first of all cite this lemma.15 Though, a scale-invariant Sobolev metric is proposed in [36] in §4.8 as a sensible generalization.62