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

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So Prop. 11.19 guarantees that the H 0 gradients of geometric energies haveonly a normal component w.r.to the curve — and this couples well with level setmethods.The horizontal version of H 0 is〈h 1 , h 2〉H 0,⊥ := ∫cπ N h 1 · π N h 2 ds (11.4)where π N h 1 (s) is the component of h 1 (s) that is orthogonal to c ′ (as was definedin Section 7.1); note that the action of this (semi)metric was already presentedin eqn. (11.1).So a good paradigm is to think at the metric H 0 as∫∫ c h 1 · h 2 ds on M,c π Nh 1 · π N h 2 ds on B.Remark 11.21 The horizontal space W c is isometric (thru DΠ c ) with thetangent space T [c] B: this implies that we may decide to use W c and Π c as a“local chart” for B. If the metric is G = H 0 , this brings us back the chart 11.2;with other metrics, this process instead defines a novel choice of chart.The same reasoning above holds for the H A metric (from Sec. 9.2), and forthe Finsler metrics F 1 and F ∞ (from Section 7); in this latter case, we alreadypresented the horizontally projected version of the metrics.For all above reasons, it is common thinking that “a good movement of acurve is a movement that is orthogonal to the curve”. But, when using a differentmetric (as for example, the Sobolev-type metrics) the above is not true anymore.11.11 Horizontality according to H jHorizontality according to H j is a complex matter. To study the horizontallyprojected metric, we need to express the solution ofinf ‖h +bbc′ ‖ H jin closed form, and this needs (pseudo)differential operators: see [36] for details.We just remark that in this case, “a good movement of a curve is not necessarilya movement that is orthogonal to the curve.”11.12 Horizontality is for any group actionIn the above we studied the horizontality properties of the quotient B =M/Diff(S 1 ). But the theory works in general for any other quotient. So we mayapply the same study and ideas to further quotients (such as B/E(n)).98
11.13 MomentaWhat follows comes from §2.5 in Michor and Mumford [36].Consider again the action of a group G on a Riemannian manifold M (asdefined in Sect. 3.8). Most groups that act on curves are smooth differentiablemanifolds themselves, and the group operations are smooth: so they are Liegroups. In our cases of interest, the actions are smooth as well.We want to study some differentiable properties of the action g · m; using therule (3.5), we see that we can restrict to studying the case e · m, where e ∈ G isthe identity element.11.13.1 Conservation of momentaGiven a curve c ∈ M, and a tangent vector ξ ∈ T e G (where T e G is the Liealgebra of G), we derive the action, for fixed c and e “moving” in direction ξ;the result of this derivative is a ζ = ζ ξ,c ∈ T c M (depending linearly on ξ). Thisdirection ζ is intuitively “the infinitesimal motion of c, when we infinitesimallyact in direction ξ on c”.To exemplify the above process, we provide a simple toy example.Example 11.22 (Rotation action on the plane) The group of 2D rotationsG = lR/(2π) (the real line modulus 2π, with the + operation) acts on the plane,as followsG × lR 2 → lR 2g, m ↦→ g · m = R g mwith( )cos g − sin gR g :=.sin g cos gSince T e G = lR, there is only one direction in the tangent to G, hence thederivative in direction ξ = 1 is just the standard derivative d/dg; similarlyT lR 2 = lR 2 ; by deriving the action in point g = e = 0 (the identity element), weobtain thatζ 1,m = JmwithJ :=( 0 −11 0Note that Jm is the vector (normal to m) obtained by rotating m of an angle π/2counterclockwise. The physical interpretation is that, when rotating the plane incounterclockwise direction with speed ξ = 1, each point m will move with velocityζ = Jm.In the above toy example, the result ζ is vector; but in the case of immersedcurves M, since ζ ∈ T c M, then ζ = ζ(θ) : S 1 → lR n is a vector field.).We then recall a very interesting condition by Emmy Noether99
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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
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 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 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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