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

5 Representation/embedding/quotient in the currentliterature5.1 The representation/embedding/quotient paradigmA common way to model shapes is by representation/embedding:• we represent the shape A by a function u A• and then we embed this representation in a space E, so that we canoperate on the shapes A by operating on the representations u A .Most often, representation/embedding alone do not directly provide a satisfactoryshape space. In particular, in many cases it happens that the representationis “redundant”, that is, the same shape has many different possible representations.An appropriate quotient is then introduced.There are many examples of shape spaces in the literature that are studiedby means of the representation/embedding/quotient scheme. Understanding thebasic math properties of these three operations is then a key step in understandingshape spaces and designing/improving them.We now present a rapid overview of how this scheme is exploited in thecurrent literature of shape spaces; then we will come back to some of them toexplain more in depth.5.2 Current literatureExample 5.1 (The family of all non empty compact subsets of lR N ) Astandard representation is obtained by associating each closed subset A to thedistance function u A or the signed distance function b A (that were defined inDefinition 2.1). We may then define a topology of shapes by deciding thatA n → A when u An → u A uniformly on compact sets. This convergence coincideswith the Kuratowski topology of closed sets; if we limit shapes to be compact sets,the Kuratowski topology is induced by the Hausdorff distance. See section 6.2.Example 5.2 Trouvé-Younes et al (see Glaunès et al. [22], Trouvé and Younes[59] and references therein) modeled the motion of shapes by studying a leftinvariant Riemannian metric on the family G of diffeomorphisms of the spacelR N ; to recover a true metric of shapes, a quotient is then performed w.r.to alldiffeomorphisms G 0 that do not move a template shape.But the representation/embedding/quotient scheme is also found when dealingwith spaces of curves:Example 5.3 (Representation by angle function) In the work of Klassenet al. [28], Srivastava et al. [52], Mio and Srivastava [41], smooth planar closedcurves c : S 1 → lR 2 of length 2π are represented by a pair of angle-velocityfunctions c ′ (u) = exp(φ(u) + iα(u)) (identifying lR 2 = IC) then (φ, α) areembedded as a subset N in L 2 (0, 2π) or W 1,2 (0, 2π). Since the goal is to obtain40