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

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contour continuation, 11convolution by arc parameter, 64convolutional kernel, 64cotangent space T ∗ c M, 31critical geodesic, 29, 30curvature, 6, 59in BV, 59Gâteaux differential of —, 36in L 2 , 63mean —, 6sectional —, 31signed scalar —, 6, 48, 59curve, 34classes of, 34embedded, 34freely immersed, 34image of a —, 3, 8, 34, 45immersed, 3, 34manifold of —, 3planar —, 3curve-wise parameterization invariant,92, 95, 96curvesup to pose, 52cutlocus, 50de la Vallée-Poussin, 70, 71degenerate, 91derivation with respect to the arc parameter,5, 61, 69diffeomorphism, 4, 26differentiable manifoldabstract, 25submanifold, 26dimension, 25Dirac’s delta, 32, 70directional derivative, 13, 17, 26, 31distance, 20, 28distance function, 8, 43, 46distance-based average, 9, 42distribution, 24, 32doubly traversed circle, 34dual space, 24edge detection, 11edge-based, 11, 12, 15, 86, 87Eells, 30elastic regularization, 86elastica energy, 36, 66, 86, 88Elworthy, 30embedded curves, 34embedding theorems, 26Emmy Noether, 99empirical principal component analysis,10energy, 5, 28of a path, 28equivalence classes, 4equivalence relation, 32Euclideandistance, 21invariance, 39, 61Euclidean group, 37, 100Euclidean norm, 5evaluation functional, 32exponential map, 10, 29external skeleton, 50fattened set, 44Finsler metric, 28Fréchet mean, 9, 42Fréchet distance, 51, 63Fréchet manifolds, 27Fréchet space, 23, 24freely immersed curve, 34Gâteaux differentiable, 24, 31Gâteaux differential, 13, 15, 16, 24, 35,70of the curvature, 36of the elastica energy, 36of the length, 78Gaussian, 13geodesic active contour, 12, 71–73, 82geodesic distance, 54geodesic ray, 43geometric, 4, 61geometric active contour, 19geometric curves, 4, 38geometric distance, 93geometric energy, 97geometric evolution, 11106
geometric functional, 4, 5geometric heat flow, 15, 16, 86geometric oriented curves, 4gradient ∇E(c), 31gradient descentfor curve length, 15gradient descent flow, 11, 13, 72Grassmanian manifold, 57group action, 32, 99on curves, 37group operation, 4Hadamard, 31Hausdorff, 23Hausdorff distance, 8, 44heat equation, 15Hilbert space, 22homeomorphism, 20, 25homotopy, 3homotopy-wise parameterization invariant,93, 95Hopf, 30, 43, 44horizontal projection, 94horizontal space, 94, 97horizontality, 92horizontally projected metric, 95ill-posed, 16, 17, 19, 85, 86, 88imageblack and white —, 12of a curve, 3, 8, 34, 45segmentation, 11smooth and featureless, 15synthetic noisy, 83immersed curve, 3immersed curves, 34implicit function theorem, 26induced geodesic distance, 42inflationary term, 15integral length, 28, 42integration by arc parameter, 6, 12invarianceEuclidean, 39, 61parameterization, see “reparameterizationinvariant”rescaling, 39, 61rotation, 39, 61translation, 39, 61invariant w.r.to the action of the group,33Karcher mean, 9, 42Karhunen-Loève theorem, 10kernel, 64l.c.t.v.s., 22, 24–26landmark, 9Lebesgue measure, 22length, 5, 28, 42length increasing, 87length shrinking effect, 15level set, 46level set averaging, 9level set method, 11, 12, 15, 19, 65, 90,98Lie algebra, 99Lie groups, 99lifing lemma, 97for H 0 , 90locally compact, 43locally-convex topological vector space,22M, 3manifold of (parametric) curves, 3matrix, 37, 48antisymmetric, 100, 101orthogonal, 37, 100mean curvature, 6measurable lifting, 54measurable representation, 54metric space, 21metrizable, 23minimal geodesic, 29, 42momentum, 100angular, 101linear, 101motion by mean curvature, 15neighborhood, 30, 55, 78, 84, 89Noether, Emmy, 99norm, 21, 28107
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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
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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 104 and 105: 9 Riemannian metrics of immersed cu
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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