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

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9 Riemannian metrics of immersed curves 589.1 H 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599.2 H A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599.3 Conformal metrics . . . . . . . . . . . . . . . . . . . . . . . . . . 609.4 “Rigidified” norms . . . . . . . . . . . . . . . . . . . . . . . . . . 6010 Sobolev type Riemannian metrics 6010.1 Sobolev-type metrics . . . . . . . . . . . . . . . . . . . . . . . . . 6110.1.1 Related works . . . . . . . . . . . . . . . . . . . . . . . . . 6210.1.2 Properties of H j metrics . . . . . . . . . . . . . . . . . . . 6210.2 Mathematical properties . . . . . . . . . . . . . . . . . . . . . . 6210.3 Sobolev metrics in shape optimization . . . . . . . . . . . . . . . 6310.3.1 Smoothing of gradients, coarse-to-fine flowing . . . . . . . 6510.3.2 Flow regularization . . . . . . . . . . . . . . . . . . . . . . 6610.4 ˜H j is faster than H j . . . . . . . . . . . . . . . . . . . . . . . . . 6710.5 Analysis and calculus of ˜H 1 gradients . . . . . . . . . . . . . . . 6810.6 Existence of gradient flows . . . . . . . . . . . . . . . . . . . . . . 7210.6.1 Lemmas and inequalities . . . . . . . . . . . . . . . . . . . 7310.6.2 Existence of flow for the centroid energy (2.9) . . . . . . . 7710.6.3 Existence of flow for geodesic active contour . . . . . . . . 8210.7 Regularization of energy vs regularization of flow/metric . . . . . 8310.7.1 Robustness w.r.to local minima due to noise . . . . . . . 8410.8 New shape optimization energies . . . . . . . . . . . . . . . . . . 8510.8.1 Average weighted length . . . . . . . . . . . . . . . . . . . 8510.8.2 New edge-based active contour models . . . . . . . . . . . 8610.9 New regularization methods . . . . . . . . . . . . . . . . . . . . . 8810.9.1 Elastic regularization . . . . . . . . . . . . . . . . . . . . . 8811 Mathematical properties of the Riemannian space of curves 8911.1 Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8911.2 Reparameterization to normal motion . . . . . . . . . . . . . . . 9011.3 The H 0 distance is degenerate . . . . . . . . . . . . . . . . . . . 9111.4 Existence of critical geodesics for H j . . . . . . . . . . . . . . . . 9211.5 Parameterization invariance . . . . . . . . . . . . . . . . . . . . . 9211.6 Standard and geometric distance . . . . . . . . . . . . . . . . . . 9311.7 Horizontal and vertical space . . . . . . . . . . . . . . . . . . . . 9311.8 From curve-wise parameterization to homotopy-wise . . . . . . . . 9411.8.1 Horizontal G ⊥ as length minimizer . . . . . . . . . . . . . 9611.9 A geometric gradient flow is horizontal . . . . . . . . . . . . . . . 9711.10Horizontality according to H 0 . . . . . . . . . . . . . . . . . . . . 9711.11Horizontality according to H j . . . . . . . . . . . . . . . . . . . . 9811.12Horizontality is for any group action . . . . . . . . . . . . . . . . 9811.13Momenta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9911.13.1 Conservation of momenta . . . . . . . . . . . . . . . . . . 99104
IndexC 0 , 73, 78, 82C 1 , 24, 73, 78, 82C 1 curve, 3C 1 functional between Fréchet spaces,24C 2 regular curves, 6C ∞ (I → lR n ), 24C j (I → lR n ), 23C 1,1loc , 82D s , 5, 61H 0 , 14–19, 58–61, 88, 91alternative definition, 61, 62, 64,66–68, 83–86, 88Hψ 0, 60H j , 61H j (I → lR n ), 23L ∞ metric, 51M/Diff(S 1 ), 4, 5, 41, 89, 93, 98O c , 94S 1 , 3, 33δ 0 , 32, 70˙γ(v) := ∂ v γ(v), 28∂ s , 5Diff(S 1 ), 4, 27, 37, 38, 41, 89, 93Emb(S 1 , lR n ), 34, 34Imm(S 1 , lR n ), 34, 34, 38, 62, 89Imm f (S 1 , lR n ), 34, 34, 38˜H j , 61b A , 8c ′ (θ) := ∂ θ c(θ), 3, 5d g (x, y), 54i-th mode of principal variation, 10u A , 8G acts on M, 32abstract differentiablemanifold, 25action, 4, 28, 91of a group on a distance, 33of a group on a space, 32of a group on the space of curves,37of a group on the space of curves,and momenta, 100of path in a metric, 29active contour, 11–15, 18, 19, 59, 66,73, 82, 85, 86, 88, 90arc parameter, 5arc-parameterized convolutional kernel,64arithmetic mean, 9asymmetric, 22asymmetric distance, 20atlas, 25, 39average integral, 6average shape, 9, 42average weighted length, 16, 85, 86avg, 6B, see M/Diff(S 1 )balloon model, 87Banach space, 22, 23, 78C 0 , C 1 , 73, 78, 82boundary based energies, 12Cartan, 31Cauchy–Lipschitz theorem, 78, 82center of mass, 17, 68, 70, 80centroid, 17, 68, 80centroid energy, 17, 70, 73, 76, 77Chan-Vese, 12, 18, 83, 84change of coordinatespolar, 48chart, 26charts, 25classes of curves, 34(simplified definitions), 3closed curves, 3Cohn-Vossen, 43Comparison principle, 15complete, 21, 23computer vision, 2, 5, 12, 37, 46, 90conformal metric, 58, 60consistent, 62contingent cone , 47105
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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
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3.6.1 Riemann metric, lengthDefinit
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Theorem 3.30 Suppose that M is a sm
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Example 3.38 Let M = C ∞ ([−1,
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• sometimes S 1 will be identifie
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The proof is by direct computation.
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The term preshape space is sometime
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5 Representation/embedding/quotient
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6.1.1 Length induced by a distanceI
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0101200000000000011111111111110000
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6.2.4 Applications in computer visi
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of a small ball from A. The motion
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CutΩ(The arrows represent the dist
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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
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 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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