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

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0101200000000000011111111111110000 111101010101010101010101Figure 6: Fattening of a setProperties 6.12• (Ξ, d H ) is path-metric;• d H (A, B) = supx∈lR N |u A (x) − u B (x)| .• each family of compact sets that are contained in a fixed large closed ballin lR N is compact in (Ξ, d H ); so• any closed ball D(A, ρ) := {B | d H (A, B) ≤ ρ} is compact in (Ξ, d H ), andmoreover• by Theorem 6.11, minimal geodesics exist in (Ξ, d H ).6.2.1 An alternative definitionLet D r (x) be the closed ball of center x and radius r > 0 in lR N , and D r = D r (0).We define the fattened set to beA + D r := {x + y | x ∈ A, |y| ≤ r} = ⋃D r (x) = {y | u A (y) ≤ r}.Note that the fattened set is always closed, (since the distance function u A (x) iscontinuous).Example 6.13 In figure 6 we see an example of a set A fattened to r = 1, 2;the set A is the black polygon (and is filled in), whereas the dashed lines in thedrawing are the contours of the fattened sets. 7We can then state the following equivalent definition of the Hausdorff distance:x∈Ad H (A, B) = min{δ > 0 | A ⊂ (B + D δ ), B ⊂ (A + D δ )} .7 The fattened sets are not drawn filled — otherwise they would cover A.44
6.2.2 Uncountable many geodesicsUnfortunately d H is quite “unsmooth”. There are choices of A, B compact thatmay be joined by an uncountable number of geodesics. In fact we can considerthis simple example.Example 6.14 (Duci and Mennucci [15]) Let us setA = {x = 0, 0 ≤ y ≤ 2}B = {x = 2, 0 ≤ y ≤ 1}C t = { x = 1, 0 ≤ y ≤ 2} 3 ∪ {y = 0, 1 ≤ x ≤ t}ACtBwith 1 ≤ t ≤ √ 5/2;and in the picture we represent (dashed) the fattened sets A + D √ 5/2 andB + D √ 5/2 . Note that d H(A, B) = √ 5 while d H (A, C t ) = d H (B, C t ) = √ 5/2:so C t are all midpoints that are on different geodesics between A and B.6.2.3 Curves and connected setsLet again Ξ be the collection of all compact subsets of lR N . Let Ξ c be thesubclass of compact connected subsets of lR N . We now relate the space of curvesto this metric space, by listing these properties and remarks.• Ξ c is a closed subset of (Ξ, d H );• Ξ c is the closure in (Ξ, d H ) of the class of (images of) all embedded curves.• Ξ c is Lipschitz-path-connected 8 ;• for all above reasons, it is possible to connect any two A, B ∈ Ξ c by aminimal geodesic moving in Ξ c .• So, if we try to find a minimal geodesic connecting two curves using themetric d H , we will end up finding a geodesic in (Ξ c , d H ); and similarly ifwe try to optimize an energy defined on curves.• But note that Ξ c is not geodesically convex in Ξ, that is, there exist twopoints A, B ∈ Ξ c and a minimal geodesics ξ connecting A to B in themetric space (Ξ, d), such that the image of ξ is not contained inside Ξ c .• We don’t know if (Ξ c , d H ) is path-metric.8 That is, any A, B ∈ Ξ c can be connected by a Lipschitz arc γ : [0, 1] → Ξ c45
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
Page 9 and 10: In the case of planar curves c 1 ,
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
Page 17 and 18: 2.4.5 Centroid energyWe will now pr
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 40 and 41: 5 Representation/embedding/quotient
Page 42 and 43: 6.1.1 Length induced by a distanceI
Page 46 and 47: 6.2.4 Applications in computer visi
Page 48 and 49: of a small ball from A. The motion
Page 50 and 51: CutΩ(The arrows represent the dist
Page 52 and 53: 7.3 L 1 metric and Plateau problemI
Page 54 and 55: Definition 8.3 (Flat curves) Let Z
Page 56 and 57: Proof. Fix α 0 ∈ S \ Z. Let T =
Page 58 and 59: 8.2.2 RepresentationThe Stiefel man
Page 60 and 61: 9.3 Conformal metricsYezzi and Menn
Page 62 and 63: 10.1.1 Related worksA family of met
Page 64 and 65: Definition 10.7 (Convolution) A arc
Page 66 and 67: and̂∇˜HjE(0) = ̂∇ H 0E(0),̂
Page 68 and 69: and we simply integrate twice! More
Page 70 and 71: Corollary 10.18 In particular, the
Page 72 and 73: 10.6 Existence of gradient flowsWe
Page 74 and 75: • The length functional (from C 1
Page 76 and 77: We can eventually estimate the diff
Page 78 and 79: 7.552.5-10 -7.5 -5 -2.5 2.5 5 7.5 1
Page 80 and 81: By using (i) and (ii) from lemma 10
Page 82 and 83: 10.6.3 Existence of flow for geodes
Page 84 and 85: 10.7.1 Robustness w.r.to local mini
Page 86 and 87: We already presented all the calcul
Page 88 and 89: 10.9 New regularization methodsTypi
Page 90 and 91: can be solved for k (this is not so
Page 92 and 93: 2. Now• at t = 1/2 it achieves th
Page 94 and 95:
Definition 11.10 • The orbit is O
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y eqn. (11.3), where the terms RHS
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So Prop. 11.19 guarantees that the
Page 100 and 101:
Theorem 11.23 Suppose that the Riem
Page 102 and 103:
2.2.3 Examples of geometric energy
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9 Riemannian metrics of immersed cu
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contour continuation, 11convolution
Page 108 and 109:
normal vector, 6objective function,
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References[1] Luigi Ambrosio, Giuse
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[30] Serge Lang. Fundamentals of di
Page 114 and 115:
[57] Ganesh Sundaramoorthi, Anthony
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Metrics of curves in shape optimization and analysis - Andrea Carlo ...

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