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

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10.9 New regularization methodsTypically, in active contour energies a length penalty is added to obtain regularityof the evolving contour:E(c) = E data (c) + α len(c) .It is important to note that this length penalty will regularize the curve onlyif the curve evolution is derived as a H 0 gradient descent flow. If another curvemetric is used to derive the gradient descent flow, then a priori the length penaltymay have no regularizing effect on the curve regularity (cf. 10.32).10.9.1 Elastic regularizationWe can now consider an alternative approach for regularity of curve:∫E(c) = E data (c) + α len(c) κ 2 (s) dswhere α > 0 is a fixed constant. This energy favors regularity of curve, but thisregularization does not rely on properties of the metric; and the regularizationis scale invariant. Note that the H 0 gradient flow of this energy is ill posed.The numerical results (originally presented in [56]) are in Fig. 16. In allframes, the final limit of the gradient descent flow is shown; between differentframes, the value of α is increased.c. . . . . . . . . . . .minima for α increasing → . . . . . . . . . . . .E data (c) + α len(c)(H 0 gradient descent)E data (c) +α len(c) ∫ c κ2 (s) ds(Sobolev gradient descent)Figure 16: Elastic regularization. (From [58] c○ 2008 IEEE. Reproduced with permission).88
11 Mathematical properties of the Riemannianspace of curvesIn this section we study some mathematical properties, and add some finalremarks, regarding the Riemannian manifold of geometric curves, when this isendowed with the metrics that were presented previously.11.1 ChartsLet againM = M i,f = Imm f (S 1 , lR n )be the space of all smooth freely-immersed curves.We already remarked in Remark 4.4 that, if the topology on M is strongenough then the immersed curves are an open subset of all functions. Wemoreover represent both curves c ∈ M and deformations h ∈ T c M as functionsS 1 → lR n ; this is a special structure that is not usually present in abstractmanifolds: so we can easily define “charts” for M.Proposition 11.1 (Charts in M i,f ) Given a curve c, there is a neighborhoodU c of 0 ∈ T c M such that for h ∈ U c , the curve c + h is still immersed; then thismap h ↦→ c + h is the simplest natural candidate to be a chart of Φ c : U c → M.Indeed, if we pick another curve ˜c ∈ M and the corresponding U˜c such thatU˜c ∩ U c ≠ ∅, then the equality Φ c (h) = c + h = ˜c + ˜h = Φ˜c (˜h) can be solved for hto obtain h = (˜c − c) + ˜h.The main goal of this section is to study the manifoldB = B i,f := M/Diff(S 1 )of geometric curves. Since B is an abstract object, we will actually work withM in everyday calculus. To recombine the two needs, we will identify an uniquefamily of “small deformations” inside M, that has a specific meaning in B. Acommon choice is to restrict the family of infinitesimal motions h to those suchthat h(θ) is orthogonal to the curve, that is, to c ′ (θ).Proposition 11.2 (Charts in B i,f ) Let Π be the projection from M to thequotient B. Let [c] ∈ B: we pick a curve c such that Π(c) = [c]. We represent thetangent space T [c] B as the space of all k : S 1 → lR n such that k(s) is orthogonalto c ′ (s).We choose U [c] ⊂ T [c] B a neighborhood of 0; then the chart is defined byΨ [c] : U [c] → B , Ψ [c] (k) := Π(c(·) + k(·))that is, it moves c(u) in direction k(u).If U [c] is small enough, then this chart is a smooth diffeomorphism; and, ifwe pick another curve ˜c ∈ M and the corresponding U [˜c] such that U [˜c] ∩ U [c] ≠ ∅,then the equalityΨ [c] (k) = Ψ [˜c] (˜k)89
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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.
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 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 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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