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

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and hilights features and problems of those theories as were studied up to afew years ago, so as to identify the needs and obstacles to further developments.Section 3 contains a summary of all mathematical concepts that are needed forthe rest of the notes. Section 4 coalesces all the above in more precise definitionsof spaces of curves to be used as “shape spaces”, and sets mathematicalrequirements and goals for applications in computer vision. Section 5 indexesexamples of “shape spaces” from the current literature, inserting it in a commonparadigm of “representation of shape”; some of this literature is then elaboratedupon in the following sections 6,7,8,9, containing two examples of metrics ofcompact subsets of lR n , two examples of Finsler metrics of curves, two examplesof Riemannian metrics of curves “up to pose”, and four examples of Riemannianmetrics of immersed curves. The last such example is the family of Sobolev-typeRiemannian metrics of immersed curves, whose properties are studied in Section10, with applications and numerical examples. Section 11 presents advancedmathematical topics regarding the Riemannian spaces of immersed and geometriccurves.I gratefully acknowledge that a part of the theory and many numericalexperiments exposited were developed in joint work with Prof. Yezzi (GaTech)and Prof. Sundaramoorthi (UCLA); other numerical experiments were by A.Duci and myself. I also deeply thank the organizers for inviting me to Cetraroto give the lectures that were the basis for these lecture notes.1 Shapes & curvesWhat is this course about? In the first two sections we begin by summarizing ina simpler form the definitions, reviewing the goals, and presenting some usefulmathematical tools.1.1 ShapesA wide interest for the study of shape spaces arose in recent years, in particularinside the computer vision community. Some examples of shape spaces are asfollows.• The family of all collections of k points in lR n .• The family of all non empty compact subsets of lR n .000 111000 111000 111000 111000 111• The family of all closed curves in lR n .There are two different (but interconnected) fields of applications for a goodshape space in computer vision:shape optimization where we want to find the shape that best satisfies adesign goal; a topic of interest in engineering at large;2
shape analysis where we study a family of shapes for purposes of statistics,(automatic) cataloging, probabilistic modeling, among others, and possiblycreate an a-priori model for a better shape optimization.1.2 CurvesWe will use formulæ of the form A := B to mean that the formula A is definedby the formula B.S 1 = {x ∈ lR 2 | |x| = 1} is the circle in the plane. S 1 is the template for allpossible closed curves. (Open curves will be called paths, to avoid confusion).Definition 1.1 (Classes of curves) • A C 1 curve is a continuously differentiablemap c : S 1 → lR n such that the derivative c ′ (θ) := ∂ θ c(θ) existsat all points θ ∈ S 1 .• An immersed curve is a C 1 curve c such that c ′ (θ) ≠ 0 at all pointsθ ∈ S 1 .c : S 1 → c(S 1 )✓✏↦→✒✑Note that, in our terminology , the “curve” is the function c, and not just theimage c(S 1 ) inside lR n .Most of the theory following will be developed for curves in lR n , when thisdoes not complicate the math. We will call planar curves those whose imageis in lR 2 .The class of immersed curves is a differentiable manifold. For the purposesof this introduction, we present a simple, intuitive definition.Definition 1.2 The manifold of (parametric) curves M is the set of allclosed immersed curves. Suppose that c ∈ M, c : S 1 → lR n is a closed immersedcurve.• A deformation of c is a function h : S 1 → lR n .• The set of all such h is the tangent space T c M of M at c.• An infinitesimal deformation of the curve c 0 in “direction” h will yield (onfirst order) the curve c 0 (u) + εh(u).• A homotopy C connecting c 0 to c 1 is a continuously differentiablefunction C : [0, 1] × S 1 → lR n such that c 0 (θ) = C(0, θ)and c 1 (θ) = C(1, θ).By defining γ(t) = C(t, ·) we can associate C to a path γ :[0, 1] → M in the space of curves M, connecting c 0 to c 1 .Mhc 0c 1When dealing with homotopies C = C(t, θ), we will use the “prime notation” C ′for the derivative ∂ θ C in the “curve parameter” θ, and the “dot notation” Ċ forthe derivative ∂ t C in the “time parameter” t.3
Page 1: Metrics of curves in shape optimiza
Page 5 and 6: • F (c) = F (c ◦ φ) for all cu
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
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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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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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Metrics of curves in shape optimization and analysis - Andrea Carlo ...

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