contour cont<strong>in</strong>uation, 11convolution by arc parameter, 64convolutional kernel, 64cotangent space T ∗ c M, 31critical geodesic, 29, 30curvature, 6, 59<strong>in</strong> BV, 59Gâteaux differential <strong>of</strong> —, 36<strong>in</strong> L 2 , 63mean —, 6sectional —, 31signed scalar —, 6, 48, 59curve, 34classes <strong>of</strong>, 34embedded, 34freely immersed, 34image <strong>of</strong> a —, 3, 8, 34, 45immersed, 3, 34manifold <strong>of</strong> —, 3planar —, 3curve-wise parameterization <strong>in</strong>variant,92, 95, 96<strong>curves</strong>up to pose, 52cutlocus, 50de la Vallée-Pouss<strong>in</strong>, 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 <strong>curves</strong>, 34embedd<strong>in</strong>g theorems, 26Emmy Noether, 99empirical pr<strong>in</strong>cipal component <strong>analysis</strong>,10energy, 5, 28<strong>of</strong> a path, 28equivalence classes, 4equivalence relation, 32Euclide<strong>and</strong>istance, 21<strong>in</strong>variance, 39, 61Euclidean group, 37, 100Euclidean norm, 5evaluation functional, 32exponential map, 10, 29external skeleton, 50fattened set, 44F<strong>in</strong>sler metric, 28Fréchet mean, 9, 42Fréchet distance, 51, 63Fréchet manifolds, 27Fréchet space, 23, 24freely immersed curve, 34Gâteaux differentiable, 24, 31Gâteaux differential, 13, 15, 16, 24, 35,70<strong>of</strong> the curvature, 36<strong>of</strong> the elastica energy, 36<strong>of</strong> the length, 78Gaussian, 13geodesic active contour, 12, 71–73, 82geodesic distance, 54geodesic ray, 43geometric, 4, 61geometric active contour, 19geometric <strong>curves</strong>, 4, 38geometric distance, 93geometric energy, 97geometric evolution, 11106
geometric functional, 4, 5geometric heat flow, 15, 16, 86geometric oriented <strong>curves</strong>, 4gradient ∇E(c), 31gradient descentfor curve length, 15gradient descent flow, 11, 13, 72Grassmanian manifold, 57group action, 32, 99on <strong>curves</strong>, 37group operation, 4Hadamard, 31Hausdorff, 23Hausdorff distance, 8, 44heat equation, 15Hilbert space, 22homeomorphism, 20, 25homotopy, 3homotopy-wise parameterization <strong>in</strong>variant,93, 95Hopf, 30, 43, 44horizontal projection, 94horizontal space, 94, 97horizontality, 92horizontally projected metric, 95ill-posed, 16, 17, 19, 85, 86, 88imageblack <strong>and</strong> white —, 12<strong>of</strong> a curve, 3, 8, 34, 45segmentation, 11smooth <strong>and</strong> featureless, 15synthetic noisy, 83immersed curve, 3immersed <strong>curves</strong>, 34implicit function theorem, 26<strong>in</strong>duced geodesic distance, 42<strong>in</strong>flationary term, 15<strong>in</strong>tegral length, 28, 42<strong>in</strong>tegration by arc parameter, 6, 12<strong>in</strong>varianceEuclidean, 39, 61parameterization, see “reparameterization<strong>in</strong>variant”rescal<strong>in</strong>g, 39, 61rotation, 39, 61translation, 39, 61<strong>in</strong>variant w.r.to the action <strong>of</strong> the group,33Karcher mean, 9, 42Karhunen-Loève theorem, 10kernel, 64l.c.t.v.s., 22, 24–26l<strong>and</strong>mark, 9Lebesgue measure, 22length, 5, 28, 42length <strong>in</strong>creas<strong>in</strong>g, 87length shr<strong>in</strong>k<strong>in</strong>g effect, 15level set, 46level set averag<strong>in</strong>g, 9level set method, 11, 12, 15, 19, 65, 90,98Lie algebra, 99Lie groups, 99lif<strong>in</strong>g lemma, 97for H 0 , 90locally compact, 43locally-convex topological vector space,22M, 3manifold <strong>of</strong> (parametric) <strong>curves</strong>, 3matrix, 37, 48antisymmetric, 100, 101orthogonal, 37, 100mean curvature, 6measurable lift<strong>in</strong>g, 54measurable representation, 54metric space, 21metrizable, 23m<strong>in</strong>imal geodesic, 29, 42momentum, 100angular, 101l<strong>in</strong>ear, 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
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- 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
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- Page 94 and 95: Definition 11.10 • The orbit is O
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- 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