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

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We can eventually estimate the difference |k 1 (τ) − k 2 (τ)| by using e.g. theinequality|A 2 B 2 C 2 − A 1 B 1 C 1 | ≤ |A 1 B 1 | |C 2 − C 1 | + |A 1 C 2 | |B 2 − B 1 | + |B 2 C 2 | |A 2 − A 1 |on the difference of the integrands( 12 − l 2(τ) − l 2 (θ))h 2 (θ) |c ′L} {{ 2 } {{ } }2(θ)|{{ }}B 2 C 2A 2( 1−2 − l 1(τ) − l 1 (θ))L 1} {{ }A 1to obtain that the above term is less or equal than‖h 1 ‖ 0 ‖c ′ 2 − c ′ 1‖ 0 + ‖c ′ 2‖ 0 ‖h 2 − h 0 ‖ 0 + ‖h 2 ‖ 0 ‖c ′ 2‖ 0∣ ∣∣∣ l 2 (θ) − l 2 (τ)L 2h 1 (θ) |c ′} {{ } }1(θ)|{{ }B 1 C 1− l ∣1(θ) − l 1 (τ) ∣∣∣L 1since |A i | ≤ 1/2. In turn (since the formulas defining k 1 (τ), k 2 (τ) the parametersare bound by τ − 2π ≤ θ ≤ τ) then|A 2 − A 1 | =l 2 (τ) − l 2 (θ)∣− l ∣ 1(τ) − l 1 (θ) ∣∣∣ ∫ τ∫ τ∣ θ=|c′ 2(x)| dxθ−|c′ 1(x)| dx ∣∣∣∣≤L 2L 1 ∣ L 2L 1≤ π L 1 + L 2‖c ′ 1 − c ′L 1 L2‖ 0 + π ‖c′ 1‖ 0 + ‖c ′ 2‖ 0|L 1 − L 2 | ≤2 L 1 L 2≤ 4π 2 ‖c′ 1‖ 0 + ‖c ′ 2‖ 0L 1 L 2‖c ′ 1 − c ′ 2‖ 0(by equations (ii) and (i) in lemma 10.25). Summarizing(|k 1 (τ) − k 2 (τ)| ≤ 2π ‖h 1 ‖ 0 + ‖h 2 ‖ 0 ‖c ′ 2‖ 0 4π 2 ‖c′ 1‖ 0 + ‖c ′ )2‖ 0‖c ′ 1 − c ′L 1 L2‖ 0 +2+ 2π‖c ′ 2‖ 0 ‖h 2 − h 0 ‖ 0 .If we derive, k i ′(θ) = h i(θ)|c ′ i (θ)|, so|k ′ 2(θ)−k ′ 2(θ)| ≤ |h 2(θ)| + |h 1 (θ)|2Symmetrizing we prove (10.30).|c ′ 2(θ)−c ′ 1(θ)| + |c′ 2(θ)| + |c ′ 1(θ)||h ′22(θ)−h ′ 1(θ)| .Lemma 10.27 Conversely, let c 1 , c 2 be two C 1 immersed curves, and h 1 , h 2be two differentiable fields; then (by using once again eqn. (i) from the lemma10.25)‖D c1 h 1 −D c2 h 2 ‖ 0 ≤ ‖c 1‖ 1 + ‖c 2 ‖ 12ε 2 ‖h 1 −h 2 ‖ 1 + ‖h 1‖ 1 + ‖h 2 ‖ 12ε 2 ‖c 1 −c 2 ‖ 1 (10.35)where ε = min(inf S 1 |c ′ 1|, inf S 1 |c ′ 2|).76
10.6.2 Existence of flow for the centroid energy (2.9)Theorem 10.29 Let us fix v ∈ lR n , let E(c) = 1 2 |avg c(c) − v| 2 ; let the initialcurve c 0 ∈ C 1 , then the ˜H 1 gradient descent flow of E has an unique solutionC = C(t, θ), for all t ∈ lR, and C(t, ·) ∈ C 1 .Results of a numerical simulation of the above gradient descent flow areshown in figure 12 on the following page.Before proving the theorem, let us comment on an interesting property ofthe above gradient flow.Remark 10.30 The length of the curves len(C(t, ·)) is constant during theevolution in t.Proof. Indeed it is easy to prove that ∂ t len(C(t, ·)) = 0: the Gâteaux differentialof the length of a curve c is∫D len(c; h) = (D s c · D s h) ds ;substituting the value of D s C(t, ·)) from eqn. (10.25),c∂ t len(C(t, ·)) = D(len(C))(∂ t C)∫1 〈=λL 2 D s C, ( D s C〈(avg c (C) − v), (C − avg c (C))〉 + α )〉 ds =〈∫〉 ∫1=λL 2 (avg c (C) − v), (C − avg c (C)) ds + (D s C · α) ds = 0 .We now present the proof of the theorem 10.29.Proof. We will first of all prove existence and uniqueness of the gradient flow forsmall time, using the Cauchy–Lipschitz theorem 25 in the space M of immersedcurves, seen as an open subset of the C 1 Banach space (see Definition 10.24).To this end we will prove that c ↦→ ∇˜H1E(c) is a locally Lipschitz functionalfrom C 1 into itself. Afterward, we will directly prove that the solution exists forall times.So, let us fix c 1 , c 2 ∈ C 1 ∩ M that are two curves near c 0 . Let us fix ε > 0 byε := infθ |c′ 0(θ)|/2 .By “near” we mean that, in all of the following, we will require that ‖c i −c 0 ‖ 1 < ε,for i = 1, 2. We will need the following (easy to prove) facts. (In the following,the index i will represent both 1, 2).• inf θ |c ′ i (θ)| ≥ ε, so all curves in the neighborhood are immersed.25 A.k.a. as the Picard–Lindelöf theorem77
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Metrics of curves in shape optimiza
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shape analysis where we study a fam
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• F (c) = F (c ◦ φ) for all cu
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κ > 0HNNHNHκ < 0Figure 1: Example
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In the case of planar curves c 1 ,
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(a) (b) (c) (d) (e)Figure 3: Segmen
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where φ may be chosen to beφ(x) =
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2.4.1 Example: geometric heat flowW
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2.4.5 Centroid energyWe will now pr
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shapes. Unfortunately, H 0 does not
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• If the second request is waived
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• 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 44 and 45: 0101200000000000011111111111110000
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 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
Page 96 and 97: y eqn. (11.3), where the terms RHS
Page 98 and 99: 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
Page 104 and 105: 9 Riemannian metrics of immersed cu
Page 106 and 107: contour continuation, 11convolution
Page 108 and 109: normal vector, 6objective function,
Page 110 and 111: References[1] Luigi Ambrosio, Giuse
Page 112 and 113: [30] Serge Lang. Fundamentals of di
Page 114 and 115: [57] Ganesh Sundaramoorthi, Anthony
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