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

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7.552.5-10 -7.5 -5 -2.5 2.5 5 7.5 10-2.5-5-7.5-10321-4 -3 -2 -1 1 2 3 4-1-2-3-47.552.5-10 -7.5 -5 -2.5 2.5 5 7.5 10-2.5-5-7.5-10321-4 -3 -2 -1 1 2 3 4-1-2-3-47.552.5-10 -7.5 -5 -2.5 2.5 5 7.5 10-2.5-5-7.5-10321-4 -3 -2 -1 1 2 3 4-1-2-3-47.552.5-10 -7.5 -5 -2.5 2.5 5 7.5 10-2.5-5-7.5-10321-4 -3 -2 -1 1 2 3 4-1-2-3-47.552.5-10 -7.5 -5 -2.5 2.5 5 7.5 10-2.5-5-7.5-10321-4 -3 -2 -1 1 2 3 4-1-2-3-47.552.5-10 -7.5 -5 -2.5 2.5 5 7.5 10-2.5-5-7.5-10321-4 -3 -2 -1 1 2 3 4-1-2-3-4321-4 -3 -2 -1 1 2 3 4-1-2-3-4321-4 -3 -2 -1 1 2 3 4-1-2-3-4321-4 -3 -2 -1 1 2 3 4-1-2-3-4321-4 -3 -2 -1 1 2 3 4-1-2-3-4C(0, θ) = c 0 (θ) = ( cos(θ), sin(θ)(1 + 2 cos(θ) 4 ) )t=-1.210t=-0.810t=-0.410t=04t=0.54t=1.4t=1.54t=2.4C(0, θ) = c 0 (θ) = ( cos(θ), 2 sin(θ) )t=-1.210t=-0.810t=-0.410t=04t=0.54t=1.4t=1.54t=2.4Two numerical simulation of the ˜H 1 gradient descent flow of the centroid energyE(c) = 1 |avg 2 c(c) − v| 2 , for two different initial curves c 0. The constants were set toλ = 1/(4π 2 ), v = (2, 2). Steps for numerical discretization were set to ∆t = 1/30,∆θ = π/64 (on [0, 2π]). Note that plots for t < 0 are shown in a smaller scale.Figure 12: ˜H1 gradient descent flow of the centroid energy. See 10.20 and 10.29.78
• Setting a 0 := ‖c 0 ‖ 1 + ε, we have ‖c i ‖ 1 ≤ a 0 .• By equation (ii) in lemma 10.25,len(c 0 ) − 2π‖c 0 − c i ‖ 1 ≤ len(c i ) ≤ len(c 0 ) + 2π‖c 0 − c i ‖ 1and in particular2πε ≤ len(c i ) ≤ len(c 0 ) + 2πε .Let f i = ∇˜H1E(c i ) be the gradient, whose formula was expressed in (10.24).We decompose f i using the relation T c M = lR n ⊕ D c M, as done previously:and obtainf i = avg ci(f i ) ∈ lR n , ˜fi = f − avg ci(f i ) ∈ D ci Mf i = avg ci(c i ) − v1˜f i = P ci Π ci h i , whereλL 2 ih i := D s c i 〈(avg ci(c i ) − v), (c i − avg ci(c i ))〉 , (10.36)by rewriting (10.25) in the notation of this proof.We will then exploit all the inequalities in the lemmas to prove that|f 1 − f 2 | ≤ a 1 ‖c 1 − c 2 ‖ 1 , ‖ ˜f 1 − ˜f 2 ‖ 1 ≤ a 2 ‖c 1 − c 2 ‖ 1for two constants a 1 , a 2 > 0 and all c 1 , c 2 near c 0 ; this effectively proves that|f 1 − f 2 | ≤ (a 1 + a 2 )‖c 1 − c 2 ‖ 1that is, ∇˜H1E(c) is a locally Lipschitz functional.The first term is dealt with equation (iv) in lemma 10.25, whence we obtain∫∫|f 1 − f 2 | = | c 1 (s) ds − c 2 (s) ds| ≤ a 1 ‖c 1 − c 2 ‖ 1 with a 1 := 2a2 0c 1 c 2ε 2 .By repeated applications of the inequalities listed in lemma 10.25, we provethat, in the designed neighborhood, the following inequalities hold‖h 2 − h 1 ‖ 0 ≤ a 3 ‖c 2 − c 1 ‖ 1‖h i ‖ ≤ a 4for two constants a 3 , a 4 > 0. So we can choose constants P, Q > 0 such that theinequality (10.30) holds uniformly in the neighborhood, and then‖P c1 Π c1 h 1 − P c2 Π c2 h 2 ‖ 1 ≤ P ‖c ′ 1 − c ′ 2‖ 0 + Q ‖h 2 − h 1 ‖ 0 ≤≤ (P + Qa 3 )‖c 1 − c 2 ‖ 179
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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
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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 76 and 77: We can eventually estimate the diff
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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Metrics of curves in shape optimization and analysis - Andrea Carlo ...

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