Metrics of curves in shape optimization and analysis - Andrea Carlo ...
Metrics of curves in shape optimization and analysis - Andrea Carlo ... 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
a shape space representation for shape analysis purposes, a quotient is thenintroduced on N. See Section 8.1.Example 5.4 Another representation of planar curves for shape analysis isfound in Younes [68]. In this case the angle function is considered mod(π). Thisrepresentation is both simple and very powerful at the same time. Indeed, it ispossible to prove that geodesics do exist and to explicitly show examplesof geodesics. See Section 8.2.Example 5.5 (Harmonic representation) A. Duci et al (see [16, 17]) representa closed planar contour as the zero level of a harmonic function. This novelrepresentation for contours is explicitly designed to possess a linear structure,which greatly simplifies linear operations such as averaging, principal componentanalysis or differentiation in the space of shapes.And, of course, we have in this list the spaces of embedded curves.Example 5.6 When studying embedded curves, usually, for the sake of mathematicalanalysis, the curves are modeled as immersed parametric curves; aquotient w.r.to the group of possible reparameterizations of the curve c (thatcoincides with the group of diffeomorphisms Diff(S 1 )) is applied afterward to allthe mathematical structures that are defined (such as the manifold of curves, theRiemannian metric, the induced distance, etc.).It is interesting to note this fact.Remark 5.7 (A remark on the quotienting order) When we started talkingof geometric curves, we proposed the quotient B = M/Diff(S 1 ) (the space ofcurves up to reparameterization); and this had to be followed by other quotientsw.r.to Euclidean motions and/or scaling, to obtain M/Diff(S 1 )/E(n)/lR + . Inpractice, though, the space B happens to be more difficult to study; hence mostshape space theories that deal with curves prefer a different order: the quotientM/E(n)/lR + is modeled and studied first; then the quotient M/E(n)/lR + /Diff(S 1 )is performed (often, only numerically).6 Metrics of setsWe now present two examples of Shape Theories where a shape may be a genericsubset of the plane; with particular attention to how they behave w.r.to curves.6.1 Some more math on distance and geodesicsWe start by reviewing some basic results in abstract metric spaces theory.41
- Page 1 and 2: Metrics of curves in shape optimiza
- Page 3 and 4: shape analysis where we study a fam
- Page 5 and 6: • F (c) = F (c ◦ φ) for all cu
- Page 7 and 8: κ > 0HNNHNHκ < 0Figure 1: Example
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- Page 11 and 12: (a) (b) (c) (d) (e)Figure 3: Segmen
- Page 13 and 14: where φ may be chosen to beφ(x) =
- Page 15 and 16: 2.4.1 Example: geometric heat flowW
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- Page 20 and 21: shapes. Unfortunately, H 0 does not
- Page 22 and 23: • If the second request is waived
- Page 24 and 25: • The Fréchet space of smooth fu
- Page 26 and 27: Since φ k are homeomorphisms, then
- Page 28 and 29: 3.6.1 Riemann metric, lengthDefinit
- Page 30 and 31: Theorem 3.30 Suppose that M is a sm
- Page 32 and 33: Example 3.38 Let M = C ∞ ([−1,
- Page 34 and 35: • sometimes S 1 will be identifie
- Page 36 and 37: The proof is by direct computation.
- Page 38 and 39: The term preshape space is sometime
- Page 42 and 43: 6.1.1 Length induced by a distanceI
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- Page 54 and 55: Definition 8.3 (Flat curves) Let Z
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a <strong>shape</strong> space representation for <strong>shape</strong> <strong>analysis</strong> purposes, a quotient is then<strong>in</strong>troduced on N. See Section 8.1.Example 5.4 Another representation <strong>of</strong> planar <strong>curves</strong> for <strong>shape</strong> <strong>analysis</strong> isfound <strong>in</strong> Younes [68]. In this case the angle function is considered mod(π). Thisrepresentation is both simple <strong>and</strong> very powerful at the same time. Indeed, it ispossible to prove that geodesics do exist <strong>and</strong> to explicitly show examples<strong>of</strong> geodesics. See Section 8.2.Example 5.5 (Harmonic representation) A. Duci et al (see [16, 17]) representa closed planar contour as the zero level <strong>of</strong> a harmonic function. This novelrepresentation for contours is explicitly designed to possess a l<strong>in</strong>ear structure,which greatly simplifies l<strong>in</strong>ear operations such as averag<strong>in</strong>g, pr<strong>in</strong>cipal component<strong>analysis</strong> or differentiation <strong>in</strong> the space <strong>of</strong> <strong>shape</strong>s.And, <strong>of</strong> course, we have <strong>in</strong> this list the spaces <strong>of</strong> embedded <strong>curves</strong>.Example 5.6 When study<strong>in</strong>g embedded <strong>curves</strong>, usually, for the sake <strong>of</strong> mathematical<strong>analysis</strong>, the <strong>curves</strong> are modeled as immersed parametric <strong>curves</strong>; aquotient w.r.to the group <strong>of</strong> possible reparameterizations <strong>of</strong> the curve c (thatco<strong>in</strong>cides with the group <strong>of</strong> diffeomorphisms Diff(S 1 )) is applied afterward to allthe mathematical structures that are def<strong>in</strong>ed (such as the manifold <strong>of</strong> <strong>curves</strong>, theRiemannian metric, the <strong>in</strong>duced distance, etc.).It is <strong>in</strong>terest<strong>in</strong>g to note this fact.Remark 5.7 (A remark on the quotient<strong>in</strong>g order) When we started talk<strong>in</strong>g<strong>of</strong> geometric <strong>curves</strong>, we proposed the quotient B = M/Diff(S 1 ) (the space <strong>of</strong><strong>curves</strong> up to reparameterization); <strong>and</strong> this had to be followed by other quotientsw.r.to Euclidean motions <strong>and</strong>/or scal<strong>in</strong>g, to obta<strong>in</strong> M/Diff(S 1 )/E(n)/lR + . Inpractice, though, the space B happens to be more difficult to study; hence most<strong>shape</strong> space theories that deal with <strong>curves</strong> prefer a different order: the quotientM/E(n)/lR + is modeled <strong>and</strong> studied first; then the quotient M/E(n)/lR + /Diff(S 1 )is performed (<strong>of</strong>ten, only numerically).6 <strong>Metrics</strong> <strong>of</strong> setsWe now present two examples <strong>of</strong> Shape Theories where a <strong>shape</strong> may be a genericsubset <strong>of</strong> the plane; with particular attention to how they behave w.r.to <strong>curves</strong>.6.1 Some more math on distance <strong>and</strong> geodesicsWe start by review<strong>in</strong>g some basic results <strong>in</strong> abstract metric spaces theory.41
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