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

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2.2.3 Examples of geometric energy functionals for segmentation 122.3 Geodesic active contour method . . . . . . . . . . . . . . . . . . . 132.3.1 The “geodesic active contour” paradigm . . . . . . . . . . 132.3.2 Example: geodesic active contour edge model . . . . . . . 132.3.3 Implicit assumption of H 0 inner product . . . . . . . . . . 142.4 Problems & goals . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4.1 Example: geometric heat flow . . . . . . . . . . . . . . . . 152.4.2 Short length bias . . . . . . . . . . . . . . . . . . . . . . . 152.4.3 Average weighted length . . . . . . . . . . . . . . . . . . . 162.4.4 Flow computations . . . . . . . . . . . . . . . . . . . . . . 162.4.5 Centroid energy . . . . . . . . . . . . . . . . . . . . . . . 172.4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Basic mathematical notions 203.1 Distance and metric space . . . . . . . . . . . . . . . . . . . . . . 203.1.1 Metric space . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Banach, Hilbert and Fréchet spaces . . . . . . . . . . . . . . . . . 213.2.1 Examples of spaces of functions . . . . . . . . . . . . . . . 223.2.2 Fréchet space . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.3 More examples of spaces of functions . . . . . . . . . . . . 233.2.4 Dual spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.5 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.6 Troubles in calculus . . . . . . . . . . . . . . . . . . . . . 253.3 Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3.1 Submanifold . . . . . . . . . . . . . . . . . . . . . . . . . 263.4 Tangent space and tangent bundle . . . . . . . . . . . . . . . . . 263.5 Fréchet Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.6 Riemann & Finsler geometry . . . . . . . . . . . . . . . . . . . . 273.6.1 Riemann metric, length . . . . . . . . . . . . . . . . . . . 283.6.2 Finsler metric, length . . . . . . . . . . . . . . . . . . . . 283.6.3 Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.6.4 Minimal geodesics . . . . . . . . . . . . . . . . . . . . . . 293.6.5 Exponential map . . . . . . . . . . . . . . . . . . . . . . . 293.6.6 Hopf–Rinow theorem . . . . . . . . . . . . . . . . . . . . . 303.6.7 Drawbacks in infinite dimensions . . . . . . . . . . . . . . 303.7 Gradient in abstract differentiable manifold . . . . . . . . . . . . 313.8 Group actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.8.1 Distances and groups . . . . . . . . . . . . . . . . . . . . 334 Spaces and metrics of curves 334.1 Classes of curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 Gâteaux differentials in the space of immersed curves . . . . . . . 354.3 Group actions on curves . . . . . . . . . . . . . . . . . . . . . . . 374.4 Two categories of shape spaces . . . . . . . . . . . . . . . . . . . 374.5 Geometric curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.5.1 Research path . . . . . . . . . . . . . . . . . . . . . . . . . 38102
4.6 Goals (revisited) . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.6.1 Well posedness . . . . . . . . . . . . . . . . . . . . . . . . 394.6.2 Are the goal properties consistent? . . . . . . . . . . . . . 395 Representation/embedding/quotient in the current literature 405.1 The representation/embedding/quotient paradigm . . . . . . . . 405.2 Current literature . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 Metrics of sets 416.1 Some more math on distance and geodesics . . . . . . . . . . . . 416.1.1 Length induced by a distance . . . . . . . . . . . . . . . . 426.1.2 Minimal geodesics . . . . . . . . . . . . . . . . . . . . . . 426.1.3 Existence of geodesics and of average . . . . . . . . . . . . 426.1.4 Geodesic rays . . . . . . . . . . . . . . . . . . . . . . . . . 436.1.5 Hopf–Rinow , Cohn-Vossen Theorem . . . . . . . . . . . . 436.2 Hausdorff metric . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.2.1 An alternative definition . . . . . . . . . . . . . . . . . . . 446.2.2 Uncountable many geodesics . . . . . . . . . . . . . . . . 456.2.3 Curves and connected sets . . . . . . . . . . . . . . . . . . 456.2.4 Applications in computer vision . . . . . . . . . . . . . . . 466.3 A Hausdorff-like distance of compact sets . . . . . . . . . . . . . 466.3.1 Analogy with the Hausdorff metric, L p vs L ∞ . . . . . . 466.3.2 Contingent cone . . . . . . . . . . . . . . . . . . . . . . . 476.3.3 Riemannian metric . . . . . . . . . . . . . . . . . . . . . . 486.3.4 Polar coordinates of smooth convex sets . . . . . . . . . . 486.3.5 Smooth deformations of a convex set . . . . . . . . . . . . 496.3.6 Pullback of the metric on convex boundaries . . . . . . . 496.3.7 Pullback of the metric on smooth contours . . . . . . . . 506.3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 507 Finsler geometries of curves 517.1 Tangent and normal . . . . . . . . . . . . . . . . . . . . . . . . . 517.2 L ∞ metric and Fréchet distance . . . . . . . . . . . . . . . . . . . 517.3 L 1 metric and Plateau problem . . . . . . . . . . . . . . . . . . . 528 Riemannian metrics of curves up to pose 528.1 Shape Representation using direction functions . . . . . . . . . . 528.1.1 Multiple measurable representations . . . . . . . . . . . . 548.1.2 Continuous vs measurable lifting — no winding . . . . . . 558.2 A metric with explicit geodesics . . . . . . . . . . . . . . . . . . . 568.2.1 Curves up to rotation ↔ Grassmanian . . . . . . . . . . . 578.2.2 Representation . . . . . . . . . . . . . . . . . . . . . . . . 588.2.3 Quotient w.r.to representation . . . . . . . . . . . . . . . 58103
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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
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
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 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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