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

Metrics of curves in shape optimization andanalysisAndrea C. G. MennucciLevel set and PDE based reconstruction methods:applications to inverse problems and image processingCIME, Cetraro, 2008Key words: Shape space, shape optimization, shape analysis, computer vision,Riemannian geometry, manifold of curves, hyperspace of compact sets, edgedetection, image segmentation, visual tracking, curve evolution, gradient flow,active contour, sobolev active contour.IntroductionIn these lecture notes we will explore the mathematics of the space of immersedcurves, as is nowadays used in applications in computer vision. In this field,the space of curves is employed as a “shape space”; for this reason, we will alsodefine and study the space of geometric curves, that are immersed curves upto reparameterizations. To develop the usages of this space, we will considerthe space of curves as an infinite dimensional differentiable manifold; we willthen deploy an effective form of calculus and analysis, comprising tools such as aRiemannian metric, so as to be able to perform standard operations: minimizinga goal functional by gradient descent, or computing the distance between twocurves. Along this path of mathematics, we will also present some currentliterature results. (Another common and interesting example of “shape spaces”is the space of all compact subsets of lR n — we will briefly discuss this optionas well, and relate it to the aforementioned theory).These lecture notes aim to be as self-contained as possible, so as to be accessibleto young mathematicians and non-mathematicians as well. For this reason,many examples are intermixed with the definitions and proofs; in presentingadvanced and complex mathematical ideas, the rigorous mathematical definitionsand proofs were sometimes sacrificed and replaced with an intuitive description.These lecture notes are organized as follows. Section 1 introduces the definitionsand some basilar concepts related to immersed and geometric curves.Section 2 overviews the realm of applications for a shape space in computervision, that we divide in the fields of “shape optimization” and “shape analysis”;1

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

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

Definition 1.9 (Integration by arc parameter) We define the integrationby arc parameter of a function g : S 1 → lR k along the curve c by∫∫g(s) ds := g(θ)|c ′ (θ)| dθ .cS 1and the average integral∫cg(s) ds := 1 ∫g(s) dslen(c) cand we will sometimes shorten this as avg c (g).The above integrations are geometric quantities (according to the first Definition1.5).1.4.1 CurvatureSuppose moreover that the curve is immersed and is C 2 regular (that means thatit is twice differentiable, and the second derivative is continuous); in this casewe may define curvatures, that indicate how much the curve bends. There aretwo different definitions of curvature of an immersed curve: mean curvature Hand signed curvature κ, that is defined when c is planar. H and k are extrinsiccurvatures, they are properties of the embedding of c into lR n .Definition 1.10 (H) If c is C 2 regular and immersed, we can define the meancurvature H of c asH = ∂ s ∂ s c = ∂ s Twhere ∂ s = 1|c ′ | ∂ θ is the derivation with respect to the arc parameter. It enjoysthe following properties.Properties 1.11 • It is easy to prove that H ⊥ T .• H is a geometric quantity.˜H = H ◦ φ.If ˜c = c ◦ φ and ˜H is its curvature, then• 1/|H| is the radius of a tangent circle that best approximates the curve tosecond order.Definition 1.12 (N) When the curve c is immersed and planar, we can definea normal vector N to the curve, by requiring that |N| = 1, N ⊥ T and N isrotated π/2 radians anticlockwise with respect to T .Definition 1.13 (κ) If c is planar and C 2 regular, then we can define a signedscalar curvature κ = 〈H, N〉, so that∂ s T = κN = H and ∂ s N = −κT .See fig. 1 on the following page. The above equality implies that κ is a geometricquantity, according to the second Definition 1.5.6

κ > 0HNNHNHκ < 0Figure 1: Example of a regular curve and its curvature.2 Shapes in applicationsA number of methods have been proposed in shape analysis to define distancesbetween shapes, averages of shapes and statistical models of shapes. At thesame time, there has been much previous work in shape optimization, for exampleimage segmentation via active contours, 3D stereo reconstruction viadeformable surfaces; in these later methods, many authors have defined energyfunctionals F (c) on curves (or on surfaces), whose minima represent the desiredsegmentation/reconstruction; and then utilized the calculus of variations to derivecurve evolutions to search minima of F (c), often referring to these evolutionsas gradient flows. The reference to these flows as gradient flows implies a certainRiemannian metric on the space of curves; but this fact has been largelyoverlooked. We call this metric H 0 , and properly define it in eqn. (2.4).2.1 Shape analysisMany method and tools comprise the shape analysis. We may list• distances between shapes,• averages for shapes,• principle component analysis for shapes and• probabilistic models of shapes.We will present a short overview of the above, in theory and in applications. Webegin by defining the distance function and signed distance function, two toolsthat we will use often in this theory.Definition 2.1 Let A, B ⊂ lR n be compact sets.7

• u A (x) := inf y∈A |x − y| is the distance function,u Au Bx 2ABx 1• b A (x) := u A (x) − u lR n \A(x) is the signed distance function.b Ab Bx 2ABx 12.1.1 Shape distancesA variety of distances have been proposed for measuring the difference betweentwo given shapes. Two examples follows.Definition 2.2 The Hausdorff distance( )d H (A, B) := u B (x)supx∈A(∨supx∈B)u A (x)where A, B ⊂ lR n are compact, and u A (x) is the distance function.(2.1)The above can be used for curves; in this case a distance may be defined byassociating to the two curves c 1 , c 2 the Hausdorff distance of the image of thecurvesd H (c 1 (S 1 ), c 2 (S 1 )) .If c 1 , c 2 are immersed and planar, we may otherwise use d H (˚c 1 ,˚c 2 ) where˚c 1 ,˚c 2 denote the internal region enclosed by c 1 , c 2 .Definition 2.3 Let A, B ⊂ lR n be two measurable sets, letA∆B := (A \ B) ∪ (B \ A)be the set symmetric difference; let |A| be the Lebesgue measure of A. Wedefine the set symmetric distance asd(A, B) = ∣ ∣ A∆B∣ ∣ ,(where it is to be intended that two sets are considered “equal” if they differ by ameasure zero set).8

In the case of planar curves c 1 , c 2 , we can apply to above idea to define2.1.2 Shape averagesd(c 1 , c 2 ) = ∣ ∣˚c1 ∆˚c 2∣ ∣ .Many definitions have also been proposed for the average ¯c of a finite set ofshapes c 1 , . . . , c N . There are methods based on the arithmetic mean (that arerepresentation dependent).• One such case is when the shapes are defined by a finite family of Nparameters; so we can define the parametric or landmark averaging¯c(p) = 1 NN∑c n (p)n=1where p is a common parameter/landmark.• More in general, we can define the signed distance level set averagingby using as a representative of a shape its signed distance function, andcomputing the average shape by¯c = { x | ¯b(x) = 0 } , where ¯b(x) = 1 NN∑b cn (x)where b cn is the signed distance function of c n ; or in case of planar curves,of the part of plane enclosed by c n .Then there are non parametric, representation independent, methods. The(possibly) most famous one is theDefinition 2.4 The distance-based average. 2 Given a distance d M betweenshapes, an average shape ¯c may be found by minimizing the sum of its squareddistances.N∑¯c = arg min d M (c, c n ) 2cn=1It is interesting to note that in Euclidean spaces there is an unique minimum,that coincides with the arithmetic mean; while in general there may be nominimum, or more than one.2.1.3 Principal component analysis (PCA)Definition 2.5 Suppose that X is a random vector taking values in lR n ; letX = E(X) be the mean of X. The principal component analysis is therepresentation of X asn∑X = X + Y i S ii=12 Due to Fréchet, 1948; but also attributed to Karcher.n=19

where S i are constant vectors, and Y i are uncorrelated real random variableswith zero mean, and with decreasing variance. S i is known as the i-th mode ofprincipal variation.The PCA is possible in general in any finite dimensional linear space equippedwith an inner product. In infinite dimensional spaces, or equivalently in case ofrandom processes, the PCA is obtained by the Karhunen-Loève theorem.Given a finite number of samples, it is also possible to define an empiricalprincipal component analysis, by estimating the expectation and variance of thedata.In many practical cases, the variances of Y i decrease quite fast: it is thensensible to replace X by a simplified variable˜X := X +k∑Y i S iwith k < n.So, if shapes are represented by some “linear” representation, the PCA isa valuable tool to study the statistics, and it has then become popular forimposing shape priors in various shape optimization problems. For more generalmanifold representations of shapes, we may use the exponential map (defined in3.29), replacing S n by tangent vectors and following geodesics to perform thesummation.We present here an example in applications.Example 2.6 (PCA by signed distance representation) In these pictureswe see an example of synergy between analysis and optimization, from Tsai et al.[60]. The figure 2 contains a row of pictures that represent the (empirical)PCA of signed distance function of a family of plane shapes. The figure 3 onthe following page contains a second row of images that represent: an image ofa plane shape (a) occluded (b) and noise degraded (c), an initialization for theshape optimizer (d) and the final optimal segmentation (e).i=1(a) (b) (c) (d) (e) (f) (g)} {{ }mean} {{ }first mode} {{ }second mode} {{ }third modeFigure 2: PCA of plane shapes.(From Tsai et al. [60] c○ 2001 IEEE. Reproduced with permission).10

(a) (b) (c) (d) (e)Figure 3: Segmentation of occluded plane shape.(From Tsai et al. [60] c○ 2001 IEEE. Reproduced with permission).2.2 Shape optimization & active contours2.2.1 A short history of active contoursIn the late 20th century, the most common approach to image segmentationwas a combination of edge detection and contour continuation. With edgedetection [5], small edge elements would be identified by a local analysis ofthe image; then a continuation method would be employed to reconstructfull contours. The methods were suffering from two drawbacks, edge detectionbeing too sensitive to noise and to photometric features (such as, sharp shadows,reflections) that were not related to the physical structure of the image; andcontinuation was in essence a NP-complete algorithm.Active contours were introduced by Kass et al. [26] as a simpler approachto the segmentation problem. The idea is to minimize an energy F (c) (wherethe variable c is a contour i.e. a curve), that contains an edge-based attractionterm and a smoothness term, which becomes large when the curve is irregular.To search for the minimum of F (c), a curve evolution method was derived fromthe calculus of variations. The evolving curve would then be attracted to thefeatures of interest, while mantaining the property of being a closed smoothcontour.An unjustified feature of the model of [26] is that the evolution is dependenton the way the contour is parameterized. Hence Caselles et al. [6], Malladi et al.[33] revisited the idea of [26] in the new form of a geometric evolution, whichcould be implemented by the level set method invented by Osher and Sethian[44]. Thereafter, Kichenassamy et al. [27] Caselles et al. [7] presented the activecontour approach to the edge detection problem in the form of the minimizationof a geometric energy that is a generalization of Euclidean arc length. Again, theauthors derived the gradient descent flow in order to minimize the geometricenergy.In contrast to the edge-based approaches for active contours (mentionedabove), region-based energies for active contours have been proposed in Ronfard[46] Zhu et al. [69] Yezzi et al. [64] Chan and Vese [8]. In these approaches, anenergy is designed to be minimized when the curve partitions the image intostatistically distinct regions. This kind of energy has provided many desirablefeatures; for example, it provides less sensitivity to noise, better ability to capture11

concavities of objects, more dependence on global features of the image, and lesssensitivity to the initial contour placement.In Mumford and Shah [42, 43], the authors introduced and rigorously studieda region-based energy that was designed to both extract the boundary of distinctregions while also smoothing the image within these regions. Subsequently, Tsaiet al. [61] Vese and Chan [63] implemented a level set method for the curveevolution minimizing the energy functional considered by Mumford&Shah.2.2.2 Energies in computer visionA variational approach to solving shape optimization problems is to define andconsequently minimize geometric energy functionals F (c) where c is a planarcurve. We may identify two main families of energies,• the region based energies:∫∫RF (c) = f in (x) dx + f out (x) dx2RR cwhere ∫ · · · dx is area element integral, and R and R care the interior and exterior areas outlined by c; and• the boundary based energies:∫F (c) = φ(c(s)) dscR c R cwhere φ is designed to be small on the salient features of the image, and ∫ C · · · dsis the integration by arc parameter defined in 1.9. Note that φ(x) ds may beinterpreted as a conformal metric on lR 2 , so minima curves are geodesics w.r.tothe conformal metric.2.2.3 Examples of geometric energy functionals for segmentationSuppose I : Ω → [0, 1] is the image (in black & white). We propose two examplesof energies taken from the above families.• The Chan-Vese segmentation energy [63]F (c) =∫Rwhere avg R I = 1|R|∣∣I(x) − avg R (I) ∣ 2 dx +∫R∫ ∣ ∣∣I(x) − avgR c(I) ∣ 2 dxR cI(x)dx is the average of I on the region R.• The geodesic active contour, (Caselles et al. [7],Kichenassamy et al. [27])∫F (c) = φ(c(s)) ds (2.2)c12

where φ may be chosen to beφ(x) =11 + |∇I(x)| 2that is small on sharp discontinuities of I (∇I is the gradient of I w.r.tox). Since real images are noisy, the function φ in practice would be moreinfluenced by the noise than by the image features; for this reason, usuallythe function φ is actually defined byφ(x) =11 + |∇G ⋆ I(x)| 2where G is a smoothing kernel, such as the Gaussian.2.3 Geodesic active contour method2.3.1 The “geodesic active contour” paradigmThe general procedure for geodesic active contours goes as follows.1. Choose an appropriate geometric energy functional, E.2. Compute the directional derivative 3DE(c; h) =d dt E(c + th) ∣∣∣∣t=0where c is a curve and h is an arbitrary perturbation of c.3. Manipulate DE(c; h) into the form∫h(s) · v(s) ds .c4. Consider v to be the “gradient”, the direction which increases E fastest.5. Evolve c = c(t, θ) by the differential equation ∂ t c = −v; this is called thegradient descent flow.2.3.2 Example: geodesic active contour edge modelWe propose an explicit computation starting from the classical active contourmodel.1. Start from an energy that is minimal along image contours:∫E(c) = φ(c(s)) dscwhere φ is defined to extract relevant features; for examples as discussedafter eqn. (2.2).3 The directional derivative is the basis for the definition 3.13 of the Gâteaux differential,that will be then discussed in Section 3.7 and 4.2.13

2. Calculate the directional derivative:∫DE(c; h) = ∇φ(c) · h + φ(c)(D s h · D s c) dsc∫= ∇φ(c) · h − ( ∇φ(c) · D s c ) (h · D s c) − φ(c)(h · D ss c) dsc∫= h · ((∇φ(c)· N)N − φ(c)κN ) ds . (2.3)3. Deduce the “gradient”:4. Write the gradient flowc∇E = −φκN + (∇φ · N)N .∂c= φκN − (∇φ · N)N .∂t(Note that the flow of geometric energies moves only in orthogonal direction w.r.tthe curve — this phenomenon will be explained in in Section 11.10.)2.3.3 Implicit assumption of H 0 inner productWe have made a critical assumption in going from the directional derivative∫DE(c; h) = h(s) · v(s) dscto deducing that the gradient of E is ∇E(c) = v. Namely, the definition of thegradient 4 is based on the following equality∫〈h(s), ∇E〉 = h(s) · v(s) ds,c }{{} }{{}h 1=h h 2=∇Ethat needs an inner-product structure.This implies that we have been presuming all along that curves are equippedwith a H 0 -type inner-product defined as follows∫〈 〉h1 , h 2 := hH 0 1 (s) · h 2 (s) ds . (2.4)2.4 Problems & goalscWe will now briefly list some examples that show some limits of the usual activecontour method.4 The precise definition of what the gradient is is in section 3.7.14

2.4.1 Example: geometric heat flowWe first review one of the most mathematically studied gradient descent flows ofcurves. By direct computation, the Gâteaux differential of the length of a closedcurve is∫∫D len(c; h) = ∂ s h · T ds = − h · H ds (2.5)cLet C = C(t, θ) be an evolving family of curves. The geometric heat flow(also known as motion by mean curvature) is∂C∂t = Hwhere H := ∂ s ∂ s C is the mean curvature. In the H 0 inner-product, this flow isthe gradient descent for curve length. 5This flow has been studied deeply by the mathematical community, and isknown to enjoy the following properties.Properties 2.7• Embedded planar curves remain embedded.• Embedded planar curves converge to a circular point.• Comparison principle: if two embedded curves are used as initialization,and the first contains the second, then it will contain the second for alltime of evolution. This is important for level set methods.• The flow is well posed only for positive times, that is, for increasing t(similarly to the usual heat equation).For the proofs, see Gage and Hamilton [21], Grayson [23].2.4.2 Short length biasThe usual edge-based active contour energy is of the form∫E(c) = g(c(s)) dscsupposing that g is reasonably constant in the region where c is running, thenE(c) ≃ g len(c), due to the integral being performed w.r.to the arc parameter ds.So one way to reduce E is to shrink the curve. Consequently, when the image issmooth and featureless (as in medical imaging), the usual edge based energieswould often drive the curve to a point.To avoid it, an inflationary term νN (with ν > 0) was usually added tothe curve evolution, to obtain∂c= φκN − ∇φ + νN ,∂tsee [7, 27]. Unfortunately, this adds a parameter that has to be tuned to matchthe characteristics of each specific image.5 A different gradient descent flow for curve length will be discussed in Remark 10.32.c15

2.4.3 Average weighted lengthAs an alternative approach we may normalize the energy to obtain a morelength-independent energy, the average weighted lengthavg c (g) := 1 ∫g(c(s)) ds (2.6)len(c)but this generates an ill-posed H 0 -flow, as we will now show.2.4.4 Flow computationsHere we write L := len(c). Let g : lR n → lR k ; letavg c (g(c)) := 1 ∫g(c(s)) ds = 1 ∫g(c(θ))|c ′ (θ)| dθ ;L c L S 1then the Gâteaux differential isD[avg c (g(c))](c; h) = 1 ∫∇g(c) · h + g(c)(∂ s h · T ) ds − (2.7)L c− 1 ∫ ∫L 2 g(c) ds ∂ s h · T ds =cc∫= ∇g(c) · h + ( g(c) − avg c (g(c)) ) (∂ s h · T ) ds .cIn the above, we omit the argument (s) for brevity; ∇g is the gradient of g.If the curve is planar, we define the normal and tangent vectors N and T asby 1.12 and 1.13; then, integrating by parts, the above becomesD[avg c (g(c))](c; h) = 1 ∫∇g(c) · h − (∇g(c) · T )(h · T ) − (2.8)L=cc− ( g(c) − avg c (g(c)) ) (h · Dsc) 2 ds =∫ (∇g(c) · N − κ ( g(c) − avg c (g(c)) )) (h · N) ds .cSuppose now that we have a shape optimization functional E including a termof the form avg c (g(c)); let C = C(t, θ) be an evolving family of curves trying tominimize E; this flow would contain a term of the form∂C∂t = . . . ( g(c(s)) − avg c (g) ) κN . . .unfortunately the above flow is ill defined: it is a backward-running geometricheat flow on roughly half of the curve. We will come back to this example inSection 10.8.1; more examples and methods for computing Gâteaux differentialsare in Section 4.2.16

2.4.5 Centroid energyWe will now propose another simple example where the above phenomenon isagain evident.Example 2.8 (Centroid-based energy) Let us fix a target point v ∈ lR n .We recall that avg c (c) is the center of mass of the curve. The energyE(c) := 1 2 |avg c(c) − v| 2 (2.9)penalizes the distance from the center of mass to v. Let in the following c =avg c (c) for simplicity. The directional derivative of E in direction h is〈〉DE(c; h) = c − v, D(c)(c; h)where in turn (by eqn. (2.7) and (2.8))∫D(c)(c; h) = h + (c − c)(D s h · D s c) ds = (2.10)c∫= h − D s c (h · D s c) − (c − c)(h · Dsc) 2 dssupposing that the curve is planar, thench − D s c (h · D s c) = N(h · N)soDE(c; h) =∫〈c − v, N〉(h · N) − 〈c − v, c − c〉κ(h · N) ds .The H 0 gradient descent flow is then∂c∂t = −∇ H 0E(c) = 〈(v − c), N〉N − κN〈 (c − c), (v − c) 〉 .The first term 〈(v − c) · N〉N in this gradient descentflow moves the whole curve towards v.PvLet P := {w : 〈(w − c) · (v − c) 〉 ≥ 0} be the half plane“on the v side” . The second term −κN 〈 (c − c) · (v − c) 〉in this gradient descent flow tries to decrease the curvelength out of P and increase the curve length in P , andthis is ill posed.ccWe will come back to this example in Proposition 10.20.17

2.4.6 ConclusionsMore recent works that use active contours for segmentation are not only basedon information from the image to be segmented (edge-based or region-based),but also prior information, that is information known about the shape of thedesired object to be segmented. The work of Leventon et al. [31] showed how toincorporate prior information into the active contour paradigm. Subsequently,there have been a number of works, for example Tsai et al. [62], Rousson andParagios [47], Chen et al. [12], Cremers and Soatto [13], Raviv et al. [45], whichdesign energy functionals that incorporate prior shape information of the desiredobject. In these works, the main idea is designing a novel term of the energythat is small when the curve is close, in some sense, to a pre-specified shape.The need for prior information terms arose from several factors such as• the fact that some images contain limited information,• the energies functions considered previously could not incorporate complexinformation,• the energies had too many extraneous local minima, and the gradient flowsto minimize these energies allowed for arbitrary deformations that gaverise to unlikely shapes; as in the example in Fig. 4 of segmentation usingthe Chan-Vese energy, where the flowing contour gets trapped into noise.Figure 4: H 0 gradient descent flow of Chan–Vese energy, to segment a noisysquareWorks on incorporating prior shape knowledge into active contours haveled to a fundamental question on how to define distance between two curvesor shapes. Many works, for example Younes [68], Soatto and Yezzi [51], Mioand Srivastava [40], Charpiat et al. [9], Michor and Mumford [37], Yezzi andMennucci [67, 65], in the shape analysis literature have proposed different waysof defining this distance.However, [37, 65] observed that all previous works on geometric activecontours that derive gradient flows to minimize energies, which were describedearlier, imply a natural notion of Riemannian metric, given by the geometricinner product H 0 (that we just “discovered” in eqn. (2.4)). Subsequently, [37, 67]have shown a surprising property: the H 0 Riemannian metric on the space ofcurves is pathological, since the “distance” between any two curves is zero.In addition to the pathologies of the Riemannian structure induced by H 0 ,there are also undesirable features of H 0 gradient flows. Some of these featuresare listed below.18

shapes. Unfortunately, H 0 does not yield a well define metric structure, sincethe associated distance is identically zero.So to achieve our goal, we will need to devise new metrics.3 Basic mathematical notionsIn this section we provide the mathematical theory that will be needed in therest of the course. (Some of the definitions are usually known to mathematics’students; we will present them nonetheless as a chance to remark less knownfacts.)We will though avoid technicalities in the definitions, and for the most partjust provide a base intuition of the concepts. The interested reader may obtainmore details from a books in analysis, such as [2], in functional analysis, suchas [48], and in differential and Riemannian geometry, such as [14], [29] or [30].We start with a basic notion, in a quite simplified form.Definition 3.1 (Topological spaces) A topological space is a set M withassociated a topology τ of subsets, that are the open sets in M.The topology is the simplest and most general way to define what are the“convergent sequences of points” and the “continuous functions”. We will notprovide details regarding topological spaces, since in the following we will mostlydeal with normed spaces and metric spaces, where the topology is induced by anorm or a metric. We just recall this definition.Definition 3.2 A homeomorphism is an invertible continuous function φ :M → N between two topological spaces M, N, whose inverse φ −1 is againcontinuous.3.1 Distance and metric spaceDefinition 3.3 Given a set M, a distance d = d M is a functionsuch that1. d(x, x) = 0,2. if d(x, y) = 0 then x = y,3. d(x, y) = d(y, x) (d is symmetric)d : M × M → [0, ∞]4. d(x, z) ≤ d(x, y) + d(y, z) (the triangular inequality).There are some possible generalizations.• The second request may be waived, in this case d is a semidistance.• The third request may be waived: then d would be an asymmetricdistance. Most theorems we will see can be generalized to asymmetricdistances; see [34].20

3.1.1 Metric spaceThe pair (M, d) is a metric space. A metric space has a distinguished topology,generated by balls of the form B(x, r) := {y | d(x, y) < r}; according to thistopology, x n → n x iff d(x n , x) → n 0; and functions are continuous if they mapconvergent sequences to convergent sequences. We will assume in the followingthat the reader is acquainted with the concepts of “open, closed, compact sets”and “continuous functions” in metric spaces. We recall though the definition of“completeness”.Definition 3.4 A metric space (M, d) is complete iff, for any given sequence(c n ) n ⊂ M, the fact thatlimm,n→∞ d(c m, c n ) = 0implies that c n converges.Example 3.5• lR n is usually equipped with the Euclidean distance∑d(x, y) = |x − y| = √ n (x i − y i ) 2 ;i=1and (lR n , d) is a complete metric space.• Any closed subset of a complete space is complete.• If we cut away an accumulation point out of a space, the resulting space isnot complete.A complete metric space is a space without “missing points”. This is importantin optimization: if a space is not complete, any optimization method that movesthe variable and searches for the optimal solution may fail since the solutionmay, in a sense, be “missing” from the space.3.2 Banach, Hilbert and Fréchet spacesDefinition 3.6 Given a vector space E, a norm ‖ · ‖ is a functionsuch that1. ‖ · ‖ is convex2. if ‖x‖ = 0 then x = 0.3. ‖λx‖ = |λ| ‖x‖ for λ ∈ lR‖ · ‖ : E → [0, ∞]Again, there are some possible generalizations.21

• If the second request is waived, then ‖ · ‖ is a seminorm.• If the third request holds only for λ ≥ 0, then the norm is asymmetric;in this case, it may happen that ‖x‖ ≠ ‖ − x‖.Each (semi/asymmetric)norm ‖ · ‖ defines a (semi/asymmetric)distanceSo a norm induces a topology.d(x, y) := ‖x − y‖.3.2.1 Examples of spaces of functionsWe present some examples and definitions.Definition 3.7 A locally-convex topological vector space E (shortenedas l.c.t.v.s. in the following) is a vector space equipped with a collection ofseminorms ‖ · ‖ k (with k ∈ K an index set); the seminorms induce a topology,such that c n → c iff ‖c n − c‖ k → n 0 for all k; and all vector space operationsare continuous w.r.to this topology.The simplest example of l.c.t.v.s. is obtained when there is only one norm; thisgives raise to two renowned examples of spaces.Definition 3.8 (Banach and Hilbert spaces) • A Banach space is avector space E with a norm ‖ · ‖ defining a distance d(x, y) := ‖x − y‖ suchthat E is metrically complete.• A Hilbert space is a space with an inner product 〈f, g〉, that defines anorm ‖f‖ := √ 〈f, g〉 such that E is metrically complete.(Note that a Hilbert space is also a Banach space).Example 3.9 Let I ⊂ lR k be open; let p ∈ [1, ∞]. A standard example ofBanach space is the L p space of functions f : I → lR n . If p ∈ [1, ∞), the L pspace contains all functions such that |f| p is Lebesgue integrable, and is equippedwith the norm√ ∫‖f‖ L p := p |f(x)| p dx .IFor the case p = ∞, L ∞ contains all Lebesgue measurable functions f : I →lR n such that there is a constant c ≥ 0 for which |f(x)| ≤ c for all x ∈ I \ N,where N is a set of measure zero; and L ∞ is equipped with the norm‖f‖ L ∞ := supess I |f(x)|that is the lowest possible constant c.If p = 2, L 2 is a Hilbert space by inner product∫〈f, g〉 := f(x) · g(x) dx .INote that, in these spaces, by definition, f = g iff the set {f ≠ g} hasLebesgue measure zero.22

3.2.2 Fréchet spaceThe following citations [24] are referred to the first part of Hamilton’s 1982survey on the Nash&Moser theorem.Definition 3.10 A Fréchet space E is a complete Hausdorff metrizable l.c.t.v.s.;where we define that the l.c.t.v.s. E iscomplete when, for any sequence (c n ), the fact thatfor all k implies that c n converges;lim ‖c m − c n ‖ k = 0m,n→∞Hausdorff when, for any given c, if ‖c‖ k = 0 for all k then c = 0;metrizable when there are countably many seminorms associated to E.The reason for the last definition is that, if E is metrizable, then we can define adistance∞∑ 2 −k ‖x − y‖ kd(x, y) :=1 + ‖x − y‖ kk=0that generates the same topology as the family of seminorms ‖ · ‖ k ; and the viceversa is true as well, see [48].3.2.3 More examples of spaces of functionsExample 3.11 Let I ⊂ lR m be open and non empty.• The Banach space C j (I → lR n ), with associated norm‖f‖ := supi≤jsup |f (i) (t)| ;t∈Iwhere f (i) is the i-th derivative.uniformly for all i ≤ j.In this space f n → f iff f (i)n→ f (i)• The Sobolev space H j (I → lR n ), with scalar product∫〈f, g〉 H n := f(t) · g(t) + · · · + f (j) · g (j) dtIwhere f (j) is the j-th derivative 6 .6 The derivatives are computed in distributional sense, and must exists as Lebesgue integrablefunctions.23

• The Fréchet space of smooth functions C ∞ (I → lR n ).The seminorms are‖f‖ k = sup |f (k) (x)|x∈Iwhere f (k) is the k-th derivative. In this space, f n → f iff all the derivativesf n(k) (x) converge to derivatives f (k) (x), and for any fixed k the convergenceis uniform w.r.to x ∈ I.This last is the strongest topology between the topologies usually associated tospaces of functions.3.2.4 Dual spaces.Definition 3.12 Given a l.c.t.v.s. E, the dual space E ∗ is the space of alllinear functions L : E → lR.If E is a Banach space, it is easy to see that E ∗ is again a Banach space, withnorm‖L‖ E ∗ := sup‖x‖ E ≤1|Lx| .The biggest problem when dealing with Fréchet spaces, is that the dual ofa Fréchet space is not in general a Fréchet space, since it often fails to bemetrizable. (In most cases, the duals of Fréchet spaces are “quite wide” spaces;a classical example being the dual elements of smooth functions, that are thedistributions). So given F, G Fréchet spaces, we cannot easily work with “thespace L(F, G) of linear functions between F and G”.As a workaround, given an auxiliary space H, we will consider “indexedfamilies of linear maps” L : F × H → G, where L(·, h) is linear, and L is jointlycontinuous; but we will not consider L as a maph ↦→ (f ↦→ L(f, h))H → L(F, G)(3.1)3.2.5 DerivativesAn example is the Gâteaux differential.Definition 3.13 We say that a continuous map P : U → G, where F, G areFréchet spaces and U ⊂ F is open, is Gâteaux differentiable if for any h ∈ Fthe limitP (f + th) − P (f)DP (f; h) := lim= [∂ t P (f + th)]|t→0 tt=0(3.2)exists. The map DP (f; ·) : F → G is the Gâteaux differential.Definition 3.14 We say that P is C 1 if DP : U × F → G exists and is jointlycontinuous.24

(This is weaker than what is usually required in Banach spaces).The basics of the usual calculus hold.Theorem 3.15 ([24, Thm. 3.2.5]) If P is C 1 then DP (f; h) is linear in h.Theorem 3.16 (Chain rule [24, Thm. 3.3.4]) If P, Q are C 1 then P ◦ Q isC 1 andD(P ◦ Q)(f; h) = DP ( Q(f); DQ(f; h) ) .Also, the implicit function theorem holds, in the form due to Nash&Moser:see again [24] for details.3.2.6 Troubles in calculusBut some important parts of calculus are instead lost.Example 3.17 Suppose P : U ⊂ F → F is smooth, and consider the O.D.E.ddt f = P fto be solved for a solution f : (−ε, ε) → F .• if F is a Banach space, then, given initial condition f(0) = x, the solutionwill exist and be unique (for ε > 0 small enough);• but if F is a Fréchet space, then f may fail to exist or be unique.See [24, Sec. 5.6]A consequence (that we will discuss more later on) is that the exponentialmap in Riemannian manifolds may fail to be locally surjective. See [24, Sec.5.5.2].3.3 ManifoldTo build a manifold of curves, we have ahead two main definitions of “manifold”to choose from.Definition 3.18 (Differentiable Manifold) An abstract differentiablemanifold is a topological space M associated to a model l.c.t.v.s. U. It isequipped with an atlas of charts φ k : U k → V k , where U k ⊂ U are open sets,and V k ⊂ M are open sets that cover M. The maps φ k are homeomorphisms.The composition φ −11 ◦ φ 2 restricted to φ −12 (V 1 ∩ V 2 ) isusually required to be a smooth map.The dimension dim(M) of M is the dimension of U.When M is finite-dimensional, the model space is alwaysU = lR n .UU 1V 2cφ 1MV 1U 2φ 225

Since φ k are homeomorphisms, then the topology of M is “identical” to thetopology of U. The rôle of the charts is to define the differentiable structure ofM mimicking that of U. See for example the definition of directional derivativein eqn. (3.4).3.3.1 SubmanifoldDefinition 3.19 Suppose A, B are open subsets of two linear spaces. A diffeomorphismis an invertible C 1 function φ : A → B, whose inverse φ −1 is againC 1 .Let U be a fixed closed linear subspace of a l.c.t.v.s. X.Definition 3.20 A submanifold is a subset Mof X, such that, at any point c ∈ M we may finda chart φ k : U k → V k , with V k , U k ⊂ X open sets,c ∈ V k . The maps φ k are diffeomorphisms, andφ k maps U ∩ U k onto M ∩ V k .Most often, M = {Φ(c) = 0} where Φ : X → Y ; so, to prove that M is asubmanifold, we will use the implicit function theorem.Note that M is itself an abstract manifold, and the model space is U; sodim(M) = dim(U) ≤ dim(X). Vice versa any abstract manifold is a submanifoldof some large X (by well known embedding theorems).3.4 Tangent space and tangent bundleLet us fix a differentiable manifold M, and c ∈ M. We want to define thetangent space T c M of M at c.Definition 3.21 • The tangent space T c M is more easily described forsubmanifolds; in this case, we choose a chart φ k and a point x ∈ U k s.t.φ k (x) = c; T c M is the image of the linear space U under the derivativeD x φ k . T c M is itself a linear subspace in X. In figure 5 on the followingpage, we graphically represent T c M though as an affine subspace, bytranslating it to the point c.• The tangent bundle T M is the collection of all tangent spaces. If M isa submanifold of the vector space X, thenT M := {(c, h) | c ∈ M, h ∈ T c M} ⊂ X × X .The tangent bundle T M is itself a differentiable manifold; its charts are of theform (φ k , Dφ k ), where φ k are the charts for M.XU 1Mφ 1UV 1c26

T cMMV 1cφ 1xU 1UFigure 5: Tangent space3.5 Fréchet ManifoldWhen studying the space of all curves, we will deal with Fréchet manifolds,where the model space E will be a Fréchet space, and the composition of localcharts φ −11 ◦ φ 2 is smooth.Some objects that we may find useful in the following are Fréchet manifolds.Example 3.22 (Examples of Fréchet manifolds) Given two finite-dimensionalmanifolds S, R, with S compact and n-dimensional,• [24, Example 4.1.3]. the space C ∞ (S; R) of smooth maps f : S → R is aFréchet manifold. It is modeled on U = C ∞ (lR n ; R).• [24, Example 4.4.6]. The space of smooth diffeomorphisms Diff(S) of Sonto itself is a Fréchet manifold. The group operation φ, ψ → φ ◦ ψ issmooth.• But if we try to model Diff(S) on C k (S; S), then the group operation isnot even differentiable.• [24, Example 4.6.6]. The quotient of the two aboveC ∞ (S; R)/Diff(S)is a Fréchet manifold. It contains “all smooth maps from S to R, up to adiffeomorphism of S”.So the theory of Fréchet space seems apt to define and operate on the manifoldof geometric curves.3.6 Riemann & Finsler geometryWe first define Riemannian geometries, and then we generalize to Finsler geometries.27

3.6.1 Riemann metric, lengthDefinition 3.23 • A Riemannian metric G on a differentiable manifoldM defines a scalar product 〈h 1 , h 2 〉 G|c on h 1 , h 2 ∈ T c M, dependent onthe point c ∈ M. We assume that the scalar product varies smoothly w.r.toc.• The scalar product defines the norm |h| c = |h| G|c = √ 〈h, h〉 G|c.Suppose γ : [0, 1] → M is a path connecting c 0 to c 1 .∫ 1• The length is Len(γ) := | ˙γ(v)| γ(v) dv0γwhere ˙γ(v) := ∂ v γ(v).c 1∫ M1c 0• The energy (or action) is E(γ) := | ˙γ(v)| 2 γ(v) dv3.6.2 Finsler metric, lengthDefinition 3.24 We define a Finsler metric to be a function F : T M → lR + ,such that• F is continuous and,• for all c ∈ M , v ↦→ F (c, v) is a norm on T c M.We will sometimes write |v| c := F (c, v).(Sometimes F is called a “Minkowsky norm”).As for the case of norms, a Finsler metric may be asymmetric; but, for sakeof simplicity, we will only consider the symmetric case.Using the norm |v| c we can then again define the length of paths byLen F (γ) =and similarly the action.3.6.3 Distance∫ 10| ˙γ(t)| γ(t) dt =0∫ 10F (γ(t), ˙γ(t)) dtDefinition 3.25 The distance d(c 0 , c 1 ) is the infimum of Len(γ) between allC 1 paths γ connecting c 0 , c 1 ∈ M.Remark 3.26 In the following chapter, we will define some differentiable manifoldsM of curves, and add a Riemann (or Finsler) metric G on those; thereare two different choices for the model space,• suppose we model the differentiable manifold M on a Hilbert space U, withscalar product 〈, 〉 U ; this implies that M has a topology τ associated toit, and this topology, through the charts φ, is the same as that of U. Let28

now G be a Riemannian metric; since the derivative of a chart Dφ(c)maps U onto T c M, one natural hypothesis will be to assume that 〈, 〉 U and〈, 〉 G,c be locally equivalent (uniformly w.r.to small movements of c); asa consequence, the topology generated by the Riemannian distance d willcoincide with the original topology τ. A similar request will hold for thecase of a Finsler metric G, in this case U will be a Banach space with anorm equivalent to that defined by G on T c M.• We will though find out that, for technical reasons, we will initially modelthe spaces of curves on the Fréchet space C ∞ ; but in this case there cannotbe a norm on T c M that generates the same original topology (for the proof,see I.1.46 in [48]).3.6.4 Minimal geodesicsDefinition 3.27 If there is a path γ ∗ providing the minimum of Len(γ) betweenall paths connecting c 0 , c 1 ∈ M, then γ ∗ is called a minimal geodesic.The minimal geodesic is also the minimum of the action (up to reparameterization).Proposition 3.28 Let ξ ∗ provide the minimum min γ E(γ) in the class of allpaths γ in M connecting x to y. Then ξ ∗ is a minimal geodesic and its speed| ˙ξ ∗ | is constant.Vice versa, let γ ∗ be a minimal geodesic, then there is a reparameterizationξ ∗ = γ ∗ ◦ φ s.t. ξ ∗ provides the minimum min γ E(γ).A proof may be found in [34].3.6.5 Exponential mapThe action E is a smooth integral, quadratic in ˙γ, and we can compute theEuler-Lagrange equations; its minima are more regular, since they are guaranteedto have constant speed; consequently, when trying to find geodesics, we will tryto minimize the action and not the length. This also related to the idea of theexponential map.Definition 3.29 Let ¨γ = Γ( ˙γ, γ) be the Euler-Lagrange ODE of the actionE(γ) = ∫ 10 | ˙γ(v)|2 dv. Any solution of this ODE is a critical geodesic. Notethat any minimal geodesic is a critical geodesic.Define the exponential map exp c : T c M → M as exp c (η) = γ(1), whereγ is the solution of{¨γ(v) = Γ( ˙γ(v), γ(v)), γ(0) = c, ˙γ(0) = η (3.3)Solving the above ODE (3.3) is informally known as shooting geodesics.The exponential map is often used as a chart, since it is the least possiblydeforming map at the origin.29

Theorem 3.30 Suppose that M is a smooth differentiable manifold modeledon a Hilbert space with a smooth Riemannian metric. The derivative of theexponential map exp c : T c M → M at the origin is an isometry, hence exp c is alocal diffeomorphism between a neighborhood of 0 ∈ T c M and a neighborhood ofc ∈ M.(See [30], VIII §5). The exponential map can then “linearize” small portions ofM, and so it will enable us to use linear methods such as the principal componentanalysis. Unfortunately, the above result does not hold if M is modeled on aFréchet space.3.6.6 Hopf–Rinow theoremTheorem 3.31 (Hopf–Rinow) Suppose M is a finite dimensional Riemannianor Finsler manifold. The following statements are equivalent:• (M, d) is metrically complete;• the O.D.E. (3.3) can be solved for all c ∈ M, η ∈ T c M and v ∈ lR;• the map exp cis surjective;and all those imply that, ∀x, y ∈ M there exists a minimal geodesic connecting xto y.3.6.7 Drawbacks in infinite dimensionsIn a certain sense, infinite dimensional manifolds are simpler than their correspondingfinite-dimensional counterparts: indeed, by Eells and Elworthy [18],Theorem 3.32 (Eells–Elworthy) Any smooth differentiable manifold M modeledon an infinite dimensional separable Hilbert space H may be embedded asan open subset of that Hilbert space.In other words, it is always possible to express M using one single chart. (Butnote that this may not be the best way for computations/applications).When M is an infinite dimensional Riemannian manifold, though, only asmall part of the Hopf–Rinow theorem still holds.Proposition 3.33 Suppose M is infinite dimensional, modeled on a Hilbertspace, and (M, d) is complete, then the O.D.E. (3.3) of a critical geodesic canbe solved for all v ∈ lR.But other implications fails.Example 3.34 (Atkin [3]) There exists an infinite dimensional complete Hilbertsmooth manifold M and x, y ∈ M such that there is no critical geodesic connectingx to y.30

That is,• (M, d) is complete ⇏ exp c is surjective,• and (M, d) is complete ⇏ minimal geodesics exist.It is then, in general, quite difficult to prove that an infinite dimensionalmanifold admits minimal geodesics (even when it is known to be metricallycomplete). There are though some positive results, such as Ekeland [19] (thatwe cannot properly discuss for lack of space); or the following.Theorem 3.35 (Cartan–Hadamard) Suppose that M is connected, simplyconnected and has seminegative sectional curvature; then these are equivalent:• (M, d) is complete• there exists a c ∈ M such that the map η → exp c (η) is well definedand then there exists an unique minimal geodesic connecting any two points.For the proof, see Corollary 3.9 and 3.11 in sec. IX.§3 in Lang [30].3.7 Gradient in abstract differentiable manifoldLet M be a differentiable manifold, c ∈ M, and φ : U → V be a chart withc ∈ V .Definition 3.36 Let E : M → lR be a functional; we say that it is Gâteauxdifferentiable at c if for all h ∈ T c M there exists the directional derivativeat c in direction hwhere x = φ −1 (c), k = [Dφ(c)] −1 (h).DE(c; h) := limt→0E(φ(x + tk)) − E(c)t. (3.4)Fixing c (and then x), DE(c; ·) is a linear function from h ∈ T c M into lR;for this reason it is considered an element of the cotangent space T ∗ c M (thatis the dual space of T c M — and we will not discuss here further). Suppose nowthat we add a Riemannian geometry to M; this defines an inner product 〈·, ·〉 con T c M, so we can then define the gradient.Definition 3.37 (Gradient) The gradient ∇E(c) of E at c is the uniquevector v ∈ T c M such that〈v, h〉 c = DE(c; h) ∀h ∈ T c M .If M is modeled on a Hilbert space H, and the inner product 〈·, ·〉 c used onT c M is equivalent to the inner product in H (as we discussed in 3.26), then theabove equation uniquely defines what the gradient is. When M is modeled on aFréchet space, though, there is no choice of inner product that is “compatible”with M; and there are pathological situations where the gradient does not exist.31

Example 3.38 Let M = C ∞ ([−1, 1] → lR); since M is a vector space, then it istrivially a manifold, with a single identity chart; and T c M = M. Let E : M → lRbe the evaluation functional E(f) = f(0), then DE(f; h) = h(0); let〈f, g〉 =∫ 1−1f(t)g(t) dt ;then the gradient of E would be δ 0 (the Dirac’s delta), that is a distribution (ormore simply a probability measure) but not an element of M.3.8 Group actionsDefinition 3.39 Let G be a group, M a set. We say that G acts on M if thereis a mapG × M → Mg, m ↦→ g · mthat respects the group operations:• if e is the identity element then e · m = m, and• for any g, h ∈ G, m ∈ Mh · (g · m) = (h · g) · m . (3.5)If G is topological group, that is, a group with a topology such that the groupoperation is continuous, and M has a topology as well, we will require that theaction be continuous. Similarly for smooth actions between smooth manifolds.The action of a group G on M induces an equivalence relation ∼, definedbym 1 ∼ m 2 iff there exists g such that m 1 = g · m 2 .We can then define the following objects.Definition 3.40m.• The orbit [m] of m is the family of all m 1 equivalent to• The quotient M/G is the space M/ ∼ of equivalence classes.• There is a projection π : M → M/G, sending each m to its orbit π(m) =[m].We will see many examples of actions in Example 4.8.32

3.8.1 Distances and groupsLet d M (x, y) be a distance on a space M, and G a group acting on M; we maythink of defining a distance on B = M/G byd B ([x], [y]) :=inf d M (x, y) = inf d M (g · x, h · y) (3.6)x∈[x],y∈[y] g,h∈Gthat is the lowest distance between two orbits. Unfortunately, this definitiondoes not in general satisfy the triangle inequality.Proposition 3.41 If d M is invariant w.r.to the action of the group G,that isd M (g · x, g · y) = d M (x, y) ∀g ∈ G ,then the above (3.6) can be simplified toand d B is a semidistance.d B ([x], [y]) = infg∈G d M (g · x, y) . (3.7)Proof. We write d B (x, y) instead of d B ([x], [y]) for simplicity. It is easy to checkthatd B (x, y) = infg∈G d M (g · x, y) = infg∈G d M (y, g · x) = infg∈G d M (g −1 · y, x) = d B (y, x) .For the triangle inequality,d B (x, z) + d B (z, y) = inf d M (g · x, z) + inf d M (z, h · y) =g∈G h∈G= inf d M (g · x, z) + d M (z, h · y) ≥ inf d M (g · x, h · y) = d B (x, y)g,h∈G g,h∈GRemark 3.42 There is a main problem: is d B ([x], [y]) > 0 when [x] ≠ [y] ?That is, there is no guarantee, in line of principle, that the above definition (3.7)won’t simply result in d B ≡ 0.4 Spaces and metrics of curvesIn this section we review the mathematical definitions regarding the space ofcurves in full detail, and express a set of mathematical goals for the theory.4.1 Classes of curvesRemember that S 1 = {x ∈ lR 2 | |x| = 1} is the circle in the plane.We will often associate lR 2 = IC, for convenience. Consequently,33

• sometimes S 1 will be identified with lR/(2π) (that is lR modulus 2π translations).• but other times (and in particular if the curve is planar) we will associateS 1 = {e it , t ∈ lR} = {z, |z| = 1} ⊂ IC.We recall that a curve is a map c : S 1 → lR n . The image of the curve isc(S 1 ).Definition 4.1 (Classes of Curves)• Imm(S 1 , lR n ) is the class of immersed curves c, such that c ′ ≠ 0at all points.• Imm f (S 1 , lR n ) is the class of freely immersed curve, the immersedcurves c such that, moreover, if φ : S 1 → S 1 is a diffeomorphism andc(φ(t)) = c(t) for all t, then φ =Id. So, in a sense, the curve is “completely”characterized by its image.• Emb(S 1 , lR n ) are the embedded curves, maps c that are diffeomorphiconto their image c(S 1 ); and the image is an embeddedsubmanifold of lR n of dimension 1.Each class contains the one following it.non-freely immersed curve.The following is an example of aExample 4.2 We define the doubly traversed circle using the complex notationc(z) = z 2 for z ∈ S 1 ⊂ IC; or otherwise identifying S 1 = lR/(2π), and inthis casec(θ) = (cos(2θ), sin(2θ))for θ ∈ lR/(2π). Setting φ(t) = t + π, we have that c = c ◦ φ, so c is not freelyimmersed.Vice versa, the following result is a sufficient condition to assert that a curve isfreely immersed.Proposition 4.3 (Michor and Mumford [37]) If c is immersed and thereis a x ∈ lR n s.t. c(t) = x for one and only one t, then c is freely immersed.Remark 4.4 Imm(S 1 , lR n ) , Imm f (S 1 , lR n ) , Emb(S 1 , lR n ) , are open subsetof the Banach space C r (S 1 → lR n ) when r ≥ 1, so they are trivially submanifolds(the charts are identity maps); and similarly for the Fréchet space C ∞ (S 1 → lR n ).Imm(S 1 , lR n ) , Imm f (S 1 , lR n ) , Emb(S 1 , lR n ) are connected iff n ≥ 3;whereas in the case n = 2 of planar curves, they are divided in connectedcomponents each containing curves with the same winding number (see Prop. 8.1for the definition).34

4.2 Gâteaux differentials in the space of immersed curvesLet once again M be the manifold of smooth immersed curves. We will mainly beinterested in the Gâteaux derivation of operators O and functions E : M → lR k .By operator we mean any one of these possible operations.• An operatorO : M → T c Msuch that O(c) is a smooth a vector field; examples are the operations ofcomputing the tangent field T , and the curvature H.• An operatorO : M → Msuch that O(c) is an immersed curve; examples are all the group actionslisted in Section 4.3• An operatorO : M → (T c M → T c M)such that O(c) is itself an operator on vector fields along c; an example isthe derivative D s ;• or sometimesO : M → (T c M → lR n )such as the average along the curve O(c) = avg c (·).In all above cases, the evaluated operator O(c) is itself a function; for thisreason, to avoid confusion in the notation, in this section we write the Gâteauxderivation as D c,h O instead of DO(c; h).The charts in M are trivial (as noted in 4.4), so the definition (3.4) of Gâteauxdifferential simplifies toD c,h E := ∂ t E(c + th)| t=0(4.1)and similarly for an operator.These rules following are the building blocks that are used (by means ofstandard calculus rules for derivatives) to compute the derivation of all othergeometric operators usually found in computer vision applications.Proposition 4.5D c,h len(c) =∫c∫(D s h · D s c) ds = −c(h · D 2 sc) ds , (4.2)D c,h (D s O) = −(D s h · D s c)(D s O) + D s (D c,h O) , (4.3)∫∫D c,h O ds = D c,h O + O. (D s h · D s c) ds , (4.4)cc∫∫∫ ∫D c,h O ds = D c,h O + O.(D s h · D s c) ds − O ds (D s h · D s c) dscccc(4.5)35

The proof is by direct computation.For example, from (4.5) we easily obtain (2.7), that is∫D c,h avg c (c) = h + (c − avg c (c))(D s h · D s c) ds .cIf C(t, θ) is a homotopy, we can obtain a different interpretation of all previousequalities substituting formally C for O, ∂ t for D c,h and eventually ∂ t C for h.The above proposition helps in formalizing and speeding up calculus ongeometric quantities, as in this example.Example 4.6 We can Gâteaux-derive the “second arc parameter derivativeoperator D ss ” by repeated applications of (4.3), as followsD c,h (D ss g) = −(D s h · D s c)D ss g + D s [D c,h (D s g)] == −(D s h · D s c)D ss g + D s [−(D s h · D s c)(D s g) + D s (D c,h g)] == D ss (D c,h g) − 2(D s h · D s c)D ss g − [D s (D s h · D s c)](D s g)and hence obtain the Gâteaux differential of the curvatureD c,h D ss c = D ss h − 2(D s h · D s c)D ss c − [D s (D s h · D s c)]D s cand eventually obtain the Gâteaux differential of the elastica energy bysubstituting this last equality in (4.4)∫∫D c,h |D ss c| 2 ds = 2D ss c · (D c,h D ss c) + |D ss c| 2 . (D s h · D s c) ds =cc∫= 2D ss c · D ss h − 3(D s h · D s c)|D ss c| 2 dssince (D ss c · D s c) = 0.cRemark 4.7 Most objects we deal with share an important property: they arereparameterization invariant. In the case of operators, in all cases in whichO(c) is a curve or a vector field, then this means that for any diffeomorphismϕ ∈ Diff + (S 1 ) there holdsO(c ◦ φ)(θ) = O(c)(φ(θ)) . (4.6)(whereas for ϕ ∈ Diff − (S 1 ) there is a possible sign choice as explained in Definition1.5).By considering an infinitesimal perturbation a : S 1 → lR of the identity inDiff(S 1 ), we obtain the formulaD c,h O(c)(θ) = a(θ) . ∂ θ O(c)(θ) for any h(θ) = a(θ).∂ θ c(θ) (4.7)and this last may be rewritten in arc parameterD c,h O(c)(s) = a(s) . D s O(c)(s) for any h(s) = a(s).D s c(s) . (4.8)Similarly if E : M → lR is a reparameterization invariant energy, thenD c,h E(c) = 0 for any h = a.D s c . (4.9)36

It is also possible to prove that the condition (4.8) is equivalent to saying thatthe operator is reparameterization invariant; and similarly for E.4.3 Group actions on curvesLet M again be the manifold of immersed curves. An example of group actionthat we saw in Definition 1.3 is obtained when G = Diff(S 1 ); G acts on curvesby right compositionc, φ ↦→ c ◦ φand this action is the reparameterization.In general, all groups that act on lR n also act on M; many are of interestin computer vision. The action of these groups on curves is always of the formc, A ↦→ A ◦ c, where A : lR n → lR n is the group action on lR n .Example 4.8 • O(n) is the group of rotations in lR n . It is represented bythe group of orthogonal matrixesO(n) := {A ∈ lR n×n | AA t = A t A = Id}The action on v ∈ lR n is the matrix·vector multiplication Av.• SO(n) is the group of special rotations in lR n with det(A) = 1.• lR n is the group of translations in lR n . The action on v ∈ lR n is thevector sum v ↦→ v + T .• E(n) is the Euclidean group, generated by rotations and translations.• lR + is the group of rescalings.The quotient spaces M/G are the spaces of curves up to rotation and/ortranslations and/or scaling (. . . ).4.4 Two categories of shape spacesLet M be a manifold of curves. We should distinguish two different ideas ofshape spaces of curves.geometric curves up togeometric curvesposecurves up to reparameterization,rotation, translation,curves up to reparameterization,The space of shapesscaling,is modeled as M/Diff(S 1 ) , M/Diff(S 1 )/E(n)/lR + ,is well-suited to shape optimization. shape analysis.Michor and Mumford [37], Srivastava et al. [52], Mio andReferences:Yezzi-M.-Sundaramoorthi Srivastava [41], Klassen et al.[53–58] [35] Glaunès et al. [28], Younes [68], Michor et al.[22], Trouvé and Younes [59] [39]37

The term preshape space is sometimes used for the leftmost space, when bothspaces are studied in the same paper.4.5 Geometric curvesUnfortunately the quotientB i = Imm(S 1 , lR n )/Diff(S 1 )of immersed curves up to reparameterization is not a Fréchet manifold.We (re)define the space of geometric curves.Definition 4.9B i,f (S 1 , lR n ) = Imm f (S 1 , lR n )/Diff(S 1 )is the quotient of Imm f (S 1 , lR n ) (the free immersions) by the diffeomorphismsDiff(S 1 ) (that act as reparameterizations).The good news is thatProposition 4.10 (§2.4.3 in Michor and Mumford [37]) If Imm f has thetopology of the Fréchet space of C ∞ functions, then B i,f is a Fréchet manifoldmodeled on C ∞ .The bad news is that• when we add a simple Riemannian metric to B i,f , the resulting metricspace is not metrically complete; indeed, there cannot be any norm on C ∞that generates the same topology of the Fréchet space C ∞ (as we discussedin 3.26);• by modeling B i,f as a Fréchet manifold, some calculus is lost, as we saw inSection 3.2.5.Remark 4.11 It seems that this is the only way to properly define the manifold.If otherwise we choose M = C k (S 1 → lR n ) to be the manifold of curves, then ifc ∈ M, c ′ ∉ T c M. Hence we (must?) model M on C ∞ functions.4.5.1 Research pathFollowing Michor and Mumford [37] we so obtained a possible program of mathresearch:• defineB = B i,f (S 1 , lR n ) = Imm f (S 1 , lR n )/Diff(S 1 )and consider B as a Fréchet manifold modeled on C ∞ ,• define a Riemann/Finsler geometry on it, study its properties,• metrically complete the space.In the last step, we would hope to obtain a differentiable manifold; unfortunately,this is sometimes not true, as we will see in the overview of the literature.38

4.6 Goals (revisited)We formulate an abstract set of goal properties on a metric 〈h 1 , h 2 〉 G|c on spacesof curves.1. [rescaling invariance] For any λ > 0, if we rescale c to λc, then〈h 1 , h 2 〉 G|λc = λ a 〈h 1 , h 2 〉 G|c(where a ∈ lR is an universal constant);2. [Euclidean invariance] Suppose that A is an Euclidean transformation,and R is its rotational part; if we apply A to c and R to h 0 , h 1 , then〈Rh 1 , Rh 2 〉 G|Ac = 〈h 1 , h 2 〉 G|c ;3. [parameterization invariance]the metric does not depend on the parameterization of the curve, that is‖˜h‖˜c = ‖h‖ c when ˜c(t) = c(ϕ(t)) and ˜h(t) = h(ϕ(t)).If a metric satisfies the above three properties, we say that it is a geometricmetric.4.6.1 Well posednessWe also add a more basic set of requirements.0. [well-posedness and existence of minimal geodesics]• The metric induces a good distance d: that is, the distance between differentcurves is positive, and d generates the same topology that the atlas of themanifold M induces;• (M, d) is complete;• for any two curves in M, there exists a minimal geodesic connecting them.4.6.2 Are the goal properties consistent?So we state the abstract problem:Problem 4.12 Consider the space of curves M, and the family of all Riemannian(or, Finsler) Geometries F on M.Does there exist a metric F satisfying the above properties 0,1,2,3?Consider metrics F that may be written in integral form∫F (c, h) = f ( c(s), ∂ s c(s), . . . , ∂ j sc(s), h(s), . . . , ∂ i j s ih(s)) dscwhat is the relationship between the degrees i, j and the properties in this section?All this boils down to a fundamental question: can we design metrics to satisfyour needs?39

5 Representation/embedding/quotient in the currentliterature5.1 The representation/embedding/quotient paradigmA common way to model shapes is by representation/embedding:• we represent the shape A by a function u A• and then we embed this representation in a space E, so that we canoperate on the shapes A by operating on the representations u A .Most often, representation/embedding alone do not directly provide a satisfactoryshape space. In particular, in many cases it happens that the representationis “redundant”, that is, the same shape has many different possible representations.An appropriate quotient is then introduced.There are many examples of shape spaces in the literature that are studiedby means of the representation/embedding/quotient scheme. Understanding thebasic math properties of these three operations is then a key step in understandingshape spaces and designing/improving them.We now present a rapid overview of how this scheme is exploited in thecurrent literature of shape spaces; then we will come back to some of them toexplain more in depth.5.2 Current literatureExample 5.1 (The family of all non empty compact subsets of lR N ) Astandard representation is obtained by associating each closed subset A to thedistance function u A or the signed distance function b A (that were defined inDefinition 2.1). We may then define a topology of shapes by deciding thatA n → A when u An → u A uniformly on compact sets. This convergence coincideswith the Kuratowski topology of closed sets; if we limit shapes to be compact sets,the Kuratowski topology is induced by the Hausdorff distance. See section 6.2.Example 5.2 Trouvé-Younes et al (see Glaunès et al. [22], Trouvé and Younes[59] and references therein) modeled the motion of shapes by studying a leftinvariant Riemannian metric on the family G of diffeomorphisms of the spacelR N ; to recover a true metric of shapes, a quotient is then performed w.r.to alldiffeomorphisms G 0 that do not move a template shape.But the representation/embedding/quotient scheme is also found when dealingwith spaces of curves:Example 5.3 (Representation by angle function) In the work of Klassenet al. [28], Srivastava et al. [52], Mio and Srivastava [41], smooth planar closedcurves c : S 1 → lR 2 of length 2π are represented by a pair of angle-velocityfunctions c ′ (u) = exp(φ(u) + iα(u)) (identifying lR 2 = IC) then (φ, α) areembedded as a subset N in L 2 (0, 2π) or W 1,2 (0, 2π). Since the goal is to obtain40

a shape space representation for shape analysis purposes, a quotient is thenintroduced on N. See Section 8.1.Example 5.4 Another representation of planar curves for shape analysis isfound in Younes [68]. In this case the angle function is considered mod(π). Thisrepresentation is both simple and very powerful at the same time. Indeed, it ispossible to prove that geodesics do exist and to explicitly show examplesof geodesics. See Section 8.2.Example 5.5 (Harmonic representation) A. Duci et al (see [16, 17]) representa closed planar contour as the zero level of a harmonic function. This novelrepresentation for contours is explicitly designed to possess a linear structure,which greatly simplifies linear operations such as averaging, principal componentanalysis or differentiation in the space of shapes.And, of course, we have in this list the spaces of embedded curves.Example 5.6 When studying embedded curves, usually, for the sake of mathematicalanalysis, the curves are modeled as immersed parametric curves; aquotient w.r.to the group of possible reparameterizations of the curve c (thatcoincides with the group of diffeomorphisms Diff(S 1 )) is applied afterward to allthe mathematical structures that are defined (such as the manifold of curves, theRiemannian metric, the induced distance, etc.).It is interesting to note this fact.Remark 5.7 (A remark on the quotienting order) When we started talkingof geometric curves, we proposed the quotient B = M/Diff(S 1 ) (the space ofcurves up to reparameterization); and this had to be followed by other quotientsw.r.to Euclidean motions and/or scaling, to obtain M/Diff(S 1 )/E(n)/lR + . Inpractice, though, the space B happens to be more difficult to study; hence mostshape space theories that deal with curves prefer a different order: the quotientM/E(n)/lR + is modeled and studied first; then the quotient M/E(n)/lR + /Diff(S 1 )is performed (often, only numerically).6 Metrics of setsWe now present two examples of Shape Theories where a shape may be a genericsubset of the plane; with particular attention to how they behave w.r.to curves.6.1 Some more math on distance and geodesicsWe start by reviewing some basic results in abstract metric spaces theory.41

6.1.1 Length induced by a distanceIn this subsection (M, d) is a generic metric space.Definition 6.1 (Length by total variation) Define the length of a continuouscurve γ : [α, β] → M, by using the total variationn∑Len d γ := sup d ( γ(t i−1 ), γ(t i ) ) t 1t 6t 4.Ti=1t 0 t 2 t5where the supremum is carried out over all finite subsets T = {t 0 , · · · , t n } ⊂[α, β] and t 0 ≤ · · · ≤ t n .Definition 6.2 (The induced geodesic distance)d g (x, y) := infγ Lend γ (6.1)the infimum is in the class of all continuous γ in M connecting x to y.Note that d g (x, y) < ∞ iff x, y may be connected by a Lipschitz path.6.1.2 Minimal geodesicsDefinition 6.3 If in the definition (6.1) there is a curve γ ∗ providing theminimum, then γ ∗ is called a minimal geodesic connecting x to y.Proposition 6.4 In Riemann and Finsler manifolds, the integral length Len(γ)that we defined in Section 3.6.1 coincides with the total variation length Len d γthat we defined in Definition 6.1. As a consequence, d = d g : we say that d ispath-metric.In general, though, it is easy to devise examples where d ≠ d g .Example 6.5 The set M = S 1 ⊂ lR 2 is a metric space, ifassociated with d(x, y) = |x − y| (that is represented as a dottedsegment in the picture); in this case, d g (x, y) = | arg(x)−arg(y)|(that is represented as a dashed arc in the picture).6.1.3 Existence of geodesics and of averageProposition 6.6 If for a ρ > 0 the closed ballD g (x, ρ) := {y | d g (x, y) ≤ ρ}is compact, then x and any y ∈ D g may be connected by a geodesic.Proposition 6.7 If for a x ∈ M and all ρ > 0, D g (x, ρ) is compact, then thedistance-based average (that was defined in 2.4) exists.The proofs may be found in [15].t 342

6.1.4 Geodesic raysMore in general,Definition 6.8 a continuous curve γ : I → M (where I ⊂ lR is an interval) isa geodesic ray if for each t ∈ I there is a ε > 0 s.t. J = [t − ε, t + ε] ⊂ I andγ restricted to J is the geodesic between γ(t − ε) and γ(t + ε).A critical geodesic in a Riemann or Finsler smooth manifold is always ageodesic ray, as in this example.Example 6.9 The multiply traversed full circle γ(t) = (0, cos(t), sin(t)) witht ∈ lR is a geodesic ray in the sphere S 2 .6.1.5 Hopf–Rinow , Cohn-Vossen TheoremDefinition 6.10 • We recall that (M, d) is path-metric if d = d g .• Moreover, (M, d) is locally compact if small closed balls are compact.Theorem 6.11 (Hopf–Rinow , Cohn-Vossen) Suppose (M, d) is locally compactand path-metric; the following statements are equivalent:• for all x ∈ M, ρ > 0, the closed ballis compact;• (M, d) is complete;D(x, ρ) := {y | d(x, y) ≤ ρ}• any geodesic ray γ : [0, 1) → M may be extended on [0, 1];and all imply that any two points may be connected by a minimal geodesic.Note that the above theorem cannot be used in infinite dimensional differentiablemanifolds, since the the closed balls are never compact in that case (seeTheorems 1.21, 1.22 in Chapter I in [48]).6.2 Hausdorff metricLet Ξ be the collection of all compact subsets of lR N . (This is sometimescalled the “topological hyperspace” of lR N ). Let A, B ⊂ lR N compact non-empty.We already defined the distance function u A (x) := inf y∈A |x − y|, and theHausdorff distance of two sets A, B as( ) ( )d H (A, B) := u B (x) ∨ u A (x) .supx∈AWe now list some known properties.supx∈B43

0101200000000000011111111111110000 111101010101010101010101Figure 6: Fattening of a setProperties 6.12• (Ξ, d H ) is path-metric;• d H (A, B) = supx∈lR N |u A (x) − u B (x)| .• each family of compact sets that are contained in a fixed large closed ballin lR N is compact in (Ξ, d H ); so• any closed ball D(A, ρ) := {B | d H (A, B) ≤ ρ} is compact in (Ξ, d H ), andmoreover• by Theorem 6.11, minimal geodesics exist in (Ξ, d H ).6.2.1 An alternative definitionLet D r (x) be the closed ball of center x and radius r > 0 in lR N , and D r = D r (0).We define the fattened set to beA + D r := {x + y | x ∈ A, |y| ≤ r} = ⋃D r (x) = {y | u A (y) ≤ r}.Note that the fattened set is always closed, (since the distance function u A (x) iscontinuous).Example 6.13 In figure 6 we see an example of a set A fattened to r = 1, 2;the set A is the black polygon (and is filled in), whereas the dashed lines in thedrawing are the contours of the fattened sets. 7We can then state the following equivalent definition of the Hausdorff distance:x∈Ad H (A, B) = min{δ > 0 | A ⊂ (B + D δ ), B ⊂ (A + D δ )} .7 The fattened sets are not drawn filled — otherwise they would cover A.44

6.2.2 Uncountable many geodesicsUnfortunately d H is quite “unsmooth”. There are choices of A, B compact thatmay be joined by an uncountable number of geodesics. In fact we can considerthis simple example.Example 6.14 (Duci and Mennucci [15]) Let us setA = {x = 0, 0 ≤ y ≤ 2}B = {x = 2, 0 ≤ y ≤ 1}C t = { x = 1, 0 ≤ y ≤ 2} 3 ∪ {y = 0, 1 ≤ x ≤ t}ACtBwith 1 ≤ t ≤ √ 5/2;and in the picture we represent (dashed) the fattened sets A + D √ 5/2 andB + D √ 5/2 . Note that d H(A, B) = √ 5 while d H (A, C t ) = d H (B, C t ) = √ 5/2:so C t are all midpoints that are on different geodesics between A and B.6.2.3 Curves and connected setsLet again Ξ be the collection of all compact subsets of lR N . Let Ξ c be thesubclass of compact connected subsets of lR N . We now relate the space of curvesto this metric space, by listing these properties and remarks.• Ξ c is a closed subset of (Ξ, d H );• Ξ c is the closure in (Ξ, d H ) of the class of (images of) all embedded curves.• Ξ c is Lipschitz-path-connected 8 ;• for all above reasons, it is possible to connect any two A, B ∈ Ξ c by aminimal geodesic moving in Ξ c .• So, if we try to find a minimal geodesic connecting two curves using themetric d H , we will end up finding a geodesic in (Ξ c , d H ); and similarly ifwe try to optimize an energy defined on curves.• But note that Ξ c is not geodesically convex in Ξ, that is, there exist twopoints A, B ∈ Ξ c and a minimal geodesics ξ connecting A to B in themetric space (Ξ, d), such that the image of ξ is not contained inside Ξ c .• We don’t know if (Ξ c , d H ) is path-metric.8 That is, any A, B ∈ Ξ c can be connected by a Lipschitz arc γ : [0, 1] → Ξ c45

6.2.4 Applications in computer visionCharpiat et al. [9] propose an approximation method to compute len H (ξ) bymeans of a family of energies defined using a smooth integrand; the approximationis mainly based on the property ‖f‖ L p → p ‖f‖ L ∞, for any measurable functionf defined on a bounded domain; they successively devise a method to findapproximation of geodesics.6.3 A Hausdorff-like distance of compact setsIn Duci and Mennucci [15] a L p -like distance on the compact subsets of lR N wasproposed. (Here p ∈ [1, ∞).)To this end, we fix ϕ : [0, ∞) → (0, ∞) , decreasing, C 1 , with ϕ(|x|) ∈ L p . Wethen define v A (x) := ϕ(u A (x)), where u A is the distance function. We eventuallydefine the distanced(A, B) := ‖v A − v B ‖ L p . (6.2)Remark 6.15 This shape space is a perfect example of the representation/embedding/quotient scheme. Indeed, this shape space is represented as N c ={v A | A compact} and embedded in L p . Given v ∈ N c , we recover the shapeA = {v = ϕ −1 (0)} (that is a level set of v).Example 6.16 A simple example (that works for all p) is given by ϕ(t) = e −t ,so that v A (x) = exp (−u A (x)); in this case, A = {v = 1}.The distance d of eqn. (6.2) enjoys the following properties.Properties 6.17• It is Euclidean invariant;• it is locally compact but not path-metric;• the topology induced is the same as that induced by d H ;• minimal geodesics do exist, since D g := {B | d g (A, B) ≤ ρ} is compact.The proofs are in [15]. We present a numerical computation of a minimal geodesic(by A. Duci) in Figure 7 on the next page.6.3.1 Analogy with the Hausdorff metric, L p vs L ∞We recall that d H (A, B) = ‖u A (x) − u B (x)‖ L ∞ ; whereas instead now we areproposing d(A, B) := ‖v A −v B ‖ L p . The idea being that this distance of compactsets is modeled on L p , whereas the Hausdorff distance is “modeled” on L ∞ .(Note that the Hausdorff distance is not really obtained by embedding, sinceu A ∉ L ∞ ). L p is more regular than L ∞ , as shown by this remark.Remark 6.18 Given any f, g ∈ L p with p ∈ (1, ∞), the segment connectingf to g is the unique minimal geodesic connecting them. Suppose now that thedimension of L ∞ (Ω, A, µ) is greater than 1. Given generically f, g ∈ L ∞ , thereis an uncountable number of minimal geodesics connecting them. 99 “Generically” is meant in the Baire sense: the set of exceptions is of first category.46

Figure 7: Example of a minimal geodesic(For the proof see 2.11 & 2.13 in [15]). The above result suggests that it maybe possible to shoot geodesics in the “Riemannian metric” associated to thisdistance.6.3.2 Contingent coneLet again N c = {v A | A compact} be the family of all representations. Given av ∈ N c , let T v N c ⊂ L p be the contingent coneT v N c := {lim t n (v n − v) | t n > 0, v n ∈ N c , v n → v}n{}v n − v= λ lim | λ ≥ 0, v n → v ,n ‖v n − v‖ L pwhere it is intended that the above limits are in the sense of strong convergencein L p .T v N c contains all directions in which it is possible “in the L p sense” toinfinitesimally deform a compact set. For example, if Φ(x, t) : lR N × (−ε, ε) →lR N is smooth diffeomorphical motion of lR N , and A t := Φ(A, t), then v At is(locally) Lipschitz, so it is differentiable for almost all t, and the derivativeis in T N c . This includes all perspective, affine, and Euclidean deformations.Unfortunately the contingent cone is not capable of expressing some shapedeformations.Example 6.19 We consider the removing motion; let A be compact, andsuppose that x is in the internal part of A; let A t := A \ B(x, t) be the removal47

of a small ball from A. The motion v At inside N c is Lipschitz, but the limitlimt→0+v At − v A‖v At − v A ‖ L pdoes not exist in L p . (Morally, if p = 1, the limit would be the measure δ x ).6.3.3 Riemannian metricLet now p = 2. The set N c may fail to be a smooth submanifold of L 2 ; yetwe will, as much as possible, pretend that it is, in order to induce a sort of“Riemannian metric” on N c from the standard L 2 metric.Definition 6.20 We define the “Riemannian metric” on N c simply by〈h, k〉 := 〈h, k〉 L 2for h, k ∈ T v N c and correspondingly a norm bywhere T v N c is the contingent cone.|h| := √ 〈h, h〉Proposition 6.21 The distance induced by this “Riemannian metric” coincideswith the geodesically induce distance d g .The proof is in 3.22 in Duci and Mennucci [15].To conclude, we propose an explicit computation of the Riemannian Metricfor the case of compact sets in the plane with smooth boundaries; we then pullback the metric to obtain a metric of closed embedded planar curves. We startwith the case of convex sets.We fix p = 2, N = 2.6.3.4 Polar coordinates of smooth convex setsLet Ω ⊂ lR 2 be a convex set with smooth boundary; let y(θ) : [0, L] → ∂Ω bea parameterization of the boundary (by arc parameter), ν(θ) the unit vectornormal to ∂Ω and pointing external to Ω. The following “polar” change ofcoordinates ψ holds:ψ : lR + × [0, L] → lR \ Ω , ψ(ρ, θ) = y(θ) + ρν(θ) (6.3)see figure 8 on the following page. We suppose that y(θ) moves on ∂Ω inanticlockwise direction; so ν = J∂ s y, ∂ ss y = −κν; where J is the rotation matrix(of angle −π/2), κ is the curvature, and ∂ s y is the tangent vector.We can then express a generic integral through this change of coordinates as∫∫f(x) dx = f(ψ(ρ, s))|1 + ρκ(s)| dρ dslR 2 \Ω∫lR + ∂Ωwhere s is arc parameter, and ds is integration in arc parameter.48

y(θ)ν(θ)y(θ) + ρν(θ)Figure 8: Polar coordinates around a convex set6.3.5 Smooth deformations of a convex setWe want to study a smooth deformation of Ω, that we call Ω t ; then the bordery(θ, t) depends on a time parameter t. Suppose also that κ(θ) > 0, that is, thatthe set is strictly convex: then for small smooth deformations, the set Ω t willstill be strictly convex. We suppose that the border of Ω t moves with orthogonalspeed α; more precisely, we assume that (∂ t y) ⊥ ∂ s y, that is, (∂ t y) = αν withα = α(t, θ) ∈ lR. Since this deformation is smooth, we expect that it will beassociated to a vector h α ∈ T v N c , defined by h α := ∂ t v Ωt . We now show brieflyhow to explicitly compute it.Suppose that x is a fixed point in the plane, x ∉ Ω t , and express it usingpolar coordinates x = ψ(ρ, θ), with ρ = ρ(t), θ = θ(t). With some computations,ρ ′ = −α. Now, for x ∉ Ω t , u Ωt (x) = ρ(t) hence we obtain the explicit formulafor h αh α := ∂ t v Ωt (x) = −ϕ ′ (u Ωt (x))α ;whereas h α (x) = 0 for x ∈ ˚Ω t .6.3.6 Pullback of the metric on convex boundariesLet us fix two orthogonal smooth vector fields α(s)ν(s), β(s)ν(s), that representtwo possible deformations of ∂Ω; those correspond to two vectors h α , h β ∈ T v N c ;so the Riemannian Metric that we defined in 6.20 can be pulled back on ∂Ω, toprovide the metric∫∫〈α, β〉 := h α (x)h β (x)dx = h α (x)h β (x)dx =lR 2 lR 2 \Ω∫ [∫]=(ϕ ′ (ρ)) 2 (1 + ρκ(s)) dρ α(s)β(s)ds∂Ω lR +that is,with∫〈α, β〉 =∂Ω(a + bκ(s))α(s)β(s)ds (6.4)∫∫a := (ϕ ′ (ρ)) 2 dρlR + , b := (ϕ ′ (ρ)) 2 ρ dρ .lR +49

CutΩ(The arrows represent the distance R(s) from y(s) ∈ ∂Ω to the cutlocus).Figure 9: Smooth contour and cutlocus.6.3.7 Pullback of the metric on smooth contoursIf Ω is smooth but not convex, then the above formula holds up to the cutlocus.Definition 6.22 The cutlocus ( a.k.a. the external skeleton) is the set ofpoints x ∉ Ω such that there are two (or more) different points y 1 , y 2 ∈ Ω ofminimum distance from x to Ω, that is,|x − y 1 | = |x − y 2 | = u Ω (x) .We define a function R(s) : [0, L] → [0, ∞] that measures the distance from ∂Ωto the cutlocus Cut; in turn, we can parameterize Cut asCut = {ψ(R(s), s) | s ∈ [0, L], R(s) < ∞} ;R(s) is locally Lipschitz where it is finite (by results in Itoh and Tanaka [25], Liand Nirenberg [32]). The “polar” change of coordinates ψ (defined in (6.3)) is adiffeomorphism between the sets{(ρ, s) s ∈ [0, L], 0 < ρ < R(s)} ↔ lR 2 \ (Ω ∪ Cut) .See fig. 9.The pullback metric for deformations of the contour ∂Ω is∫ [ ∫ ]R(s)〈h, k〉 =(ϕ ′ (ρ)) 2 (1 + ρκ(s)) dρ α(s)β(s) ds∂Ω06.3.8 ConclusionIn this case we have found (a posteriori) a Riemannian metric of closed embeddedplanar curves, and we know the structure of the completion, and the completionadmits minimal geodesics. On the down side, the completion is not really “asmooth Riemannian manifold”. For example, it is difficult to study the minimalgeodesics, and to prove any property about them. Anyway, the fact that N c islocally compact and the regularity property 6.18 of L p spaces suggest that itmay be possible to “shoot geodesics” (in some weak form). Unfortunately thetopology has too few open sets to be used for shape optimization: many simpleenergy functionals are not continuous.50

7 Finsler geometries of curvesWe present two examples of Finsler geometries of curves that have been used(sometimes covertly) in the literature.7.1 Tangent and normalLet v, T ∈ lR N , letN = T ⊥ = {w ∈ lR N : w · T = 0}be the space orthogonal to T . Usually T will be the tangent to a curve c. 10Definition 7.1 We define the projection onto the normal space N = T ⊥and the projection on the tangent Tπ N : lR N → lR N , π N v = v − 〈v, T 〉Tπ T : lR N → lR N , π T v = 〈v, T 〉Tso π N v + π T v = v and |π N v| 2 + |π T v| 2 = |v| 2 .7.2 L ∞ metric and Fréchet distanceIf we wish to define a norm on T c M that is modeled on the norm of the Banachspace L ∞ (S 1 → lR N ), we defineF ∞ (c, h) := ‖π N h‖ L ∞= sup |π N h(θ)|θwhere N = (D s c) ⊥ . This metric weights the maximum normal motion of thecurve. (The rôle of π N will be properly explained in §11.10). This Finsler metricis geometric. The length of a homotopy isLen ∞ (C) :=∫ 1Definition 7.2 (Fréchet distance)0d f (c 0 , c 1 ) := infφsup |π N ∂ t C(t, θ)| dt .θ∈S 1sup |c 1 (φ(u)) − c 0 (u)|uwhere u ∈ S 1 and φ is chosen in the class of diffeomorphisms of S 1 .This distance is induced by the Finsler metric F ∞ (c, h).Theorem 7.3 d f (c 0 , c 1 ) coincides with the infimum of the length Len ∞ (C) forC connecting c 0 to (a reparameterization of) c 1 .For the proof, see theorem 15 in [35].10 There is a slight abuse of notation here, since in the definition N = T ⊥ given for planarcurves in 1.12, we defined N to be a “vector” and not the “vector space orthogonal to T ”.51

7.3 L 1 metric and Plateau problemIf we wish to define a geometric norm on T c M that is modeled on the norm ofthe Banach space L 1 (S 1 → lR N ), we may define the metric∫F 1 (c, h) = ‖π N h‖ L 1 = |π N h(θ)||c ′ (θ)|dθ ;the length of a homotopy is then∫∫Len(C) = |π N ∂ t C(t, θ)||C ′ (t, θ)| dt dθwhich coincides with∫∫Len(C) =|∂ t C(t, θ) × ∂ θ C(t, θ)| dθ dtThis last is easily recognizable as the surface area of the homotopy (up tomultiplicity); the problem of finding a minimal geodesic connecting c 0 and c 1 inthe F 1 metric may be reconducted to the Plateau problem of finding a surfacewhich is an immersion of I = S 1 × [0, 1] and which has fixed borders to thecurves c 0 and c 1 . The Plateau problem is a wide and well studied subject uponwhich Fomenko expounds in the monograph [20].8 Riemannian metrics of curves up to poseA particular approach to the study of shapes is to define a shape to be a curve upto reparameterization, rotation, translation, scaling; to abbreviate, we call thisthe shape space of curves up to pose. We present two examples of Riemannianmetrics.8.1 Shape Representation using direction functionsKlassen, Mio, Srivastava et al in [28, 41] represent a planar curve c by a pair ofvelocity-angle functions (φ, α) through the identityc ′ (u) = exp(φ(u) + iα(u))(identifying lR 2 = IC), and then defining a metric on the velocity-angle functionspace. They propose models of spaces of curves where the metrics involve higherorder derivatives in [28].We review here the simplest such model, where φ = 0. We consider in thefollowing planar curves ξ : S 1 → lR 2 of length 2π and parameterized by arclength. (Note that such curves are automatically Lipschitz continuous).Proposition 8.1 (Continuous lifting, winding number) If ξ ∈ C 1 and parameterizedby arc parameter, then ξ ′ is continuous and |ξ ′ | = 1, so there existsa continuous function α : lR → lR satisfyingξ ′ (s) = ( cos(α(s)), sin(α(s)) ) (8.1)52

130−2Figure 10: Examples of curves of different winding number.and α(s) is unique, up to adding the constant k2π with k ∈ Z.Moreover α(s + 2π) − α(s) = 2πI, where I is an integer, known as windingnumber, or rotation index of ξ. This number is unaltered if ξ is homotopicallydeformed in a smooth way.See the examples in Fig. 10.The addition of a real constant to α(s) is equivalent to a rotation of ξ. Wethen understand that we may represent arc parameterized curves ξ(s), up totranslation, scaling, and rotations, by considering a suitable class of liftings α(s)for s ∈ [0, 2π].Two spaces are defined in Klassen et al. [28]; we present the case of “ShapeRepresentation using Direction Functions”;Definition 8.2 let L 2 := L 2 ([0, 2π] → lR); Φ : L 2 → lR 3 is defined byΦ 1 (α) :=∫ 2π0α(s) ds , Φ 2 (α) :=∫ 2π0cos α(s) ds , Φ 3 (α) :=∫ 2πand the space of (pre)-shapes is defined as the closed subset S of L 2 ,S := { α ∈ L 2 | Φ(α) = (2π 2 , 0, 0) } ;0sin α(s) ds .the condition Φ 1 (α) = 2π 2 forces a choice of rotation, while Φ 2 = 0, Φ 3 = 0ensure that α represents a closed curve.For any α ∈ S, it is possible to reconstruct the curve by integratingξ(s) =∫ s0(cos(α(t)), sin(α(t)))dt (8.2)This means that α identifies an unique curve (of length 2π, and arc parameterized)up to rotations and translations, and to the choice of the base point ξ(0); forthis last reason, S is called in [28] a preshape space.Inside the family of arc parameterized curves, the reparameterization 11 of ξis restricted to the transformation ˆξ(u) = ξ(u − s 0 ), that is a different choice ofbase point for the parameterization along the curve. Inside the preshape spaceS, the same operation is encoded by the relation ˆα(s) = α(s − s 0 ).To obtain a model of immersed curves up to reparameterization, rotation,translation, scaling we need to quotient S by the relation α ∼ ˆα, for all possibles 0 ∈ lR. We do not discuss this quotient here.We now prove that S \ Z is a smooth manifold. We first define Z.11 We are considering only reparameterizations in Diff + (S 1 ).53

Definition 8.3 (Flat curves) Let Z be the set of all α ∈ L 2 ([0, 2π]) such thatα(s) = a + k(s)π where k(s) ∈ Z and k is measurable, a = 2π − ∫ k ∈ lR, and|{k(s) = 0 mod 2}| = |{k(s) = 1 mod 2}| = π• Z is closed (by thm. 4.9 in Brezis [4]).• S \ Z contains the (representation α by continuous lifting of) all smoothimmersed curves.• Z contains the (representations α of) flat curves ξ, that is, curves ξ whoseimage is contained in a line.Example 8.4 An example of a flat curve is{s/ √ {2 s ∈ [0, π]ξ 1 (s) = ξ 2 (s) =(2π − s)/ √ 2 s ∈ (π, 2π] , α = π/2 s ∈ [0, π]3π/2 s ∈ (π, 2π]Proposition 8.5 S \ Z is a smooth immersed submanifold of codimension 3 inL 2 .Proof. By the implicit function theorem. Indeed, suppose by contradictionthat ∇Φ 1 , ∇Φ 2 , ∇Φ 3 are linearly dependant at α ∈ S, that is, there existsa ∈ lR 3 , a ≠ 0 s.t.a 1 cos(α(s)) + a 2 sin(α(s)) + a 3 = 0for almost all s; then, by integrating, a 3 = 0, therefore a 1 cos(α(s))+a 2 sin(α(s)) =0 that means that α ∈ Z.The manifold S \ Z inherits a Riemannian structure, induced by the scalarproduct of L 2 ; (critical) geodesics may be prolonged smoothly as long as theydo not meet Z.Even if S may not be a manifold at Z, we may define the geodesic distanced g (x, y) in S as the infimum of the length of Lipschitz paths γ : [0, 1] → L 2 goingfrom x to y and whose image is contained in S; 12 since d g (x, y) ≥ ‖x − y‖ L 2,and S is closed in L 2 , then the metric space (S, d g ) is metrically complete.We don’t know if (S, d g ) admits minimal geodesics, or if it falls in the samecategory as the Atkin example 3.34.8.1.1 Multiple measurable representationsWe may represent any Lipschitz closed arc parameterized curve ξ using a measurableα ∈ S.Definition 8.6 A measurable lifting is a measurable function α : lR → lRsatisfying (8.1). Consequently, a measurable representation is a measurablelifting α satisfying conditions Φ(α) = (2π 2 , 0, 0) (from Definition 8.2).12 It seems that S is Lipschitz-arc-connected, so d g (x, y) < ∞; but we did not carry on adetailed proof54

We remark that• a measurable representation always exists; for example, letbe the inverse ofarc : S 1 → [0, 2π)α ↦→ (cos(α), sin(α))when α ∈ [0, 2π). arc() is a Borel function; then ((arc ◦ ξ ′ )(s) + a) ∈ S, fora choice of a ∈ lR.• The measurable representation is never unique: for example, given anymeasurable A, B ⊂ [0, 2π] with |A| = |B|,will as well represent ξ.α(s) + 2π1 A (s) − 2π1 B (s)This implies that the family A ξ of measurable α ∈ S that represent the samecurve ξ is infinite. It may be then advisable to define a quotient distance ˆdas follows:ˆd(ξ 1 , ξ 2 ) := inf d(α 1 , α 2 ) (8.3)α 1∈A ξ1 ,α 2∈A ξ2where d(α 1 , α 2 ) = ‖α 1 − α 2 ‖ L 2, or alternatively d = d g is the geodesic distanceon S.8.1.2 Continuous vs measurable lifting — no windingIf ξ ∈ C 1 , we have an unique 13 continuous representation α ∈ S; but notethat, even if ξ 1 , ξ 2 ∈ C 1 , the infimum (8.3) may not be given by the continuousrepresentations α 1 , α 2 of ξ 1 , ξ 2 . Moreover there are rectifiable curves ξ that donot admit a continuous representation α, as for example the polygons.A problem similar to the above is expressed by this proposition.Proposition 8.7 For any h ∈ Z, the set of closed smooth curves ξ with rotationindex h, when represented in S using the continuous lifting, is dense in S \ Z.It implies that we cannot properly extend the concept of rotation index to S.The proof is based on this lemma.Lemma 8.8 Suppose that ξ is not flat, let τ be one of the measurable liftings ofξ. There exists a smooth projection π : V → S defined in a neighborhood V ⊂ L 2of τ such that, if f ∈ L 2 ∩ C ∞ then π(f) is in L 2 ∩ C ∞ .This is the proof of both the lemma and the proposition.13 Indeed, the continuous lifting is unique up to addition of a constant to α(s), which isequivalent to a rotation of ξ; and the constant is decided by Φ 1 (α) = 2π 255

Proof. Fix α 0 ∈ S \ Z. Let T = T α0 S be the tangent at α 0 . T is the vectorspace orthogonal to ∇φ i (α 0 ) for i = 1, 2, 3. Let e i = e i (s) ∈ L 2 ∩ Cc∞ be near∇φ i (α 0 ) in L 2 , so that the map (x, y) : T × lR 3 → L 2(x, y) ↦→ α = α 0 + x +3∑e i y i (8.4)is an isomorphism. Let S ′ be S in these coordinates; by the Implicit FunctionTheorem (5.9 in Lang [30]), there exists an open set U ′ ⊂ T , 0 ∈ U ′ , an openV ′ ⊂ lR 3 , 0 ∈ V ′ , and a smooth function F : U → lR 3 such that the local partS ′ ∩ (U ′ × V ′ ) of the manifold S ′ is the graph of y = F (x).We immediately define a smooth projection π : U ′ × V ′ → S ′ by settingπ ′ (x, y) = (x, F (x)); this may be expressed in the original L 2 space; let(x(α), y(α)) be the inverse of (8.4) and U = x −1 (U ′ ); we define the projectionπ : U → S by settingThenπ(α)(s) − α(s) =π(α) = α 0 + x +i=13∑e i F i (x(α))i=13∑e i (s)a i , a i := (F i (x(α)) − y i (α)) ∈ lR (8.5)i=1so if α(s) is smooth, then π(α)(s) is smooth.Let α n be smooth functions such that α n → α in L 2 , then π(α n ) → α 0 ; ifwe choose them to satisfy α n (2π) − α n (0) = 2πh, then, by the formula (8.5),π(α)(2π) − π(α)(0) = 2πh so that π(α n ) ∈ S and it represents a smooth curvewith the assigned rotation index h.8.2 A metric with explicit geodesicsA similar method has been proposed recently in Michor et al. [39], based onan idea originally in [68]. We consider immersed planar curves and again weidentify lR 2 = IC.Proposition 8.9 (Continuous lifting of square root) If ξ : [0, 2π] → IC isan immersed planar curve, then ξ ′ is continuous and ξ ′ ≠ 0, so there exists acontinuous function α :→ IC satisfyingξ ′ (θ) = α(θ) 2 (8.6)and α is uniquely identified up to multiplying by ±1.If the rotation index of ξ is even then α(0) = α(2π), whereas if the rotationindex of ξ is odd then α(0) = −α(2π).56

We will use this lifting to obtain, as a first step, a representation of curves upto rotation, translation, scaling. To this end, let ξ be an immersed planar closedcurve of len(ξ) = 2 (not necessarily parameterized by arc parameter); let α bethe square root lifting of the derivative ξ ′ . Let e, f be the real and imaginarypart of α, that is, α = e + if. The condition that ξ is a closed curve translatesinto ∫ 2π0 (e + if)2 dθ = 0 (where equality is in IC), hence we have two equalities(for real and imaginary part)∫ 2π0e 2 − f 2 dθ = 0,∫ 2πwhile the condition that len(ξ) = 2 translates into∫ 2π0|ξ ′ | dθ =∫ 2π0|e + if| 2 dθ =0∫ 2π0ef dθ = 0(e 2 + f 2 ) dθ = 2 .With some algebra we obtain that the above conditions are equivalent to∫ 2π0e 2 dθ = 1,∫ 2π0f 2 dθ = 1,∫ 2π0ef dθ = 0.Let L 2 = L 2 ([0, 2π] → lR); let S ⊂ L 2 × L 2 be defined byS :={(e, f) |∫ 2π0e 2 dθ = 1 =∫ 2π0f 2 dθ,∫ 2π0}ef dθ = 0.S is the Stiefel manifold of orthonormal pairs in L 2 . S is a smooth manifold,and inherits the (flat) metric of L 2 × L 2 . What is most surprising is thatTheorem 8.10 (2.2 in [39]) Let δξ be a small deformation of ξ, and δe, δfbe the corresponding small deformation of the representation e, f. Then∫∫ 2π|D s δξ| 2 ds = (δe) 2 + (δf) 2 dθξ0that is, the (geometric) Riemannian metric in M is mapped into the (flat andparametric) metric in S.It is then natural to “embed” closed planar curves (up to translation andscaling) into S.8.2.1 Curves up to rotation represented as a GrassmanianWe note that rotation of ξ by an angle τ is equivalent to rotation of the frame(e, f) by an angle τ/2. So the orbit of all rotations of ξ is associated to the planegenerated by (e, f) in L 2 : the space of curves up to rotation/translation/scalingis represented by the Grassmanian manifold of 2-planes in L 2 . A theoremby Neretin then applies, that provides a closed form formula for critical andminimal geodesics. See section 4 in [39].57

8.2.2 RepresentationThe Stiefel manifold is a complete smooth Riemannian manifold; it contains the(representation of) all closed rectifiable parametric planar curves, up to scalingand translation. So with this choice of metric and representation, we obtainthat the completion of the Fréchet manifold of smooth curves is a Riemanniansmooth manifold. There are two problems left.• The quotient w.r.to reparameterization. This is studied in [39], where itis proven that unfortunately geodesics in the quotient space may developsingularities in the reparameterizations at the end times of the geodesic.• But there is also the quotient w.r.to representation.8.2.3 Quotient w.r.to representationAs in the previous section, we may define a measurable square root lifting; thislifting will not be unique. Indeed, let (e, f) be the square root representation ofξ, that is ξ ′ = (e + if) 2 ; choose any function a : S 1 → {−1, 1} arbitrarily (butmeasurable); then (ae, af) represents the same curve. So again we may define aquotient metric ˆd(ξ 1 , ξ 2 ) as we did in eqn. (8.3); similar comments to those atthe end of Section 8.1.2 hold.9 Riemannian metrics of immersed curvesMetrics of “geometric” curves have been studied by Michor and Mumford [37, 36,38] and Yezzi and Mennucci [65, 66, 67]; more recently, Yezzi-M.-Sundaramoorthi[53–58, 35] have studied Sobolev-like metrics of curves and shown many goodproperties for applications to shape optimization; similar results have also beenshown independently by Charpiat et al. [9, 10, 11].We now discuss some Riemannian metrics on immersed curves.• The H 0 metric∫〈h 1 , h 2 〉 H 0 = 〈h 1 (s), h 2 (s)〉 ds .• Michor and Mumford [37]’s metric∫〈h 1 , h 2 〉 H A = (1 + A|κ c | 2 )〈h 1 , h 2 〉 ds .• Yezzi and Mennucci [67] conformal metric• Charpiat et al. [10] rigidified normscc∫〈h 1 , h 2 〉 H 0 = φ(c) 〈hφ1 , h 2 〉 ds .c58

• Sundaramoorthi et al. [53] Sobolev type metrics∫∫〈h 1 , h 2 〉 H n = 〈h 1 , h 2 〉 ds + len(c) 2n 〈∂s n h 1 , ∂s n h 2 〉 ds〈h 1 , h 2 〉 ˜Hn=c〈 ∫ h 1 ds,c∫cc〉 ∫h 2 ds + len(c) 2n 〈∂s n h 1 , ∂s n h 2 〉 ds .cWe will now present a quick overview of all metrics (but for the latter, thatis discussed in the next section).9.1 H 0The H 0 inner product was defined in eqn. (2.4) as∫〈 〉h1 , h 2 := hH 0 1 (s) · h 2 (s) ds ; (9.1)cit is possibly the simplest geometric inner product that we may imagine toapply to curves. We already noted that the minimizing flows implemented intraditional active contour methods are “gradient flows” only if we use this H 0inner product.We will show in Sec. 11.3 that the H 0 -induced distance is identically zero. In[67] there is a result that shows that the distance is non degenerate, and minimalgeodesics exists, when the shape space is restricted to curves with uniformlybounded curvature.9.2 H AMichor and Mumford [37] propose the metric H A〈h 1 , h 2 〉 H A =|c∫ L0(1 + A|κ c | 2 )〈h 1 , h 2 〉 dswhere κ c is the curvature of c, and A > 0 is fixed.Properties 9.1• The induced distance is non degenerate.• The completion M (intended in the metric sense) isBV 2 ⊂ M ⊂ Lipwhere BV 2 are the curves that admit curvature as a measure and Lip arethe rectifiable curves.• There are compactness bounds.59

9.3 Conformal metricsYezzi and Mennucci [67] proposed to change the metric, from H 0 to a conformalmetric∫〈h 1 , h 2 〉 H 0 = ψ(c) 〈hψ1 , h 2 〉 dscwhere ψ(c) associates to each curve c a positive number. Then the gradientdescent flow of an energy E defined on curves C(t, ·) is∂C∂t = −∇ψ E(C) = − 1ψ(C) ∇E(C)where ∇E(C) is the gradient for the H 0 metric. This is equivalent to a changeof time variable t in the gradient descent flow. So all properties of the flows areunaffected if we switch from a H 0 to a conformal-H 0 metric.Properties 9.2 Consider a conformal metric where ψ(c) depends (monotonically)on the length len(c) of the curve.• If ψ(c) ≥ len(c), the induced metric is non degenerate;• unfortunately, according to a result by Shah [50], when ψ(c) ≡ L(c) veryfew (minimal) geodesics do exist (only “grassfire” geodesics, moving byconstant normal speed).9.4 “Rigidified” normsCharpiat et al.[10] consider norms that favor pre-specified rigid motions. Theydecompose a motion h using the H 0 projection ash = h rigid+ h restwhere h rigidcontains the rigid part of the motion; then they choose λ large, andconstruct the norm‖h‖ 2 rigid = λ‖h rigid ‖2 H 0 + ‖h rest ‖2 H 0.Note that these norms are equivalent to the H 0 -type norm; as a result theinduced distance is (again) degenerate.10 Sobolev type Riemannian metricsIn this part we discuss the Sobolev norms, with applications and experiments.What follows summarizes [55, 57, 58, 35].60

10.1 Sobolev-type metricsRecently in [53–58, 35] Yezzi–M.–Sundaramoorthi studied a family of Sobolevtypemetrics. Let D s := 1|c ′ | ∂ θ be the derivative with respect to arc parameter.Let j ≥ 1 be an integer 14 and∫〈h 1 , h 2 〉 Hj := 〈Dsh j 1 , Dsh j 2 〉 ds (10.1)0cwhere ∫ · · · ds was defined in 1.9. Let λ > 0 a fixed constant; we define thecSobolev-type metrics∫〈h 1 , h 2 〉 H j := 〈h 1 , h 2 〉 ds + λL 2j 〈h 1 , h 2 〉 Hj(10.2)〈h 1 , h 2 〉 ˜Hj:=c〈 ∫ h 1 ds,c∫c〉h 2 ds + λL 2j 〈h 1 , h 2 〉 Hj0where L = len(c). Notice that these metrics are geometric:0(10.3)• they are easily seen to be invariant w.r.to rotations and translations, (in astronger sense than in page 39, indeed in this case〈h 1 , h 2 〉 Ac = 〈h 1 , h 2 〉 c , 〈Rh 1 , Rh 2 〉 c = 〈h 1 , h 2 〉 cfor any Euclidean transformation A and rotation R);• they are reparameterization invariant due to the usage of the arc parameterin derivatives and integrals;• they are scale invariant, since the normalizing factors L 2j make them0-homogeneous w.r.to rescaling of the curve.For this last reason, we redefine the H 0 metric to be∫〈h 1 , h 2〉H := h 1 · h 2 ds (10.4)0so that it is again 0-homogeneous.This is a conformal version of the H 0 metricdefined in eqn. (2.4), so a gradient descent flow is a time reparameterization ofthe flow for the original H 0 metric; and the induced distance is again degenerate.When we will present a shape optimization energy E and we will minimizeE using a gradient descent flow driven by the H j or ˜H j gradient of E, we willcall the resulting algorithm a Sobolev active contour method (abbreviatedas SAC in the following).14 It is though possible to define Sobolev metrics for any j ∈ lR, j > 0; see Prop. 3.1 in [57].c61

10.1.1 Related worksA family of metrics similar to H j above (but for the length dependent scalefactors 15 ) was studied (independently) in Michor and Mumford [36]: the Sobolevtypeweak Riemannian metric on Imm(S 1 , R 2 )∫〈h, k〉 Gjc=j∑〈Dsh, i Dsk〉 i ds ;c i=0in that paper the geodesic equation, horizontality, conserved momenta, lower andupper bounds on the induced distance, and scalar curvatures are computed. Notethat this metric is locally equivalent to the above metrics defined in equation(10.2), (10.3).Charpiat et al in [10, 11] studied (again independently) some generalizedmetrics and relative gradient flows; in particular they defined the Sobolev-typemetric∫〈h 1 , h 2 〉 + 〈D s h 1 , D s h 2 〉 ds .10.1.2 Properties of H j metricscThis is a list of important properties that will be discussed in the followingsections.• Flow regularization: Sobolev gradient flows are smoother than H 0 flows.• PDE order reducing property: Sobolev gradient flows are lower order thanH 0 flows.• SAC does not require derivatives of the curve to be defined for manycommonly used region-based and edge-based energies.• Coarse-to-fine deformation property: a SAC automatically favors coarsescaledeformations before moving to fine-scale deformations; this is idealfor visual tracking.• Sobolev-type norms induce a well-defined distance on space of curves;• moreover the structure of the completion of the space of immersed curvesw.r.to H 1 and H 2 norm is fairly well understood. So they offer a consistenttheory of shape optimization and shape analysis.10.2 Mathematical propertiesWe start summarizing the main mathematical properties, that were presented in[35] mostly. We first of all cite this lemma.15 Though, a scale-invariant Sobolev metric is proposed in [36] in §4.8 as a sensible generalization.62

Lemma 10.1 (Poincaré inequality) Pick h : [0, l] → lR n , weakly differentiable,with h(0) = h(l) (so h is periodically extensible); let ĥ = ∫ 1 llh(x) dx;0thensup |h(u) − ĥ| ≤ 1u2∫ l0|h ′ (x)| dx . (10.5)This is proved as Lemma 18 in [35]; it is one of the main ingredients for thefollowing propositions, whose full proofs are in [35].Proposition 10.2 The H j and ˜H j distances are equivalent:√1 + (2π)d ˜Hj≤ d H j ≤2j λ(2π) 2j λd ˜Hjwhereas d H j ≤ d H k for j < k.Proposition 10.3 The H j and ˜H j distances are lower bounded (with appropriateconstants depending on λ) by the Fréchet distance (defined in 7.2).Proposition 10.4 c ↦→ len(c) is Lipschitz in M with H j metric, that is,| len(c 0 ) − len(c 1 )| ≤ d H j (c 0 , c 1 )Theorem 10.5 (Completion of B i w.r.to H 1 ) let d H 1 be the distance inducedby H 1 ; the metric completion of the space of curves is equal to the spaceof all rectifiable curves.Theorem 10.6 (Completion of B i w.r.to H 2 ) Let E(c) := ∫ |D 2 sc| 2 ds bedefined on non-constant smooth curves; then E is locally Lipschitz in w.r.to d H 2.Moreover the completion of C ∞ (S 1 ) according to the metric H 2 is the space ofcurves that admit curvature D 2 sc ∈ L 2 (S 1 ).The hopeful consequence of the above theorem would be a possible solution tothe problem 4.12; it implies that the space of geometric curves B with the H 2Riemannian metric completes onto the usual Hilbert space H 2 . 16 This resultin turn would ease a (possible) proof of existence geodesics.Concluding, it seems that, to have a complete Riemannian manifold ofgeometric (freely) immersed curves, a metric should penalize derivatives of 2ndorder (at least).10.3 Sobolev metrics in shape optimizationWe first present a definition.16 The detailed proof has not yet been written...63

Definition 10.7 (Convolution) A arc-parameterized convolutional kernel Kalong the curve c of length L is a L-periodic function K : lR → lR. Given avector field f : S 1 → lR n and a kernel K, we define the convolution by arcparameter 17 formally as∫(f ⋆ K)(s) := K(s − ŝ)f(ŝ) dŝ . (10.6)By defining the run-length function l : lR → lRl(τ) :=c∫ τ0|c ′ (x)| dxwe can rewrite the above eqn. (10.6) explicitly in θ parameter as(f ⋆ K)(θ) :=∫ 2π0K ( l(θ) − l(τ) ) f(τ)|c ′ (τ)| dτ . (10.7)Recall the definition 3.37 of gradient ∇E by means of the identity〈∇E, h〉 c = DE(c; h) ∀h ∈ T c M .Let f = ∇ H 0E, g = ∇ H 1E be the gradients w.r.to the inner products H 0 andH 1 ; by the definition of gradient, we obtain that〈f, h〉 H0 ,c = 〈g, h〉 H1 ,c∀h ∈ T c Mthat is 18 ∫∫h · f ds = h · g + λL 2 (D s h · D s g) dsccby integrating by parts this becomes∫h · (f − g + λL 2 Dsg) 2 ds = 0 ∀hthen we conclude thatc∀h∇ H 0E = ∇ H 1E − λL 2 D 2 s(∇ H 1E) . (10.8)With similar computation, for ˜H 1 we obtain that∫∇ H 0E = ∇˜H1E ds − λL 2 D s(∇˜H1E) 2 . (10.9)cGiven ∇ H 0E, both equations can be solved for ∇ H 1E and ∇˜H1E, by meansof suitable convolution kernels ˜K λ , K λ , that is, we have the formulas∇ H 1E = ∇ H 0E ⋆ K λ , ∇˜H1E = ∇ H 0E ⋆ ˜K λ ;17 Note this definition is different from the eqn. (13) in [55].18 We use the definition (10.4) of H 0 .64

65λ=0.01λ=0.04λ=0.08108λ=0.01λ=0.04λ=0.0846432201−20−0.5 0 0.5−4−0.5 0 0.5Figure 11: Plots of K λ (left) and ˜K λ (right) for various λ with L = 1. The plotsshow the kernels over one period.The kernels ˜K λ , K λ are known in closed form:˜K λ (s) = 1 L( ) s− L√ 2λLcoshK λ (s) =2L √ (λ sinh(12 √ λ1 + (s/L)2 − (s/L) + 1/62λ), for s ∈ [0, L], (10.10)and K λ , ˜K λ are periodically extended to all of lR. See Fig. 11.), s ∈ [0, L]. (10.11)The above idea extends to any j: it is possible to obtain ∇ H j E and ∇˜HjEfrom ∇ H 0E, by convolution. But there is also a way to compute ∇˜HjE withoutresolving to convolutions, see Prop. 10.10.The bad news is that, when shape optimization is implemented using a levelset method (as is usually the case), the curve must be traced out to computegradients. 19 This in particular leads to some problems when a curve undergoesa topological change, since the Sobolev gradient depends on the global shape ofthe curve; but those problems are easily bypassed when the optimization energyis continuous across topological changes of the curves: see Sec. 5.1 in [55].10.3.1 Smoothing of gradients, coarse-to-fine flowingIn Sundaramoorthi et al. [57] it was noted that the regularizing properties maybe explained in the Fourier domain: indeed, if we calculate Sobolev gradients∇ H j E of an arbitrary energy E in the frequency domain, then̂∇ H j E(l) =̂∇ H 0E(l)1 + λ(2πl) 2j for l ∈ Z (10.12)19 Though, an alternate method that does not need the tracing of the curve is described inSec. 5.3 in [55] – but it was not successively used, since it depends on some difficult-to-tuneparameters.65

and̂∇˜HjE(0) = ̂∇ H 0E(0),̂∇˜HjE(l) = ̂∇ H 0E(l)λ(2πl) 2j for l ∈ Z\{0}, (10.13)It is clear from the previous expressions that high frequency components of∇ H 0E(c) are increasingly less pronounced in the various forms of the H j gradients.So H j and ˜H j gradients are much smoother than H 0 gradients. Note that usinga SAC method smooths the gradients, so it induces smoother minimization flows,but it does not smooth the curves themselves.The above phenomenon may be also explained by the following argument.The gradient satisfies the following property.Proposition 10.8 If DE(c) ≠ 0, the gradient ∇E(c) is the vector v in T c Mthat provides the maximum inDE(c)(v)|DE(c)(h)|= sup. (10.14)‖v‖ c h∈T cM\{0} ‖h‖ cThus, the gradient is the most efficient perturbation, i.e., it maximizeschange in energy by moving in direction hcost of moving in direction hBy constructing ‖ · ‖ c to penalize “unfavorable” directions, we have controlover the path taken to minimize E without changing the energy E. 20Since higher frequencies are more penalized in H 1 than in H 0 , this explainswhy a SAC prefers to move the curves in a coarser scale w.r.to a traditionalactive contour method.10.3.2 Flow regularizationIt is also possible to show mathematically that the Sobolev–type gradientsregularize the flows of well known energies, by reducing the degree of the P.D.E.,as shown in this example.Example 10.9 (Elastica) In the case of the elastic energy∫ ∫E(c) = κ 2 ds = |Dsc| 2 2 ds ,ccthe H 0 gradient is 21 ∇ H 0E = LD s (2D 3 sc + 3|D 2 sc| 2 D s c)20 In a sense, the focus of research in active contours has been mostly on the numerator ineqn. (10.14) — whereas we now focus on the denominator.21 We use the definition (10.4) of H 0 ; the directional derivative was computed in 4.666

that includes fourth order derivatives; whereas the ˜H 1 -gradient is− 2λL D2 sc + 3L(|D 2 sc| 2 D s c) ⋆ (D s ˜Kλ ) (10.15)that is an integro-differential second order P.D.E.A practical positive outcome of this phenomenon will be shown in Section 10.9.1.10.4 ˜Hj is faster than H jWe will now show that the metric ˜H 1 (that is metrically equivalent to H 1 , byProp. 10.2) leads to a simpler calculus for gradients; with benefits in numericalapplications as well.∫We recall that avg c (f) := 1len(c)f(s) ds.cLet E(c) be an energy of curves. Gradients are implicitly defined by thefollowing relationsDE(c; h) = 〈h, ∇ H 0E〉 H 0 = 〈h, ∇ H 1E〉 H 1 = 〈h, ∇˜H1E〉 ˜H1∀h .As we saw in equations (10.8) and (10.9), the above leads to the ODEsNormODEH 1 ∇ H 0E = ∇ H 1E − λL 2 D 2 s(∇ H 1E)˜H 1 ∇ H 0E = avg c (∇˜H1E) − λL 2 D 2 s(∇˜H1E)The second ODE is much easier to solve.Proposition 10.10 Let f = ∇ H 0E, g = ∇˜H1E, then g is derived from f by thefollowing closed form formulasg(s) = g(0) + sg ′ (0) − 1 ∫ sλL 2 (s − ŝ)(f(ŝ) − avg c (f)) dŝg ′ (0) = − 1 ∫ LλL 3 s(f(s) − avg c (f)) dsg(0) =∫ L00f(s) ˜K λ (s) ds .where ˜K λ was defined in (10.11).Proof. The ODEf = avg c (g) − λL 2 D 2 sgimmediately tells that avg c (f) = avg c (g); so we rewrite it asλL 2 D 2 sg = −f + avg c (f)067

and we simply integrate twice! Moreover, exploiting the identity∫ s(∫ t) ∫ sh(r) dr dt = (s − r) h(r) dr ,00and with some extra computations, we obtain the result.In the end we obtain that the ˜H 1 gradient need not be computed as a convolution;so the ˜H 1 gradient enjoys nearly the same computational speed as the H 0gradient; moreover the resulting gradient flow is more stable, so it works fine innumerical implementations for a larger choice of time step discretization. Forthese reasons, ˜H1 Sobolev active contours are very fast.10.5 Analysis and calculus of ˜H 1 gradientsWe write h ∈ T c M as h = avg c (h) + ˜h; this decomposes0T c M = lR n ⊕ D c M (10.16)withD c M :={}h : S 1 → lR n | avg c (h) = 0 .If we assign to lR n its usual Euclidean norm, and to D c M the H j 0 norm22 ,then‖h‖ 2˜Hj = |avg c (h)| 2 lR + n λL2j ‖˜h‖ 2 .H j 0This means that the two spaces lR n and D c M are orthogonal w.r.to ˜H j .A remark on this decomposition is due.Remark 10.12 In the above, lR n is akin to be the space of translations andD c M the space of non-translating deformations.That labeling is not rigorous, though! since the subspace of T c M of deformationsthat do not move the center of mass avg c (c) is not D c M, but rather{ ∫h : h + ( c − avg c (c) ) }〈D s h · T 〉 ds = 0 ,as is easily deduced from equation (2.8).cThe decomposition (10.16) is quite useful in the calculus when analyticallycomputing ˜H 1 gradients and proving existence of flows. Indeed, let f = ∇ H 0E,g = ∇˜H1E; decomposeTo solvewe have to solve two equations:f = avg c (f) + ˜f , g = avg c (g) + ˜g .f = avg c (g) − λL 2 D 2 sg22 H j 0 was defined in (10.1). Note that Hj 0 is a seminorm on TcM, since its value is zero onconstant fields; but H j 0 is a norm on DcM, by Poincaré inequality (10.5).68

avg c (g) = avg c (f) λL 2 Ds 2 ˜g = − ˜flR n ⊕ D c M .We will now show how to properly solve these coupled equations.We will need to define some useful objects.Definition 10.13 We define the projection operatorΠ c : T c M → D c Mh ↦→ h − ∫ c h ds (10.17)Definition 10.14 When we consider the derivation with respect to the arcparameter as a linear operatorD c : D c M → D c Mh ↦→ D s hthen it admits the inverse, that is the primitive operator(10.18)P c : D c M → D c M (10.19)Proof. The proof is just based on noting that, for any smooth h ∈ T c M, h ∈ D c Miff there is a smooth k ∈ T c M with h = D s k. Vice versa, two primitives differby a constant, hence when h ∈ D c M there is exactly one primitive k in D c Msuch that h = D s k.Example 10.15 The tangent vector field D s c is in D c M, and its primitive isP c D s c = c − avg c (c) .Proposition 10.16 Fix a curve c, let L = len(c). We can extend the primitiveoperator by composing it with the projection; the resulting composite operatorcan be expressed in convolutional form asfor any continuous h ∈ T c M; where the kernel K P cP c Π c h = K P c ⋆ h (10.20)isK P c (s) := − s L + 1 2for s ∈ [0, L) (10.21)and K P c (s) is extended periodically to s ∈ lR (note that K P c (s) jumps at allpoints of the form s = nL, n ∈ Z).The above properties lead to this theorem, that can be used to shed a newlight to what was expressed in 10.10 in the previous section.Theorem 10.17 Let f = ∇ H 0E, g = ∇˜H1E; the solution ofcan be expressed asf = avg c (g) − λL 2 D 2 sgg = avg c (f) − 1λL 2 P cP c Π c f = avg c (f) − 1λL 2 KP c ⋆ K P c ⋆ f .69

Corollary 10.18 In particular, the kernels defined in eqn. (10.10), (10.21) arerelated by˜K λ = 1 L − 1λL 2 KP c ⋆ KcPBy deriving,D s KcP = δ 0 − 1 Lwhere δ 0 is the Dirac’s delta; so we obtain the relationsWe also recall this Lemma.D s ˜Kλ = − 1λL 2 KP cD ss ˜Kλ = − 1λL 2 δ 0 + 1λL 3 .(10.23)Lemma 10.19 (De la Vallée-Poussin) Suppose that f : S 1 → lR n is integrableand satisfies∫k · f ds = 0cfor all k ∈ D c M smooth; then f is almost everywhere equal to a constant.For the proof, see Chapter 13 in [1], or Lemma VIII.1 in [4].We now compute the gradient of the centroid energy (2.9).Proposition 10.20 (Gradient of the centroid-based energy) Let in thefollowing again, for simplicity of notation, c = avg c (c) be the center of mass ofthe curve. LetE(c) = 1 2 |c − v|2 .The ˜H 1 gradient of E is∇˜H1E(c) = c − v + 1λL 2 P cΠ c(Ds c〈(c − v), (c − c)〉 ) . (10.24)Proof. We already computed the Gâteaux differential of E; but this time weprefer to use the first form of eqn. (2.10), so as to write∫DE(c; h) = (c − v) · h + 〈c − v, c − c〉(D s h, D s c) ds .Let f = ∇˜H1E(c) be the ˜H 1 gradient of E. The equalitycDE(c; h) = 〈h, f〉 ˜H1∀hbecomes〈c − v − f, h〉 +∫cD s h · (〈c− v, c − c〉D s c − λL 2 D s f ) ds = 0, ∀h .70

Since h is arbitrary, using de la Vallée-Poussin Lemma, we obtain thatf = c − v , λL 2 D s f = D s c〈(c − v), (c − c)〉 + α , (10.25)where the constant α is the unique α ∈ lR n such that the rightmost term is inD c M. In conclusion we obtain (10.24).We similarly compute the gradient of the active contour energy.Proposition 10.21 (Gradient of geodesic active contour model) We consideronce again the geodesic active contour model [7, 27] (that was presentedin Section 2.2) where the energy is∫E(c) = φ(c(s)) dscwith φ : lR 2 → lR + appropriately designed. The gradient with respect to ˜H 1 is∇˜H1E(c) = Lavg c (∇φ(c)) − 1λL P cP c Π c(∇φ(c))+1λL P cΠ c(φ(c)Ds c ) . (10.26)Proof. Let us note 23 (recalling eqn. (2.3)) that∇ H 0E = L∇φ(c) − LD s (φ(c)D s c) , (10.27)so the above equation (10.26) can be obtained by the relation in Theorem10.17.Alternatively, we know that∫DE(c; h) = L ∇φ(c) · h + φ(c)(D s h · D s c) ds ;let f = ∇˜H1E(c) be the ˜H 1 gradient of E; the equalitycDE(c; h) = 〈h, f〉 ˜H1∀hbecomes ∫(L∇φ(c) − avg c (f)) · h + D s h · (Lφ(c)D s c − λL 2 D s f) ds = 0cimitating the proof of the previous proposition, we obtain (10.26).We remark a few things.• Note that the first term in the formula (10.26) is in lR n while the othertwo are in D c M.• Using the kernel ˜Kλ that was defined in (10.10), we can rewrite∇˜H1E(c) = L ˜K λ ⋆ (∇φ(c)) + 1λL P cΠ c(φ(c)Ds c ) (10.28)• The formula for the gradient does not require that the curve be twicedifferentiable: we will use this fact to prove an existence result for thegradient flow, in Theorem 10.31.23 We use the definition (10.4) of H 0 .71

10.6 Existence of gradient flowsWe recall this definition (that was already presented informally in the introduction).Definition 10.22 (Gradient descent flow) Given a differentiable energy E :M → lR, and a metric 〈, 〉 c , let ∇E(c) be the gradient. Let us fix moreoverc 0 : S 1 → lR n , c 0 ∈ M. The gradient descent flow of E is the solutionC = C(t, θ) of the initial value P.D.E.{∂t C = −∇E(C)C(0, θ) = c 0 (θ)We present an example computation for the geodesic active contour modelon a radially symmetric “image”.Example 10.23 LetC(t, θ) = r(t) (cos θ, sin θ) ,φ(x) = d(|x|)with r, d : lR → lR + ; the energy takes the form∫E(C) = φ(C(t, s)) ds = 2π r(t) d(r(t))Then C is the ˜H 1 gradient descent iffCr ′ (t) = − 1 (d(r(t)) + r(t) d ′ (r(t)) ) .λ2πwhereas C is the H 0 gradient descent iffr ′ (t) = −2π ( d(r(t)) + r(t)d ′ (r(t)) ) ,where we use the definition (10.4) of H 0 .Note also thatProof. Indeed,∂ t E(C) = r ′ (t) ( d(r(t)) + r(t)d ′ (r(t)) ) .ds = r(t) dθD s =1r(t) D θlen(C) = 2πr(t)φ(C) = d(r(t))∇φ(x) = x|x| d′ (|x|) for x ≠ 0∇φ(C) = d ′ (r(t))(cos θ, sin θ)avg c (∇φ(C)) = 072

P c Π C ∇φ(C) = P c ∇φ(C) = r(t) d ′ (r(t)) (sin θ, − cos θ)P c P c Π C ∇φ(C) = −r(t) 2 d ′ (r(t)) (cos θ, sin θ)φ(C)D s C = d(r(t))(− sin θ, cos θ)P c (φ(C)D s C) = r(t)d(r(t)) (cos θ, sin θ)∇˜H1E(C) = Lavg c (∇φ(C)) − 1λL P ( ) 1CP C Π C ∇φ(C) +λL P (CΠ C φ(C)Ds C ) ==1 (d(r(t)) + r(t) d ′ (r(t)) ) (cos θ, sin θ) ;λ2πwhereas for the ˜H 0 gradient descent we use the formula (10.27)∇ H 0E = L∇φ(C) − LD s (φ(C)D s C) == 2π ( r(t)d ′ (r(t)) + d(r(t)) ) (cos θ, sin θ)We will show in the following theorems how to prove that ˜H 1 gradient flowsof common energies are well defined; to this end, we will provide a detailed prooffor the centroid energy E (2.9) discussed in the previous section, and for thegeodesic active contour model [7, 27] (that was presented in Section 2.2); thoseproofs illustrates methods that may be used for many other energies.10.6.1 Lemmas and inequalitiesLet us also prepare the proof by presenting some useful inequalities in threeLemma.Definition 10.24 We recall from 3.11 that C 0 = C 0 (S 1 → lR n ) is the space ofcontinuous functions, that is a Banach space with norm‖c‖ 0 :=sup |c(θ)| .θ∈[0,2π]Similarly, C 1 = C 1 (S 1 → lR n ) is the space of continuously differentiable functions,that is a Banach space with norm‖c‖ 1 := ‖c‖ 0 + ‖c ′ ‖ 0where c ′ (θ) = ∂c∂θ(θ) is the usual parametric derivative of c.Lemma 10.25 • We will use repeatedly the two following inequalities. Ifa 1 , a 2 , b 1 , b 2 ∈ lR then|a 1 b 1 − a 2 b 2 | ≤ |b1|+|b2|2|a 1 − a 2 | + |a1|+|a2|2|b 1 − b 2 || a1b 1− a2b 2| ≤ |b1|+|b2|2|b |a 1b 2| 1 − a 2 | + |a1|+|a2|2|b 1b 2||b 1 − b 2 |(i)73

• The length functional (from C 1 to lR) is Lipschitz, sincelen(c 1 ) − len(c 2 ) ≤ 2π‖c ′ 1 − c ′ 2‖ 0 ;(ii)• if h 1 , h 2 : S 1 → lR n are continuous fields, by (i)|∫h 1 (s) ds −c 1∫h 2 (s) ds|c 2≤ π(‖h 1 ‖ 0 + ‖h 2 ‖ 0 )‖c ′ 1 − c ′ 2‖ 0 ++ π(‖c ′ 1‖ 0 + ‖c ′ 2‖ 0 )‖h 1 − h 2 ‖ 0 ; (iii)• similarly|∫h 1 (s) ds −c 1∫h 2 (s) ds| ≤ A‖c ′ 1 − c ′ 2‖ 0 + ‖h 1 − h 2 ‖ 0c 2(iv)whereA := 2π2 (‖h 1 ‖ 0 + ‖h 2 ‖ 0 )(‖c ′ 1‖ 0 + ‖c ′ 2‖ 0 )len(c 0 ) len(c 1 ). (v)• Let then Π c be the projection operator (as in definition (10.17)); note that(Π c h) ′ = h ′ ; from all above inequalities we obtain that‖Π c1 h 1 − Π c2 h 2 ‖ 0 ≤ A‖c ′ 1 − c ′ 2‖ 0 + 2‖h 1 − h 2 ‖ 0‖Π c1 h 1 − Π c2 h 2 ‖ 1 = ‖Π c1 h 1 − Π c2 h 2 ‖ 0 + ‖(Π c1 h 1 ) ′ − (Π c2 h 2 ) ′ ‖ 0 ≤≤ A‖c ′ 1 − c ′ 2‖ 0 + 2‖h 1 − h 2 ‖ 0 + ‖h ′ 1 − h ′ 2‖ 0≤ A‖c 1 − c 2 ‖ 1 + 2‖h 1 − h 2 ‖ 1 . (vi)We will now prove the local Lipschitz regularity of the operators P c (h), D c (h)(that were defined in 10.14) in both the two variables h and c. Unfortunatelyto prove the theorem 10.29 we will need to also compare the action of P c fordifferent values of c; since P c h is properly defined only when h ∈ D c M (and ingeneral D c1 M ≠ D c2 M), we will actually need to study the composite operatorP c Π c h.Lemma 10.26 Let Π c the projector from T c M to D c M (that we defined ineqn. (10.17)). Let c, c 1 , c 2 be C 1 immersed curves, and h, h 1 , h 2 be continuousfields; then‖P c Π c h‖ 1 ≤ (2π + 2)‖c ′ ‖ 0 ‖h‖ 0 (10.29)and‖P c1 Π c1 h 1 − P c2 Π c2 h 2 ‖ 1 ≤ P ‖c ′ 1 − c ′ 2‖ 0 + (10.30)+ Q ‖h 2 − h 0 ‖ 0whereQ := (1/2 + π)(‖c ′ 2‖ 0 + ‖c ′ 1‖ 0 ) .74

and similarly P is the evaluationof the polynomialP = p ( ‖h 1 ‖ 0 , ‖h 2 ‖ 0 , ‖c ′ 1‖ 0 , ‖c ′ 2‖ 0 , 1/(len(c 1 ) len(c 2 )) )p(x 1 , x 2 , x 3 , x 4 , x 5 ) = (π +1/2)(x 1 +x 2 )+2π 3 (x 1 x 3 +x 2 x 4 )(x 3 +x 4 )x 5 (10.31)(that has constant positive coefficients).Proof. Fix c i immersed, and h i ∈ T ci M for i = 1, 2; let L i = len(c i ); letk i := P ci Π ci h i (s)for simplicity. We rewrite this in the convolutional form∫k i (s) := K i (s − ŝ)h i (ŝ) dŝ (10.32)c iwhen integrals are performed in arc parameter, and the kernel (following (10.21))isK i (s) := − s + 1 for s ∈ [0, L i ]L i 2and K i is extended periodically. By substituting and integrating on only oneperiod of K i ,∫ s( 1k i (s) =s−L i2 − s − ŝ )h i (ŝ) dŝ .L iWe can then prove easily (10.29): indeed by the convolutional representation(10.32)∫ len(c)|(P c Π c h)(t)| ≤ ‖h‖ 0 ∣ − slen(c) + 1 2∣ ds ≤ len(c)‖h‖ 0and instead, deriving and applying (iv) from lemma 10.25,0|(P c Π c h) ′ (θ)| = |Π c h(θ)| |c ′ (θ)| ≤ 2‖h‖ 0 ‖c ′ ‖ 0 .To prove (10.30), we write (10.32) in θ parameter, as was done in theDefinition 10.7; we need the run-length functions l i : lR → lRl i (τ) :=∫ τ0|c ′ i(x)| dxso that, setting s = l i (τ) and ŝ = l i (θ) we can writek i (τ) =∫ ττ−2π( 12 − l i(τ) − l i (θ))h i (θ)|c ′Li(θ)| dθi75

We can eventually estimate the difference |k 1 (τ) − k 2 (τ)| by using e.g. theinequality|A 2 B 2 C 2 − A 1 B 1 C 1 | ≤ |A 1 B 1 | |C 2 − C 1 | + |A 1 C 2 | |B 2 − B 1 | + |B 2 C 2 | |A 2 − A 1 |on the difference of the integrands( 12 − l 2(τ) − l 2 (θ))h 2 (θ) |c ′L} {{ 2 } {{ } }2(θ)|{{ }}B 2 C 2A 2( 1−2 − l 1(τ) − l 1 (θ))L 1} {{ }A 1to obtain that the above term is less or equal than‖h 1 ‖ 0 ‖c ′ 2 − c ′ 1‖ 0 + ‖c ′ 2‖ 0 ‖h 2 − h 0 ‖ 0 + ‖h 2 ‖ 0 ‖c ′ 2‖ 0∣ ∣∣∣ l 2 (θ) − l 2 (τ)L 2h 1 (θ) |c ′} {{ } }1(θ)|{{ }B 1 C 1− l ∣1(θ) − l 1 (τ) ∣∣∣L 1since |A i | ≤ 1/2. In turn (since the formulas defining k 1 (τ), k 2 (τ) the parametersare bound by τ − 2π ≤ θ ≤ τ) then|A 2 − A 1 | =l 2 (τ) − l 2 (θ)∣− l ∣ 1(τ) − l 1 (θ) ∣∣∣ ∫ τ∫ τ∣ θ=|c′ 2(x)| dxθ−|c′ 1(x)| dx ∣∣∣∣≤L 2L 1 ∣ L 2L 1≤ π L 1 + L 2‖c ′ 1 − c ′L 1 L2‖ 0 + π ‖c′ 1‖ 0 + ‖c ′ 2‖ 0|L 1 − L 2 | ≤2 L 1 L 2≤ 4π 2 ‖c′ 1‖ 0 + ‖c ′ 2‖ 0L 1 L 2‖c ′ 1 − c ′ 2‖ 0(by equations (ii) and (i) in lemma 10.25). Summarizing(|k 1 (τ) − k 2 (τ)| ≤ 2π ‖h 1 ‖ 0 + ‖h 2 ‖ 0 ‖c ′ 2‖ 0 4π 2 ‖c′ 1‖ 0 + ‖c ′ )2‖ 0‖c ′ 1 − c ′L 1 L2‖ 0 +2+ 2π‖c ′ 2‖ 0 ‖h 2 − h 0 ‖ 0 .If we derive, k i ′(θ) = h i(θ)|c ′ i (θ)|, so|k ′ 2(θ)−k ′ 2(θ)| ≤ |h 2(θ)| + |h 1 (θ)|2Symmetrizing we prove (10.30).|c ′ 2(θ)−c ′ 1(θ)| + |c′ 2(θ)| + |c ′ 1(θ)||h ′22(θ)−h ′ 1(θ)| .Lemma 10.27 Conversely, let c 1 , c 2 be two C 1 immersed curves, and h 1 , h 2be two differentiable fields; then (by using once again eqn. (i) from the lemma10.25)‖D c1 h 1 −D c2 h 2 ‖ 0 ≤ ‖c 1‖ 1 + ‖c 2 ‖ 12ε 2 ‖h 1 −h 2 ‖ 1 + ‖h 1‖ 1 + ‖h 2 ‖ 12ε 2 ‖c 1 −c 2 ‖ 1 (10.35)where ε = min(inf S 1 |c ′ 1|, inf S 1 |c ′ 2|).76

10.6.2 Existence of flow for the centroid energy (2.9)Theorem 10.29 Let us fix v ∈ lR n , let E(c) = 1 2 |avg c(c) − v| 2 ; let the initialcurve c 0 ∈ C 1 , then the ˜H 1 gradient descent flow of E has an unique solutionC = C(t, θ), for all t ∈ lR, and C(t, ·) ∈ C 1 .Results of a numerical simulation of the above gradient descent flow areshown in figure 12 on the following page.Before proving the theorem, let us comment on an interesting property ofthe above gradient flow.Remark 10.30 The length of the curves len(C(t, ·)) is constant during theevolution in t.Proof. Indeed it is easy to prove that ∂ t len(C(t, ·)) = 0: the Gâteaux differentialof the length of a curve c is∫D len(c; h) = (D s c · D s h) ds ;substituting the value of D s C(t, ·)) from eqn. (10.25),c∂ t len(C(t, ·)) = D(len(C))(∂ t C)∫1 〈=λL 2 D s C, ( D s C〈(avg c (C) − v), (C − avg c (C))〉 + α )〉 ds =〈∫〉 ∫1=λL 2 (avg c (C) − v), (C − avg c (C)) ds + (D s C · α) ds = 0 .We now present the proof of the theorem 10.29.Proof. We will first of all prove existence and uniqueness of the gradient flow forsmall time, using the Cauchy–Lipschitz theorem 25 in the space M of immersedcurves, seen as an open subset of the C 1 Banach space (see Definition 10.24).To this end we will prove that c ↦→ ∇˜H1E(c) is a locally Lipschitz functionalfrom C 1 into itself. Afterward, we will directly prove that the solution exists forall times.So, let us fix c 1 , c 2 ∈ C 1 ∩ M that are two curves near c 0 . Let us fix ε > 0 byε := infθ |c′ 0(θ)|/2 .By “near” we mean that, in all of the following, we will require that ‖c i −c 0 ‖ 1 < ε,for i = 1, 2. We will need the following (easy to prove) facts. (In the following,the index i will represent both 1, 2).• inf θ |c ′ i (θ)| ≥ ε, so all curves in the neighborhood are immersed.25 A.k.a. as the Picard–Lindelöf theorem77

7.552.5-10 -7.5 -5 -2.5 2.5 5 7.5 10-2.5-5-7.5-10321-4 -3 -2 -1 1 2 3 4-1-2-3-47.552.5-10 -7.5 -5 -2.5 2.5 5 7.5 10-2.5-5-7.5-10321-4 -3 -2 -1 1 2 3 4-1-2-3-47.552.5-10 -7.5 -5 -2.5 2.5 5 7.5 10-2.5-5-7.5-10321-4 -3 -2 -1 1 2 3 4-1-2-3-47.552.5-10 -7.5 -5 -2.5 2.5 5 7.5 10-2.5-5-7.5-10321-4 -3 -2 -1 1 2 3 4-1-2-3-47.552.5-10 -7.5 -5 -2.5 2.5 5 7.5 10-2.5-5-7.5-10321-4 -3 -2 -1 1 2 3 4-1-2-3-47.552.5-10 -7.5 -5 -2.5 2.5 5 7.5 10-2.5-5-7.5-10321-4 -3 -2 -1 1 2 3 4-1-2-3-4321-4 -3 -2 -1 1 2 3 4-1-2-3-4321-4 -3 -2 -1 1 2 3 4-1-2-3-4321-4 -3 -2 -1 1 2 3 4-1-2-3-4321-4 -3 -2 -1 1 2 3 4-1-2-3-4C(0, θ) = c 0 (θ) = ( cos(θ), sin(θ)(1 + 2 cos(θ) 4 ) )t=-1.210t=-0.810t=-0.410t=04t=0.54t=1.4t=1.54t=2.4C(0, θ) = c 0 (θ) = ( cos(θ), 2 sin(θ) )t=-1.210t=-0.810t=-0.410t=04t=0.54t=1.4t=1.54t=2.4Two numerical simulation of the ˜H 1 gradient descent flow of the centroid energyE(c) = 1 |avg 2 c(c) − v| 2 , for two different initial curves c 0. The constants were set toλ = 1/(4π 2 ), v = (2, 2). Steps for numerical discretization were set to ∆t = 1/30,∆θ = π/64 (on [0, 2π]). Note that plots for t < 0 are shown in a smaller scale.Figure 12: ˜H1 gradient descent flow of the centroid energy. See 10.20 and 10.29.78

• Setting a 0 := ‖c 0 ‖ 1 + ε, we have ‖c i ‖ 1 ≤ a 0 .• By equation (ii) in lemma 10.25,len(c 0 ) − 2π‖c 0 − c i ‖ 1 ≤ len(c i ) ≤ len(c 0 ) + 2π‖c 0 − c i ‖ 1and in particular2πε ≤ len(c i ) ≤ len(c 0 ) + 2πε .Let f i = ∇˜H1E(c i ) be the gradient, whose formula was expressed in (10.24).We decompose f i using the relation T c M = lR n ⊕ D c M, as done previously:and obtainf i = avg ci(f i ) ∈ lR n , ˜fi = f − avg ci(f i ) ∈ D ci Mf i = avg ci(c i ) − v1˜f i = P ci Π ci h i , whereλL 2 ih i := D s c i 〈(avg ci(c i ) − v), (c i − avg ci(c i ))〉 , (10.36)by rewriting (10.25) in the notation of this proof.We will then exploit all the inequalities in the lemmas to prove that|f 1 − f 2 | ≤ a 1 ‖c 1 − c 2 ‖ 1 , ‖ ˜f 1 − ˜f 2 ‖ 1 ≤ a 2 ‖c 1 − c 2 ‖ 1for two constants a 1 , a 2 > 0 and all c 1 , c 2 near c 0 ; this effectively proves that|f 1 − f 2 | ≤ (a 1 + a 2 )‖c 1 − c 2 ‖ 1that is, ∇˜H1E(c) is a locally Lipschitz functional.The first term is dealt with equation (iv) in lemma 10.25, whence we obtain∫∫|f 1 − f 2 | = | c 1 (s) ds − c 2 (s) ds| ≤ a 1 ‖c 1 − c 2 ‖ 1 with a 1 := 2a2 0c 1 c 2ε 2 .By repeated applications of the inequalities listed in lemma 10.25, we provethat, in the designed neighborhood, the following inequalities hold‖h 2 − h 1 ‖ 0 ≤ a 3 ‖c 2 − c 1 ‖ 1‖h i ‖ ≤ a 4for two constants a 3 , a 4 > 0. So we can choose constants P, Q > 0 such that theinequality (10.30) holds uniformly in the neighborhood, and then‖P c1 Π c1 h 1 − P c2 Π c2 h 2 ‖ 1 ≤ P ‖c ′ 1 − c ′ 2‖ 0 + Q ‖h 2 − h 1 ‖ 0 ≤≤ (P + Qa 3 )‖c 1 − c 2 ‖ 179

By using (i) and (ii) from lemma 10.25 once again, we can conclude that‖ ˜f 1 − ˜f 2 ‖ 1 ≤ a 2 ‖c 1 − c 2 ‖ 1for a constant a 2 > 0. The Cauchy–Lipschitz theorem is now invoked to guaranteethat the gradient descent flow does exist and is unique for small times. Let thenC(t, θ) be the solution, that will exist for t ∈ (t − , t + ), the maximal interval.In the following, given any g = g(t, θ) we will simply write ‖g‖ 0 instead ofand similarly‖g(t, ·)‖ 0 = sup |g(t, θ)|θ‖g‖ 1 = ‖g‖ 0 + ‖∂ θ g‖ 0 = sup(|g(t, θ)| + |∂ θ g(t, θ)|) ;θwe will also write len(C) = len(C(t, ·)) for the length of the curve at time t.We want to prove that the maximal interval is actually lR, that is, t + =−t − = ∞. The base and rough idea of the proof is assuming that t + or t − arefinite, and derive a contradiction by showing that ∇E(C) does not blow up whent ↘ t − or t ↗ t + .More precisely, we will show that, if t − > −∞ thenlim sup ‖∇E(C)‖ 1 < ∞ (∗∗)t↘t −this implies that the flow C(t, ·) admits a limit (inside the Banach space C 1 ) ast → t − , and then it may continued (contradicting the fact that t − is the lowesttime limit of existence of the flow). A similar result may be derived when t ↗ t +(but we will omit the proof, that is actually simpler).One key step in showing that (∗∗) holds is to consider the two fundamentalquantitiesI(t) := infθ |∂ θC(t, θ)| , N(t) := ‖C‖ 1and prove thatlim inf I(t) > 0 , lim sup N(t) < ∞when t − > −∞ and t ↘ t − . 26 . We proceed in steps.• By remark 10.30, we know that len(C) is constant (for as long as the flowis defined), and then equal to len(c 0 ).• At any fixed time, by (10.5),|C − C| ≤ len(C)/2 = len(c 0 )/2 (10.37)where C is the center of mass of C(t, ·); and then|C| ≤ |C − C| + |v − C| + |v| ≤ len(c 0 )/2 + √ 2E(C) + |v| (10.38)26 Actually, by tracking the first part of the proof in detail, it possible to prove that all otherconstants a 1 , a 2 , a 3 , a 4 , P, Q may be bounded in terms of these two quantities I(t), N(t)80

• During the flow, the value of the energy changes with rate∂ t E(C) = −‖∇˜H1E(C)‖ 2˜H1 =using (10.37), we can bound= −|C − v| 2 1−λ len(C)∫C2 〈C − v, C − C〉 2 ds|∂ t E(C)| ≤ E(C)(2 + 1/λ)so from Gronwall inequality we obtain that E(C) does not blow up; consequentlyby (10.38), ‖C‖ 0 does not blow up as well.• LetH := D s C〈(C − v), (C − C)〉 ,from the above we obtain that H does not blow up as well, since‖H(t, ·)‖ 0 ≤ len(c 0 ) √ 2E(C) (10.39)• The parameterization of the curves changes according to the lawsothis proves thatand∂ t log(|∂ θ C| 2 ) = 2〈D s ∂ t C, D s C〉 = 2H · D sCλ len(c 0 ) 2|∂ t log(|∂ θ C| 2 )| ≤ 2√ 2E(C)λ len(c 0 )lim supt↘t −lim inft↘t − I(t) > 0sup |∂ θ C(t, θ)| < ∞ .θ(As a consequence, ∂ θ C(t, θ) ≠ 0 at all times: curves will always beimmersed)• So by (10.29)does not blow up as well.‖P C Π C H‖ 1 ≤ (2π + 2)‖C ′ ‖ 0 ‖H‖ 0• Since both ‖C‖ 0 and ‖∂ θ C‖ 0 do not blow up, then ‖C‖ 1 does not blow up.• Decomposing F = −∂ t C = ∇˜H1E(C) (as was done in (10.36)) we write‖ ˜F ‖ 1 =1λ len(c 0 ) 2 ‖P CΠ C H‖ 1and from all above ‖ ˜F ‖ 1 does not blow up; but also|F | = |avg C (C) − v| ≤ √ 2E(C)as well; but then ‖∇˜H1E(C)‖ 1 itself does not blow up, as we wanted toprove.81

10.6.3 Existence of flow for geodesic active contourTheorem 10.31 Let once again∫E(c) = φ(c(s)) dscbe the geodesic active contour energy. Suppose that φ ∈ C 1,1loc(that is, φ ∈ C1and its derivative is Lipschitz on compact subsets of lR n ), let the initial curve bec 0 ∈ C 1 ; then the gradient flowdc= −∇dt˜H1E(c)exists and is unique in C 1 for small times.Proof. We rapidly sketch the proof, that is quite similar to the proof of theprevious theorem. We show that ∇˜H1E(c) is locally Lipschitz in the C 1 Banachspace (see Definition 10.24). Indeed, since φ ∈ C 1,1loc, the mapsc ↦→ ∇φ(c)c ↦→ φ(c)D s care locally Lipschitz as maps from C 1 to C 0 , so (using Lemmas 10.25 and 10.26)we obtain thatc ↦→ avg c (∇φ(c))is loc.Lip. from C 1 to lR n , andc ↦→ P c P c Π c(∇φ(c))c ↦→ P c Π c(φ(c)Ds c )are locally Lipschitz as maps from C 1 to C 1 ; combining all together (and usingthe Lemma 10.25 again)∇˜H1E(c) = Lavg c (∇φ(c)) − 1λL P cP c Π c(∇φ(c))+1λL P cΠ c(φ(c)Ds c )is locally Lipschitz as maps from C 1 to C 1 ; so by the Cauchy–Lipschitz theoremwe know that the gradient flow exists for small times.Remark 10.32 (Gradient descent of curve length) Of particular interestis the case when φ = 1, that is E = L, the length of the curve. In this case, wealready know that the H 0 flow is the geometric heat flow. By 10.15, the ˜H 1gradient instead reduces to∇ ˜H1L = c − avg c(c)λL. (10.40)So the ˜H 1 gradient flow constitutes a simple rescaling of the curve about itscentroid (!).82

It is interesting to notice that the H 1 and ˜H 1 gradient flows are well-posedfor both ascent and descent while the H 0 gradient flow is only well-posed fordescent. This is related to the fact that the H 0 gradient descent flow smoothsthe curve, whereas the ˜H 1 gradient descent (or ascent) has neither a beneficialnor a detrimental effect on the regularity of the curve.10.7 Regularization of energy vs regularization of flow/metricImagine an energy E minimized on curves (numerically sampled to p points).Suppose it is not satisfactory in applications (it may be ill posed, or not robustto noise). We have two solutions available.• Add a regularization term R(c) and minimize E(c)+εR(c) by H 0 gradientdescent. The numerical complexity of computing ∇ H 0R is of order O(p).• Minimize E(c) by ˜H 1 gradient descent. The numerical complexity ofcomputing ∇˜H1E using the equations in Prop. 10.10 is of order O(p).The numerical complexity of the two approaches is similar; but in the secondcase we are minimizing the original energy. This can bring evident benefits, asshown in the following example (originally presented in the 2006 IMA conferenceon shape spaces).Example 10.33 We consider a region-based segmentation of a synthetic noisyimage, where the energy is the Chan-Vese energyE(c) =∫(I − u) 2 dA +c in∫(I − v) 2 dAc outplus the regularization terms α len(c) for H 0 flows.When flowing according to −∇ H 0E + ακN and a small value of α > 0, weobtain the following evolution of the curve, where the curve gets trapped in alocal minima due to noise.For a larger value of α, the regularization term forces the curve away fromthe square shape.When evolving using −∇˜H1E, the curve captures the correct shape on coarsescale, and then segments the noisy boundary of the square further.83

10.7.1 Robustness w.r.to local minima due to noiseThe concept of local depends on the norm.Definition 10.34 A curve c 0 is a local minimum of an energy E iff ∃ɛ > 0such that if ‖h‖ c0 < ɛ then E(c 0 ) ≤ E(c 0 + h).Note that Sobolev-type norms dominate H 0 -type norm:‖h‖ H 0 ≤ ‖h‖ Sobolev,and the norms are not equivalent. As a result the neighborhood of critical pointsin Sobolev-type space is different than in H 0 space.We present an experimental demonstration of what “local” means (originallypresented in [57]).Example 10.351. We initialize a contour in a noisy image.2. We run the H 0 gradient flow on energyE(c) = E cv (c) + α len(c),where E cv is the Chan-Vese energy. Let’s call the convergedcontour c 0 ; it is shown in the picture on the right.3. We adjust c 0 at one sample point by one pixel, and call the modificationĉ 0 . Note: ĉ 0 is an H 0 (but also “rigidified”) local perturbation of c 0 .4. We run and compare H 0 , “rigidified,” and Sobolev gradient flows initializedwith ĉ 0 .The results are in Figure 13 on the next page. As we can see, the SACevolution can escape from the local minimum that is induced by noise. Surprisingly,in the numerical experiments, the same result holds for SAC even withoutperturbing the local minimum: this is due to the effect of numerical noise.Many other examples, on synthetic and on real images, are present in [55,57, 58].84

H 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . time → . . . . . . . . . . . . . . . . . . . . . . . . . . . . .“Rigidified”(Translationfavored). . . . . . . . . . . . . . . . . . . . . . . . . . . . . time → . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . time → . . . . . . . . . . . . . . . . . . . . . . . . . . . . .SobolevActiveContoursFigure 13: Comparison of minimization flows when initialized near a localminimum; see Example 10.35. (From [57] c○ 2008 IEEE. Reproduced with permission).10.8 New shape optimization energiesMost of what follows was originally presented in [56, 58].There are many useful active contour energies that cannot be optimized usingH 0 gradient flow. There are mainly two reasons why this happens.• Some energies result in ill-posed H 0 flows.• Some energies result in high-order PDE and are difficult to implementnumerically.In many cases, we can optimize these energies using Sobolev active contoursand avoid ill-posed problems and reduce order of PDE.We remark that other gradients (e.g. the “rigidified” active contours ofCharpiat et al. [10]) cannot be used in the examples we will present: theminimization flows still are ill posed or high order. Similarly global methods(e.g. graph cuts) cannot deal with these types of energies.10.8.1 Average weighted lengthA simple example that falls in the first category is the average weightedlength energy that we presented in equation (2.6) in the introduction:∫E 1 (c) = φ(c(s)) ds = avg c (φ) ;cthis energy was introduced since it does not present the short length bias (thatwas discussed in §2.4.2).Let L φ = ∫ c φ ds and L = len(c) so that E 1(c) = L φ /L, then the gradient is∇E 1 = 1 L ∇L φ − L φL 2 ∇L .85

We already presented all the calculus needed to compute the H 0 and ˜H 1gradients of E 1 (see Section 2.4.4 and Example 10.21); let us summarize all theresults.• In the case of H 0 gradient descent flow, the second term is ill posed , sinceL φ /L 2 > 0 and ∇ H 0L is the driving term in the geometric heat flow.This is also clear from the explicit formula∇ H 0E 1 (c) = [∇φ · N − κ(φ − avg c (φ))]N (10.41)since avg c (φ) is the average value of φ, therefore φ − avg c (φ) is negativeon roughly half of the curve; so the second term tries to increase length onhalf of curve using a geometric heat flow (and this is ill posed); whereasthe first term ∇φ does not stabilize the flow.• In the case of the Sobolev gradient, we saw in the proof of Theorem 10.31that the gradients ∇˜H1L φ and ∇˜H1L are locally Lipschitz in C 1 , so, by usingLemma 10.25 we conclude that the gradient ∇˜H1E 1 is locally Lipschitz aswell, hence the gradient flow is well defined.10.8.2 New edge-based active contour modelsThe average weighted length idea can be used to build new energies with interestingapplications in tracking.Let us call∫E old (c) = φ(c(s)) dsthe traditional edge-based active contour energy.When using a Sobolev–type norm, we can consider non-shrinking edge-basedmodels, as in the following examples.Example 10.36 Let us consider the energy∫E new (c) = φ(c(s)) ( L −1 + αLκ 2 (s) ) dswherecc• the first term is edge-detection without shrinking bias (nor regularity),• and the second term is a kind of elastic regularization (that will bediscussed also in the next section) that is moreover relaxed near edges;the length terms render the whole energy scale invariant.The frames in Figure 14 on the following page show initialization and finalresults obtained with different norms and energies.86

Example 10.37 Let us consider a length increasing edge-based model. Inthis case, we maximize the energy∫∫E inc (c) = φ(c(s)) ds − α κ 2 (s) dscwhere φ > 0 is high near edges. We compare numerical results with a typicaledge-based balloon model∫∫E bal (c) = φ(c(s)) ds − α φ(x) dxcRwhere R is the area enclosed by c. See figure 15.cInitial E old , H 0 E old , Sobolev E new , SobolevFigure 14: Comparison of segmentations obtained with different energies, asexplained in Example 10.36. (From [58] c○ 2008 IEEE. Reproduced with permission).Figure 15: Left to right we see the initial contour, the minima of E bal forα = 0.2, 0.25, 0.4 using H 0 ; the maximum for E inc with α = 0.1 using ˜H 1flowing. See Example 10.37. (From [58] c○ 2008 IEEE. Reproduced with permission).87

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

can be solved for k (this is not so easy to prove: see Michor and Mumford [37],or 4.4.7 and 4.6.6 in Hamilton [24]).11.2 Reparameterization to normal motionIn the preceding proposition we decided to use “orthogonal motion” as distinguishedchart for the manifold B. This choice leads also to a “lifting”.Lemma 11.3 Given any smooth homotopy C of immersed curves, there existsa reparameterization given by a parameterized family of diffeomorphisms Φ :[0, 1] × S 1 → S 1 , so that setting˜C(t, θ) := C(t, Φ(t, θ)) ;we have that ∂ t ˜C is orthogonal to ∂θ ˜C at all points; more precisely,π ˜T∂ t ˜C = 0 , πÑ∂ t ˜C = πN ∂ t C(where the last equality is up to the reparameterization Φ).For the proof, see Thm. 3.10 in [67] or §2.5 in [37].This explains what is commonly done in the level set method, where thetangent part of the flow is discarded.It is unclear that this choice is actually the best possible choice for computervision application. Consider the following example.Example 11.4 Suppose that c(θ) = (cos(θ), sin(θ)) is a planar circle. LetC(t, θ) = c(θ) + vt be an uniform translation of c. Let ˜C be as in the previousproposition; the Figure 17 shows the two motions. The two motions C and˜C coincide in B but are represented differently in M. Which one is best? Both“translation” and “orthogonal motions” seem a natural idea at first glance.C C ~The dotted line represents the trajectory of a point on the curve.Figure 17: Translation of a circle as a normal-only motion, by Prop. 11.3.90

11.3 The H 0 distance is degenerateIn §C in [67], or §3.10 in [37], the following theorem is proved.Theorem 11.5 The H 0 -induced distance is degenerate: the distance betweenany two curves is 0.This results is generalized in [38] to L 2 -type metrics of submanifolds of anycodimension.Here we will sketch the main idea of the proof for a very simple case: it ispossible to connect two segmentsc 0 (u) = (u, 0) , c 1 (u) = (u, 1)with a family of “zigzag” homotopies C k (t, θ) so that the H 0 action is infinitesimalwhen k → ∞. A snapshot of the curves along the homotopy, for t = 0, 1/8 . . . 1and k = 5, is in Fig. 18.t = 1/2t = 1/4t = 1/8t = 0t = 1t = 7/8t = 3/4t = 1/2case t ∈ [0, 1/2] 1/k∂C∂θα∂C∂vcase t ∈ [1/2, 1]Figure 18: Zig-zag homotopyIn the following, let C = C k for k large. We recall that the H 0 action isE(C) =and we also defineE ⊥ (C) =∫ 10∫∫ 10C∫C|∂ t C| 2 ds dt =|π N ∂ t C| 2 ds dt =∫ 1 ∫ 100∫ 1 ∫ 100|∂ t C| 2 |∂ θ C| dθ dt ;|π N ∂ t C| 2 |∂ θ C| dθ dt (11.1)(this “action” will be explained in Sec. 11.10). The argument goes as follows.1. The angle α is (the absolute value of) the angle between ∂ θ C and thevertical direction (that is also the direction of ∂ t C). Note that• α = α(k, t), and91

2. Now• at t = 1/2 it achieves the maximum α(k, 1/2) = arctan(1/(2k)), andalso• |∂ θ C| = 1/ sin(α).|π N ∂ t C| 2 |∂ θ C| = |∂ t C| 2 |∂ θ C| sin 2 (α) ∼ sin(α)so if k is large, then the angle α is small and then E ⊥ (C) < ε ∼ 1/k.3. We now smooth C just a bit to obtain Ĉ, so that E⊥(Ĉ) < 2ε.4. We then apply Prop. 11.3 to Ĉ to finally obtain ˜C, that moves only in thenormal direction, soas we wanted to show.E ⊥ (Ĉ) = E ⊥( ˜C) = E( ˜C) < 2ε11.4 Existence of critical geodesics for H jIn contrast, it is possible to show that a Sobolev-type metric does admit criticalgeodesics.Theorem 11.6 (4.3 in Michor and Mumford [36]) Consider the Sobolevtype metric∫〈h, k〉 G n c= 〈h, k〉 + 〈Ds n h, Ds n k〉 dscwith n ≥ 1. Let k ≥ 2n + 1; suppose c, h are in the (usual and parametric) H kSobolev space (that is, c, h admit k-th (weak)derivative, that is square integrable).Then it is possible to shoot the geodesic, starting from c and in direction h, forshort time.In §4.8 in [36] it is suggested that the theorem may similarly hold for Sobolevmetrics with length-dependent scale factors.11.5 Parameterization invarianceLet M be the manifold of (freely)immersed curves. Let 〈h 1 , h 2 〉 c be a Riemannianmetric, for h 1 , h 2 ∈ T c M; let ‖h‖ c = √ 〈h, h〉 c be its associated norm. LetE(γ) := ∫ 10 ‖ ˙γ(v)‖2 γ(v)dv be the action.Let C : [0, 1] × S 1 → S 1 be a smooth homotopy. We recall the definition thatwe saw in Section 4.6, and add a second stronger version.Definition 11.7 A Riemannian metric is• curve-wise parameterization invariant when the metric does not dependon the parameterization of the curve, that is ‖˜h‖˜c = ‖h‖ c when˜c(t) = c(ϕ(t)) and ˜h(t) = h(ϕ(t));92

• homotopy-wise parameterization invariant if, for any ϕ : [0, 1] ×S 1 → S 1 smooth, ϕ(t, ·) a diffeomorphism of S 1 for all fixed t, let ˜C(t, θ) =C(t, ϕ(t, θ)), then E( ˜C) = E(C).It is not difficult to prove that the second condition implies the first. Thereis an important remark to note: a homotopy-wise-parameterization-invariantRiemannian metric cannot be a proper metric.Proposition 11.8 Suppose that a Riemannian metric is homotopy-wise parameterizationinvariant; if h 1 , h 2 ∈ T c M and h 1 is tangent to c at all points, then〈h 1 , h 2 〉 c = 0. So the Riemannian metric in this case is actually a semimetric(and ‖ · ‖ c is not a norm, but rather a seminorm in T c M).Proof. Let C(t, u) = c(ϕ(t, u)) with ϕ(t, ·) be a time-varying family of diffeomorphismsof S 1 . So E(C) = E(c) = 0, that is∫ 10‖c ′ ∂ t φ‖ 2 c dt = 0so ‖c ′ ∂ t φ‖ c = 0; but note that, by choosing appropriately φ, we may representany h ∈ T c M that is tangent to c as h = c ′ ∂ t φ. By polarization, we obtain thethesis.11.6 Standard and geometric distanceThe standard distance d(c 0 , c 1 ) in M is the infimum of Len(γ) where γ is anyhomotopy that connects c 0 , c 1 ∈ M (cf. Definition 3.25).We are though interested in studying metrics and distances in the quotientspace B := M/Diff(S 1 ).We suppose that the metric G is “curve-wise parameterization invariant”;then G may be projected to B := M/Diff(S 1 ). So in the following we will use adifferent definition of “geometric distance”.Definition 11.9 (geometric distance) d G (c 0 , c 1 ) is the infimum of the lengthLen(C) in the class of all homotopies C connecting the curve c 0 to any reparameterizationc 1 ◦ φ of c 1 .This implements the quotienting formula that we saw in Definition 3.6 (inthis case, the group is G = Diff(S 1 )): so we will use d G as a distance on B.At the same time, note that in writing d G (c 0 , c 1 ) we are abusing notation:d G is not a distance in the space M, but rather it is a semidistance, since thedistance between c and a reparameterization c ◦ φ is zero.11.7 Horizontal and vertical spaceConsider a metric G (curve-wise parameterization invariant) on M. Let Πonce again be projection from M := Imm f (S 1 ) to the quotient B = B i,f =M/Diff(S 1 ).We present a list of definitions (see also the figure 19 on the next page).93

Definition 11.10 • The orbit is O c = [c] = {c ◦ φ | φ} = Π −1 ({c}).• The vertical space V c is the tangent to O c :V c := T c O c ⊂ T c Mthat can be explicitly written asV c := {h = b(s)c ′ (s) | b : S 1 → lR}i.e. all the vector fields h where h(s) is tangent to c.• The horizontal space W c is the orthogonal complement inside T c MW c := V ⊥c .Note that W c depends on G, but V c does not.MO cWccVcFigure 19: The action of Diff(S 1 ) on M. The orbits O c are dotted, the spacesW c and V c are dashed.The figure 20 may also help in understanding. The whole orbit O c is projectedto [c]. The vertical space V c is the kernel of DΠ c , so it is projected to 0.11.8 From curve-wise parameterization invariant to homotopywiseparameterization invariantIn the following sections, we just aim to give an overview of the theory; we willnot dwelve in depicting the most general hypotheses such that all presented resultsdo hold for a generic metric G; we will though, case by case, present referencesto how the results can be proven for the metrics discussed in this paper, and inparticular the Sobolev-type metrics.We suppose that this theorem holds.Theorem 11.11 Given any h ∈ T c M, there is an unique minimum point forminx∈W c‖x − h‖ Gand the minimum is called the horizontal projection of h.94

MO cW ccBVcΠ[c]Figure 20: The vertical and horizontal spaces, and the projection Π from M toB.This theorem is verified when G is a Sobolev-type metrics, see §4.5 in [36]; andit is trivially verified by H 0 .Definition 11.12 The horizontally projected metric G ⊥ is defined bywhere ˜h, ˜k are the projections of h, k to W c .〈h, k〉 G ⊥ ,c := 〈˜h, ˜k〉 G,c (11.2)Proposition 11.13 Equivalently the norm ‖h‖ G ⊥can be defined by‖h‖ G ⊥where the infimum is in the class of smooth b : S 1 → lR.Proof. Indeed by the projection theorem,= infb ‖h + bc′ ‖ G (11.3)inf ‖h +bbc′ ‖ G = ‖˜h‖ Gwhere ˜h is the projections of h to W c . By polarization we obtain (11.2).It is easy to prove that G ⊥ is curve-wise parameterization invariant, but moreoverProposition 11.14 G ⊥ is homotopy-wise parameterization invariant.Proof. Let ˜C(t, θ) = C(t, ϕ(t, θ)), then∂ t ˜C = ∂t C + C ′ ϕ ′so at any given time t‖∂ t ˜C(t, ·)‖G ⊥ = ‖∂ t C + C ′ ϕ ′ ‖ G ⊥ = ‖∂ t C‖ G ⊥95

y eqn. (11.3), where the terms RHS are evaluated at (t, ϕ(t, ·)); since G ⊥ iscurve-wise parameterization invariant, then‖∂ t ˜C(t, ·)‖G ⊥ = ‖∂ t C(t, ·)‖ G ⊥ .Consequently,Corollary 11.15 G is homotopy-wise parameterization invariant if and only iffor all b.‖h‖ G = ‖h + bc ′ ‖ GSummarizing all above theory, we obtain a method to understand/study/designthe metric G on B, following these steps:1. choose a metric G on M that is curve-wise parameterization invariant2. G generates the horizontal space W = V ⊥3. project G on W to define G ⊥ that is homotopy-wise parameterizationinvariant.11.8.1 Horizontal G ⊥ as length minimizerAs we defined in 11.9, to define the distance we minimize the length of the pathsγ : [0, 1] → M that connect a curve c 0 to a reparameterization c 1 ◦ φ of the curvec 1 . Consider the following sketchy example.Example 11.16 Consider two paths γ 1 and γ 2 connectingc 0 to reparameterizations of c 1 . Recall that the spaceW c is orthogonal to the orbits, the space V c is tangent tothe orbits. The path γ 1 (that moves in some tracts alongthe orbits, with ˙γ 1 ∈ V γ ) is longer than the path γ 2 .The above example explains the following lemma.W cc 0cVcγγ12Lemma 11.17 Let us choose G to be a curve-wise parameterization invariantmetric (so, by Prop. 11.8, it cannot be homotopy-wise parameterizationinvariant).1. Let ˜C(t, θ) = C(t, ϕ(t, θ)) where ϕ(t, ·) is a diffeomorphism for all fixed t.Minimizemin E G ( ˜C)ϕThen at the minimum ˜C ∗ , ∂ t ˜C∗ is horizontal at every point, andE G ( ˜C ∗ ) = E G ⊥( ˜C ∗ ) .O ccc 11 o φo φ’96

2. So the distance can be equivalently computed as the infimum of Len G ⊥ orof Len G ; and distances are equal, d G ⊥ = d G .In practice, when computing minimal geodesic (possibly by numerical methods),it is usually best to minimize E G , since it also penalizes the vertical part of themotion, and hence the minimization will produce a “smoother” geodesic.The proof of the above lemma is based on the validity of “the lifting Lemma”:Lemma 11.18 (lifting Lemma) Given any smooth homotopy C of immersedcurves, there exists a reparameterization given by a parameterized family ofdiffeomorphisms Φ : [0, 1] × S 1 → S 1 , so that setting˜C(t, θ) := C(t, Φ(t, θ))we have that ∂ t ˜C is in the horizontal space WC at all times t.For the metric H 0 , this reduces to lemma 11.3 in this paper; for Sobolev-typemetrics, the proof is in §4.6 in [36].11.9 A geometric gradient flow is horizontalProposition 11.19 If E is a geometric energy of curves, in the sense that Eis invariant w.r.to reparameterizations φ ∈ Diff + (S 1 ); then ∇E is horizontal.Proof. Let c be a smooth curve; suppose for simplicity (and with no loss ofgenerality) that it has len(c) = 2π and that it is parameterized by arc parameter,so that |c ′ | ≡ 1 and c ′ = D s c. Let b : S 1 → lR be smooth, and define φ :lR × S 1 → S 1 by solving the b flow, that is the ODE∂ t φ(t, θ) = b(φ(t, θ))with initial data φ(0, θ) = θ. Note that, for all t, φ(t, ·) ∈ Diff + (S 1 ). LetC(t, θ) = c(φ(t, θ)), note that∂ t C(t, θ) = b(φ(t, θ))c ′ (φ(t, θ))then E(C) = E(c), so that (deriving and setting t = 0)and we conclude by arbitrariness of b.∂ t E(C) = 0 = 〈∇E(c), bc ′ 〉11.10 Horizontality according to H 0Proposition 11.20 The horizontal space W c w.r.to H 0 isW c := {h : h(s) ⊥ c ′ (s)∀s}that is the space of vector fields orthogonal to the curve.97

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

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

2.2.3 Examples of geometric energy functionals for segmentation 122.3 Geodesic active contour method . . . . . . . . . . . . . . . . . . . 132.3.1 The “geodesic active contour” paradigm . . . . . . . . . . 132.3.2 Example: geodesic active contour edge model . . . . . . . 132.3.3 Implicit assumption of H 0 inner product . . . . . . . . . . 142.4 Problems & goals . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4.1 Example: geometric heat flow . . . . . . . . . . . . . . . . 152.4.2 Short length bias . . . . . . . . . . . . . . . . . . . . . . . 152.4.3 Average weighted length . . . . . . . . . . . . . . . . . . . 162.4.4 Flow computations . . . . . . . . . . . . . . . . . . . . . . 162.4.5 Centroid energy . . . . . . . . . . . . . . . . . . . . . . . 172.4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Basic mathematical notions 203.1 Distance and metric space . . . . . . . . . . . . . . . . . . . . . . 203.1.1 Metric space . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Banach, Hilbert and Fréchet spaces . . . . . . . . . . . . . . . . . 213.2.1 Examples of spaces of functions . . . . . . . . . . . . . . . 223.2.2 Fréchet space . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.3 More examples of spaces of functions . . . . . . . . . . . . 233.2.4 Dual spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.5 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.6 Troubles in calculus . . . . . . . . . . . . . . . . . . . . . 253.3 Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3.1 Submanifold . . . . . . . . . . . . . . . . . . . . . . . . . 263.4 Tangent space and tangent bundle . . . . . . . . . . . . . . . . . 263.5 Fréchet Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.6 Riemann & Finsler geometry . . . . . . . . . . . . . . . . . . . . 273.6.1 Riemann metric, length . . . . . . . . . . . . . . . . . . . 283.6.2 Finsler metric, length . . . . . . . . . . . . . . . . . . . . 283.6.3 Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.6.4 Minimal geodesics . . . . . . . . . . . . . . . . . . . . . . 293.6.5 Exponential map . . . . . . . . . . . . . . . . . . . . . . . 293.6.6 Hopf–Rinow theorem . . . . . . . . . . . . . . . . . . . . . 303.6.7 Drawbacks in infinite dimensions . . . . . . . . . . . . . . 303.7 Gradient in abstract differentiable manifold . . . . . . . . . . . . 313.8 Group actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.8.1 Distances and groups . . . . . . . . . . . . . . . . . . . . 334 Spaces and metrics of curves 334.1 Classes of curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 Gâteaux differentials in the space of immersed curves . . . . . . . 354.3 Group actions on curves . . . . . . . . . . . . . . . . . . . . . . . 374.4 Two categories of shape spaces . . . . . . . . . . . . . . . . . . . 374.5 Geometric curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.5.1 Research path . . . . . . . . . . . . . . . . . . . . . . . . . 38102

4.6 Goals (revisited) . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.6.1 Well posedness . . . . . . . . . . . . . . . . . . . . . . . . 394.6.2 Are the goal properties consistent? . . . . . . . . . . . . . 395 Representation/embedding/quotient in the current literature 405.1 The representation/embedding/quotient paradigm . . . . . . . . 405.2 Current literature . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 Metrics of sets 416.1 Some more math on distance and geodesics . . . . . . . . . . . . 416.1.1 Length induced by a distance . . . . . . . . . . . . . . . . 426.1.2 Minimal geodesics . . . . . . . . . . . . . . . . . . . . . . 426.1.3 Existence of geodesics and of average . . . . . . . . . . . . 426.1.4 Geodesic rays . . . . . . . . . . . . . . . . . . . . . . . . . 436.1.5 Hopf–Rinow , Cohn-Vossen Theorem . . . . . . . . . . . . 436.2 Hausdorff metric . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.2.1 An alternative definition . . . . . . . . . . . . . . . . . . . 446.2.2 Uncountable many geodesics . . . . . . . . . . . . . . . . 456.2.3 Curves and connected sets . . . . . . . . . . . . . . . . . . 456.2.4 Applications in computer vision . . . . . . . . . . . . . . . 466.3 A Hausdorff-like distance of compact sets . . . . . . . . . . . . . 466.3.1 Analogy with the Hausdorff metric, L p vs L ∞ . . . . . . 466.3.2 Contingent cone . . . . . . . . . . . . . . . . . . . . . . . 476.3.3 Riemannian metric . . . . . . . . . . . . . . . . . . . . . . 486.3.4 Polar coordinates of smooth convex sets . . . . . . . . . . 486.3.5 Smooth deformations of a convex set . . . . . . . . . . . . 496.3.6 Pullback of the metric on convex boundaries . . . . . . . 496.3.7 Pullback of the metric on smooth contours . . . . . . . . 506.3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 507 Finsler geometries of curves 517.1 Tangent and normal . . . . . . . . . . . . . . . . . . . . . . . . . 517.2 L ∞ metric and Fréchet distance . . . . . . . . . . . . . . . . . . . 517.3 L 1 metric and Plateau problem . . . . . . . . . . . . . . . . . . . 528 Riemannian metrics of curves up to pose 528.1 Shape Representation using direction functions . . . . . . . . . . 528.1.1 Multiple measurable representations . . . . . . . . . . . . 548.1.2 Continuous vs measurable lifting — no winding . . . . . . 558.2 A metric with explicit geodesics . . . . . . . . . . . . . . . . . . . 568.2.1 Curves up to rotation ↔ Grassmanian . . . . . . . . . . . 578.2.2 Representation . . . . . . . . . . . . . . . . . . . . . . . . 588.2.3 Quotient w.r.to representation . . . . . . . . . . . . . . . 58103

9 Riemannian metrics of immersed curves 589.1 H 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599.2 H A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599.3 Conformal metrics . . . . . . . . . . . . . . . . . . . . . . . . . . 609.4 “Rigidified” norms . . . . . . . . . . . . . . . . . . . . . . . . . . 6010 Sobolev type Riemannian metrics 6010.1 Sobolev-type metrics . . . . . . . . . . . . . . . . . . . . . . . . . 6110.1.1 Related works . . . . . . . . . . . . . . . . . . . . . . . . . 6210.1.2 Properties of H j metrics . . . . . . . . . . . . . . . . . . . 6210.2 Mathematical properties . . . . . . . . . . . . . . . . . . . . . . 6210.3 Sobolev metrics in shape optimization . . . . . . . . . . . . . . . 6310.3.1 Smoothing of gradients, coarse-to-fine flowing . . . . . . . 6510.3.2 Flow regularization . . . . . . . . . . . . . . . . . . . . . . 6610.4 ˜H j is faster than H j . . . . . . . . . . . . . . . . . . . . . . . . . 6710.5 Analysis and calculus of ˜H 1 gradients . . . . . . . . . . . . . . . 6810.6 Existence of gradient flows . . . . . . . . . . . . . . . . . . . . . . 7210.6.1 Lemmas and inequalities . . . . . . . . . . . . . . . . . . . 7310.6.2 Existence of flow for the centroid energy (2.9) . . . . . . . 7710.6.3 Existence of flow for geodesic active contour . . . . . . . . 8210.7 Regularization of energy vs regularization of flow/metric . . . . . 8310.7.1 Robustness w.r.to local minima due to noise . . . . . . . 8410.8 New shape optimization energies . . . . . . . . . . . . . . . . . . 8510.8.1 Average weighted length . . . . . . . . . . . . . . . . . . . 8510.8.2 New edge-based active contour models . . . . . . . . . . . 8610.9 New regularization methods . . . . . . . . . . . . . . . . . . . . . 8810.9.1 Elastic regularization . . . . . . . . . . . . . . . . . . . . . 8811 Mathematical properties of the Riemannian space of curves 8911.1 Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8911.2 Reparameterization to normal motion . . . . . . . . . . . . . . . 9011.3 The H 0 distance is degenerate . . . . . . . . . . . . . . . . . . . 9111.4 Existence of critical geodesics for H j . . . . . . . . . . . . . . . . 9211.5 Parameterization invariance . . . . . . . . . . . . . . . . . . . . . 9211.6 Standard and geometric distance . . . . . . . . . . . . . . . . . . 9311.7 Horizontal and vertical space . . . . . . . . . . . . . . . . . . . . 9311.8 From curve-wise parameterization to homotopy-wise . . . . . . . . 9411.8.1 Horizontal G ⊥ as length minimizer . . . . . . . . . . . . . 9611.9 A geometric gradient flow is horizontal . . . . . . . . . . . . . . . 9711.10Horizontality according to H 0 . . . . . . . . . . . . . . . . . . . . 9711.11Horizontality according to H j . . . . . . . . . . . . . . . . . . . . 9811.12Horizontality is for any group action . . . . . . . . . . . . . . . . 9811.13Momenta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9911.13.1 Conservation of momenta . . . . . . . . . . . . . . . . . . 99104

IndexC 0 , 73, 78, 82C 1 , 24, 73, 78, 82C 1 curve, 3C 1 functional between Fréchet spaces,24C 2 regular curves, 6C ∞ (I → lR n ), 24C j (I → lR n ), 23C 1,1loc , 82D s , 5, 61H 0 , 14–19, 58–61, 88, 91alternative definition, 61, 62, 64,66–68, 83–86, 88Hψ 0, 60H j , 61H j (I → lR n ), 23L ∞ metric, 51M/Diff(S 1 ), 4, 5, 41, 89, 93, 98O c , 94S 1 , 3, 33δ 0 , 32, 70˙γ(v) := ∂ v γ(v), 28∂ s , 5Diff(S 1 ), 4, 27, 37, 38, 41, 89, 93Emb(S 1 , lR n ), 34, 34Imm(S 1 , lR n ), 34, 34, 38, 62, 89Imm f (S 1 , lR n ), 34, 34, 38˜H j , 61b A , 8c ′ (θ) := ∂ θ c(θ), 3, 5d g (x, y), 54i-th mode of principal variation, 10u A , 8G acts on M, 32abstract differentiablemanifold, 25action, 4, 28, 91of a group on a distance, 33of a group on a space, 32of a group on the space of curves,37of a group on the space of curves,and momenta, 100of path in a metric, 29active contour, 11–15, 18, 19, 59, 66,73, 82, 85, 86, 88, 90arc parameter, 5arc-parameterized convolutional kernel,64arithmetic mean, 9asymmetric, 22asymmetric distance, 20atlas, 25, 39average integral, 6average shape, 9, 42average weighted length, 16, 85, 86avg, 6B, see M/Diff(S 1 )balloon model, 87Banach space, 22, 23, 78C 0 , C 1 , 73, 78, 82boundary based energies, 12Cartan, 31Cauchy–Lipschitz theorem, 78, 82center of mass, 17, 68, 70, 80centroid, 17, 68, 80centroid energy, 17, 70, 73, 76, 77Chan-Vese, 12, 18, 83, 84change of coordinatespolar, 48chart, 26charts, 25classes of curves, 34(simplified definitions), 3closed curves, 3Cohn-Vossen, 43Comparison principle, 15complete, 21, 23computer vision, 2, 5, 12, 37, 46, 90conformal metric, 58, 60consistent, 62contingent cone , 47105

contour continuation, 11convolution by arc parameter, 64convolutional kernel, 64cotangent space T ∗ c M, 31critical geodesic, 29, 30curvature, 6, 59in BV, 59Gâteaux differential of —, 36in L 2 , 63mean —, 6sectional —, 31signed scalar —, 6, 48, 59curve, 34classes of, 34embedded, 34freely immersed, 34image of a —, 3, 8, 34, 45immersed, 3, 34manifold of —, 3planar —, 3curve-wise parameterization invariant,92, 95, 96curvesup to pose, 52cutlocus, 50de la Vallée-Poussin, 70, 71degenerate, 91derivation with respect to the arc parameter,5, 61, 69diffeomorphism, 4, 26differentiable manifoldabstract, 25submanifold, 26dimension, 25Dirac’s delta, 32, 70directional derivative, 13, 17, 26, 31distance, 20, 28distance function, 8, 43, 46distance-based average, 9, 42distribution, 24, 32doubly traversed circle, 34dual space, 24edge detection, 11edge-based, 11, 12, 15, 86, 87Eells, 30elastic regularization, 86elastica energy, 36, 66, 86, 88Elworthy, 30embedded curves, 34embedding theorems, 26Emmy Noether, 99empirical principal component analysis,10energy, 5, 28of a path, 28equivalence classes, 4equivalence relation, 32Euclideandistance, 21invariance, 39, 61Euclidean group, 37, 100Euclidean norm, 5evaluation functional, 32exponential map, 10, 29external skeleton, 50fattened set, 44Finsler metric, 28Fréchet mean, 9, 42Fréchet distance, 51, 63Fréchet manifolds, 27Fréchet space, 23, 24freely immersed curve, 34Gâteaux differentiable, 24, 31Gâteaux differential, 13, 15, 16, 24, 35,70of the curvature, 36of the elastica energy, 36of the length, 78Gaussian, 13geodesic active contour, 12, 71–73, 82geodesic distance, 54geodesic ray, 43geometric, 4, 61geometric active contour, 19geometric curves, 4, 38geometric distance, 93geometric energy, 97geometric evolution, 11106

geometric functional, 4, 5geometric heat flow, 15, 16, 86geometric oriented curves, 4gradient ∇E(c), 31gradient descentfor curve length, 15gradient descent flow, 11, 13, 72Grassmanian manifold, 57group action, 32, 99on curves, 37group operation, 4Hadamard, 31Hausdorff, 23Hausdorff distance, 8, 44heat equation, 15Hilbert space, 22homeomorphism, 20, 25homotopy, 3homotopy-wise parameterization invariant,93, 95Hopf, 30, 43, 44horizontal projection, 94horizontal space, 94, 97horizontality, 92horizontally projected metric, 95ill-posed, 16, 17, 19, 85, 86, 88imageblack and white —, 12of a curve, 3, 8, 34, 45segmentation, 11smooth and featureless, 15synthetic noisy, 83immersed curve, 3immersed curves, 34implicit function theorem, 26induced geodesic distance, 42inflationary term, 15integral length, 28, 42integration by arc parameter, 6, 12invarianceEuclidean, 39, 61parameterization, see “reparameterizationinvariant”rescaling, 39, 61rotation, 39, 61translation, 39, 61invariant w.r.to the action of the group,33Karcher mean, 9, 42Karhunen-Loève theorem, 10kernel, 64l.c.t.v.s., 22, 24–26landmark, 9Lebesgue measure, 22length, 5, 28, 42length increasing, 87length shrinking effect, 15level set, 46level set averaging, 9level set method, 11, 12, 15, 19, 65, 90,98Lie algebra, 99Lie groups, 99lifing lemma, 97for H 0 , 90locally compact, 43locally-convex topological vector space,22M, 3manifold of (parametric) curves, 3matrix, 37, 48antisymmetric, 100, 101orthogonal, 37, 100mean curvature, 6measurable lifting, 54measurable representation, 54metric space, 21metrizable, 23minimal geodesic, 29, 42momentum, 100angular, 101linear, 101motion by mean curvature, 15neighborhood, 30, 55, 78, 84, 89Noether, Emmy, 99norm, 21, 28107

normal vector, 6objective function, 5open sets, 20operator, 35orbit, 32, 94orthogonaldecomposition of deformation, 68parameterizationinvariance, see “reparameterizationinvariant”curve-wise, 92, 93homotopy-wise, 93parametric or landmark averaging, 9path-metric, 42, 43pathological, 18, 91paths, 3PCA, 9Picard–Lindelöf theorem, 78planar curves, 3polar change of coordinates, 48pose, 52preshape space, 38, 53primitive operator, 69principal component analysis, 9, 30principal variation, 10prior information, 18probability measure, 7, 32projection, 32projection operator, 69, 74quotient, 32quotient distance ˆd, 55quotient space, 4random processes, 10random vector, 9region based energies, 12region-based, 11, 12, 83removing, 47reparameterization, 4, 100reparameterization invariant, 4, 6, 36,37, 39, 61, 92, 95–97rescaling, 37, 82, 86, 100invariance, 39, 61Riemannian metric, 28rigidified norm, 60rigidified norms, 58Rinow, 30, 43, 44rotation, 37, 100invariance, 39, 61rotation index, 53run-length function, 64, 75SAC, 61scalar product, 5, 28scale invariant, 61, 86, 88segmentation, 83semidistance, 20, 93semimetric, 93seminorm, 22, 93set symmetric difference, 8set symmetric distance, 8shape, 18shape analysis, 3, 7, 62shape optimization, 2, 19, 62shape space, 2, 37, 40, 41, 46of curves up to pose, 52categories of —, 37shooting geodesics, 29short length bias, 15, 85signed distance function, 8signed distance level set averaging, 9signed scalar curvature, 6smooth functions, 24Sobolev active contour, 61Sobolev space, 23, 92Sobolev-type metrics, 61space of non-translating deformations,68space of translations, 68special rotation, 37square root lifting, 57standard distance, 93Stiefel manifold, 57stochastic processes, 10submanifold, 26supess, 22tangent bundle, 26tangent space, 3, 26108

tangent vector, 5topological hyperspace, 43topological space, 20topology, 20total variation length, 42translation, 37, 100invariance, 39, 61up to pose, 52vertical space, 94visual tracking, 62, 86winding number, 53109

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