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

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Theorem 11.23 Suppose that the Riemannian metric 〈·, ·〉 c on M is invariantw.r.to the action of G: this is equivalent to saying that the action is an isometry.Let γ(t) be a critical geodesic: then〈〉ζ ξ,γ(t) , ˙γ(t)(11.5)γ(t)is constant in t (for any choice of ξ ∈ T e G).The quantity (11.5) is called “the momentum of the action”.As an alternative interpretation, note that the vectors ζ that are obtainedby deriving the action are exactly all the vectors in the vertical spaces V c ofthe corresponding action G (since ζ are infinitesimal motions inside the orbit).Recall that h ∈ T c M is horizontal (that is, h ∈ W c ) iff it is orthogonal to V c〈ζ, h ′ 〉 = 0 for all ζ ∈ V c . So, as corollary of Emmy Noether’s theorem, we obtainthatCorollary 11.24 if a geodesic is shot in a horizontal direction ˙γ(0), then thegeodesic will be horizontal for all subsequent times.The following are examples of momenta that are related to the actions oncurves (that we saw in Example 4.8).Example 11.25 • The rescaling group is represented by lR + , that is onedimensional, so there is only one tangent direction ξ = 1 in lR + . Theaction is l, c ↦→ lc, so there is only one direction, that is ζ = c.• The translation group is represented by lR n , that is a vector space, so thetangent directions are ξ ∈ lR n ; the action is ξ, c ↦→ ξ + c; then ζ = ξ (andnote that this is constant in θ).• The rotation group is represented by orthogonal matrixes, so we set G =O(n); the group identity e is the identity matrix; the action is matrixvectormultiplication A, c(θ) ↦→ Ac(θ); the tangent T e G is the set of theantisymmetric matrixes B ∈ lR n×n ; then ζ = Bc.• The reparameterization group is G = Diff(S 1 ); a tangent vector is a scalarfield ξ : S 1 → lR; the action is the composition φ, c ↦→ c ◦ φ; we in the endhave thatζ(θ) = ξ(θ)c ′ (θ)(where c ′ = D θ c) that is, ζ is a generic vector field parallel to the curve.For the ˜H 1 metric, this implies the following properties of a geodesic path γ(t).Proposition 11.26 • Scaling momentum:∫〈γ, ˙γ〉 ˜H1= avg c (γ) · avg c ( ˙γ) + λL 2D s γ · D s ˙γ ds = constant.100
• Linear momentum: avg c ( ˙γ) is constant, since, for all ξ ∈ lR n ,〈ξ, ˙γ〉 ˜H1= ξ · avg c ( ˙γ) = constant;since ξ is arbitrary, this means that avg c ( ˙γ) is constant in t.• Angular momentum: for any antisymmetric matrix B ∈ lR n×n ,∫〈Bγ, ˙γ〉 ˜H1= (Bavg c (γ)) · avg c ( ˙γ) + λL 2 (BD s γ) · (D s ˙γ) ds = constant.• Reparameterization momentum: for any scalar field ξ : S 1 → lR, settingζ(θ, t) = ξ(θ)γ ′ (θ, t)we get∫〈ζ, ˙γ〉 ˜H1= avg c (ξγ ′ ) · avg c ( ˙γ) + λL 2D s (ξγ ′ ) · D s ˙γ = constant;integrating by parts,avg c (ξγ ′ ) · avg c ( ˙γ) − λL 2 ∫(ξγ ′ ) · D ss ˙γ = constant;since ξ is arbitrary, this means thatis constant in t, for any θ ∈ S 1 .Contentsγ ′ · avg c ( ˙γ) − λL 2 γ ′ · D ss ˙γ1 Shapes & curves 21.1 Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Geometric curves and functionals . . . . . . . . . . . . . . . . . 41.4 Curve–related quantities . . . . . . . . . . . . . . . . . . . . . . . 51.4.1 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Shapes in applications 72.1 Shape analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.1 Shape distances . . . . . . . . . . . . . . . . . . . . . . . . 82.1.2 Shape averages . . . . . . . . . . . . . . . . . . . . . . . . 92.1.3 Principal component analysis (PCA) . . . . . . . . . . . . 92.2 Shape optimization & active contours . . . . . . . . . . . . . . . 112.2.1 A short history of active contours . . . . . . . . . . . . . . 112.2.2 Energies in computer vision . . . . . . . . . . . . . . . . . 12101
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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
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
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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
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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 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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Metrics of curves in shape optimization and analysis - Andrea Carlo ...

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