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

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1.3 Geometric curves and functionalsShapes are usually considered to be geometric objects. Representing a curveusing c : S 1 → lR n forces a choice of parameterization, that is not really partof the concept of “shape”. To get rid of this, we first summarily present whatreparameterizations are.Definition 1.3 Let Diff(S 1 ) be the family of diffeomorphisms of S 1 : all themaps φ : S 1 → S 1 that are C 1 and invertible, and the inverse φ −1 is C 1 .Diff(S 1 ) enjoys some important mathematical properties.• Diff(S 1 ) is a group, and its group operation is the composition φ, ψ ↦→φ ◦ ψ.• Diff(S 1 ) is divided in two connected components, the family Diff + (S 1 )of diffeomorphisms with derivative φ ′ > 0 at all points; and the familyDiff − (S 1 ) of diffeomorphisms with derivative φ ′ < 0 at all points.Diff + (S 1 ) is a subgroup of Diff(S 1 ).• Diff(S 1 ) acts on M, the action 1 is the right function composition c ◦ φ.The resulting curve c ◦ φ is a reparameterization of c.The action of the subgroup Diff + (S 1 ) moreover does not change the orientationof the curve.Definition 1.4 The quotient spaceB := M/Diff(S 1 )is the space of curves up to reparameterization, also called geometric curvesin the following. Two parametric curves c 1 , c 2 ∈ M such that c 1 = c 2 ◦ φ are thesame geometric curve inside B.For some applications we may choose instead to consider the quotient w.r.toDiff + (S 1 ); the quotient space M/Diff + (S 1 ) is the space of geometric orientedcurves.B is mathematically defined as the set B = {[c]} of all equivalence classes[c] of curves that are equal but for reparameterization,[c] := {c ◦ φ for φ ∈ Diff(S 1 )} ;and similarly for the quotient M/Diff + (S 1 ). More properties of these quotientswill be presented in Section 4.5.We can now define the geometric functionals (that are, loosely speakinginvariant w.r.to reparameterization of the curve).Definition 1.5 A functional F (c) defined on curves will be called geometricif one of the two following alternative definitions holds.1 We will provide more detailed definitions and properties of the “actions” in Section 3.8.4
• F (c) = F (c ◦ φ) for all curves c and for all φ ∈ Diff(S 1 ).• For all curves c and all φ, if φ ∈ Diff + (S 1 ) then F (c) = F (c ◦ φ), whereasif φ ∈ Diff − (S 1 ) then F (c) = −F (c ◦ φ)In the first case, F can be “projected” to B, that is, it may be considered as afunction F : B → lR. In the second case, F can be “projected” to M/Diff + (S 1 ).It is important to remark that “geometric” theories have often provided thebest results in computer vision.1.4 Curve–related quantitiesA good way to specify the design goal for shape optimization is to define anobjective function (a.k.a. energy) F : M → lR that is minimum in the curvethat is most fit for the task.When designing our F , we will want it to be geometric; this is easily accomplishedif we use geometric quantities to start from. We now list the mostimportant such quantities.In the following, given v, w ∈ lR n we will write |v| for the standard Euclideannorm, and 〈v, w〉 or (v · w) for the standard scalar product. We will againwrite c ′ (θ) := ∂ θ c(θ).Definition 1.6 (Derivations) If the curve c is immersed, we can define thederivation with respect to the arc parameter∂ s = 1|c ′ | ∂ θ .(We will sometimes also write D s instead of ∂ s .)Definition 1.7 (Tangent) At all points where c ′ (θ) ≠ 0, we define the tangentvectorT (θ) = c′ (θ)|c ′ (θ)| = ∂ sc(θ) .(At the points where c ′ = 0 we may define T = 0).It is easy to prove (and quite natural for our geometric intuition) that D s and Tare geometric quantities (according to the second Definition 1.5).Definition 1.8 (Length) The length of the curve c is∫len(c) := |c ′ (θ)| dθ . (1.1)S 1We can define formally the arc parameter s byds := |c ′ (θ)| dθ ;we use it only in integration, as follows.5
Page 1 and 2: Metrics of curves in shape optimiza
Page 3: shape analysis where we study a fam
Page 7 and 8: κ > 0HNNHNHκ < 0Figure 1: Example
Page 9 and 10: In the case of planar curves c 1 ,
Page 11 and 12: (a) (b) (c) (d) (e)Figure 3: Segmen
Page 13 and 14: where φ may be chosen to beφ(x) =
Page 15 and 16: 2.4.1 Example: geometric heat flowW
Page 17 and 18: 2.4.5 Centroid energyWe will now pr
Page 20 and 21: shapes. Unfortunately, H 0 does not
Page 22 and 23: • If the second request is waived
Page 24 and 25: • The Fréchet space of smooth fu
Page 26 and 27: Since φ k are homeomorphisms, then
Page 28 and 29: 3.6.1 Riemann metric, lengthDefinit
Page 30 and 31: Theorem 3.30 Suppose that M is a sm
Page 32 and 33: Example 3.38 Let M = C ∞ ([−1,
Page 34 and 35: • sometimes S 1 will be identifie
Page 36 and 37: 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
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Definition 8.3 (Flat curves) Let Z
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Proof. Fix α 0 ∈ S \ Z. Let T =
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8.2.2 RepresentationThe Stiefel man
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9.3 Conformal metricsYezzi and Menn
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10.1.1 Related worksA family of met
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Definition 10.7 (Convolution) A arc
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and̂∇˜HjE(0) = ̂∇ H 0E(0),̂
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and we simply integrate twice! More
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Corollary 10.18 In particular, the
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10.6 Existence of gradient flowsWe
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• The length functional (from C 1
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We can eventually estimate the diff
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7.552.5-10 -7.5 -5 -2.5 2.5 5 7.5 1
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By using (i) and (ii) from lemma 10
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10.6.3 Existence of flow for geodes
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10.7.1 Robustness w.r.to local mini
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We already presented all the calcul
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10.9 New regularization methodsTypi
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can be solved for k (this is not so
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2. Now• at t = 1/2 it achieves th
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Definition 11.10 • The orbit is O
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y eqn. (11.3), where the terms RHS
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So Prop. 11.19 guarantees that the
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Theorem 11.23 Suppose that the Riem
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2.2.3 Examples of geometric energy
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9 Riemannian metrics of immersed cu
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contour continuation, 11convolution
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normal vector, 6objective function,
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References[1] Luigi Ambrosio, Giuse
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[30] Serge Lang. Fundamentals of di
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[57] Ganesh Sundaramoorthi, Anthony
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