J Math Imaging VisDOI 10.1007/s10851-012-0413-4

Functional Currents: A New Mathematical Tool to Modeland Analyse Functional Shapes

Nicolas Charon · Alain Trouvé

© Springer Science+Business Media New York 2013

Abstract This paper introduces the concept of functionalcurrent as a mathematical framework to represent and treatfunctional shapes, i.e. submanifold-supported signals. It ismotivated by the growing occurrence, in medical imagingand computational anatomy, of what can be described asgeometrico-functional data, that is a data structure that in-volves a deformable shape (roughly a finite dimensionalsubmanifold) together with a function defined on this shapetaking values in another manifold. Whereas mathematicalcurrents have already proved to be very efficient theoreti-cally and numerically to model and process shapes as curvesor surfaces, they are limited to the manipulation of purelygeometrical objects. We show that the introduction of theconcept of functional currents offers a genuine solution tothe simultaneous processing of the geometric and signal in-formation of any functional shape. We explain how func-tional currents can be equipped with a Hilbertian norm thatsuccessfully combines the geometrical and functional con-tent of functional shapes under geometrical and functionalperturbations, thus paving the way for various processingalgorithms. We illustrate this potential on two problems: theredundancy reduction of functional shape representationsthrough matching pursuit schemes on functional currentsand the simultaneous geometric and functional registrationof functional shapes under diffeomorphic transport.

Keywords Computational anatomy · Shape modelling ·Currents · Diffeomorphic registration · Geometry and signal

N. Charon (�) · A. TrouvéCMLA, ENS Cachan, 91 avenue du président Wilson,94230 Cachan, Francee-mail: Nicolas.Charon@cmla.ens-cachan.fr

A. Trouvée-mail: Alain.Trouve@cmla.ens-cachan.fr

1 Introduction

Shape analysis is certainly one the most challenging prob-lems in pattern recognition and computer vision [3, 5, 14,17]. Moreover, during the last decade, shape analysis hasplayed a major role in medical imaging through the emer-gence of computational anatomy [1, 13, 18–20, 24]. Morespecifically, the quest of anatomical biomarkers through theanalysis of normal and abnormal geometrical variability ofanatomical manifolds has fostered the development of inno-vative mathematical frameworks for the representation andthe comparison of a large variety of geometrical objects.Among them, since their very first significant emergence inthe field of computational anatomy, mathematical currentshave become an increasingly used framework to representand analyze shapes of very various types, from unlabeledlandmarks to curves [11], fiber bundles [8] surfaces [12] or3D volumes.

More recently though, an increasing number of datastructures have emerged in computational anatomy that notonly involve a geometrical shape but some signal attachedto this shape, which we denote as functional shapes. Themost basic example is, of course, classical images for whichthe geometrical support is simply a rectangle on which a‘gray level’ signal is defined. In many cases however, thesupport can have a much more complex geometry like, forinstance, the activation maps on surfaces of cortex obtainedthrough fMRI scans. Signals can also include structures thatare more sophisticated than simple real values: we couldthink of a vector field on a surface as well as tensor-valuedsignals that appear in DTI imaging. Such a diversity bothin shape and signal makes it a particularly delicate issue toembed and compare all geometrico-functional objects in onecommon framework.

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As a result, recent approaches have been primarily in-vestigating methods where shape and signal are treated sep-arately, as for instance in [22]. In our opinion, there areseveral important difficulties raised by such methods. Elim-inating geometrical differences by matching geometricalsupports first requires an exact mapping between two shapesor equivalently a common coordinate system. In practice,this is neither an easy nor a canonical thing to do. In [10]and [22], who focus on the case of fMRI signals or corticalthickness on the brain, the authors propose to map brain sur-faces on a common sphere model by a smoothing process.While this provides a direct way to compare signals at corre-sponding points, it is still not sufficient to get a relevant com-parison because, for functional shapes, tangential deforma-tions or reparametrizations should also be taken into accountin order to avoid residual mismatches between signals (seefor instance the example of Fig. 4). As a result, there mustbe an additional estimation of this tangential transformation.In the aforementioned work, this is done as a following andseparate step which consists of finding a reparametrizationof the sphere that best matches the two signals. Yet, sincethis process is applied on the parameter space (the sphere),there are no guarantees that the obtained transformation hasany geometrical meaning with respect to the shapes them-selves. This is why most approaches have eventually triedto reincorporate something of the original shapes in this laststep, usually by using curvatures.

Even if interesting results can be obtained with such ap-proaches, we believe that many technicalities and arbitrari-ness are introduced incidentally, making them difficult togeneralize to a wider class of datasets. To give one exam-ple, the necessity for a common parametrization on a fixedshape (e.g a sphere) results in algorithms that are completelynon-robust to small changes of topology such as disconnec-tions at some location of the shape. Treating for instance sur-faces with many holes or fiber bundle datasets like the oneof Fig. 10 would then become dramatically difficult. Instead,the core motivation underlying this article is that exact pointto point correspondences can be avoided, provided a properrepresentation and comparison framework for geometrico-functional data structures is defined. Several attempts havebeen undertaken in this direction, notably by a direct mod-elling with currents, but these have encountered importantlimitations, which we shall describe in Sect. 2. The maincontribution of the paper is to propose a new analytical set-ting that shares some common features with mathematicalcurrents but overcomes its main limitations when dealingwith functional shapes. The core idea, developed in Sect. 3is to augment usual currents by a natural tensor product withan extra component embedding the signal values, leading toour definition of functional currents. We consider then var-ious actions on functional currents by diffeomorphic trans-port and show in Sect. 4 that kernel norms can provide a suit-able Hilbertian structure on functional currents, providing a

way to compare geometrical support and signal with one sin-gle metric. We show in what sense this metric is consistentwith the idea of comparing functional shapes with respectto deformations between them, which makes it a good ap-proach for defining data attachment terms (cf. Propositions 3and 4). We then illustrate the potential of this approach inSect. 5 on two different problems. The first illustration is theconstruction, via a matching pursuit algorithm, of a redun-dancy reduction or compression algorithm for the represen-tation of functional shapes by functional currents with a fewexamples of compression on curves and surfaces with real-valued data. The second illustration is about the potentialbenefits of functional currents in the field of computationalanatomy. In particular, we show a few basic results of dif-feomorphic inexact registration between functional shapeswith our extension of large deformation diffeomorphic met-ric mapping (LDDMM) algorithm [2] to functional currents.

2 Currents in the Modelling of Shapes

2.1 A Brief Presentation of Currents in ComputationalAnatomy

Currents were historically introduced as a generalization ofdistributions by Schwartz and then de Rham in [4]. The the-ory was then developed and connected to geometric mea-sure theory in large part by Federer [9]. At first, these resultsfound interesting applications in the calculus of variationsas well as differential equations. However, the use of cur-rents in the field of computational anatomy is more recent,initially proposed in [11]. In the following, we try to outlinethe minimum background of the theory of currents neededto describe the link between shapes and currents.

First, we fix some notation. Call E a generic Euclideanspace of dimension n. We will denote by Ω

p

0 (E) the spaceof continuous p-differential forms on E that vanish at infin-ity. Every element ω of Ω

p

0 (E) is then a continuous func-tion such that for all x ∈ E, ω(x) ∈ ΛpE∗. Since we havethe isomorphism ΛpE∗ ≈ (ΛpE)∗, we can view ω(x) bothas a p-multilinear and alternating form on E and as a lin-ear form on the

(np

)-dimensional space of p-vectors in E.

In all of the following, we will use the notation ωx(ξ) asthe evaluation of a differential form ω at point x ∈ E andon the p-vector ξ . On ΛpE a Euclidean structure can bedefined, induced by the one on E, which is such that ifξ = ξ1 ∧ · · · ∧ ξp and η = η1 ∧ · · · ∧ ηp are two simple p-vectors, 〈ξ, η〉 = det(〈ξi, ηj 〉)i,j . The norm of a simple p-vector is therefore the volume of the element. The spaceΩ

p

0 (E) is then equipped with the L∞ norm of boundedfunctions defined on E. With this notation, we define thespace of p-currents on E as the topological dual Ω

p

0 (E)′,i.e. the space of linear and continuous forms on Ω

p

0 (E).

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Note that in the special case where p = 0, the previous defi-nition is exactly the one of usual distributions on E that canbe also seen as signed measures on E. The simplest exam-ples of currents are given by the generalization of the Diracmass: if x ∈ E and ξ ∈ ΛpE, δξ

x is the current that associatesto any ω ∈ Ω

p

0 (E) its evaluation ωx(ξ).Now, the fundamental relationship between shapes and

currents lies in the fact that every d-dimensional and ori-ented submanifold X of E of finite volume can be repre-sented by an element of Ωd

0 (E)′. Indeed, we know from in-tegration theory on manifolds [9, 15] that any d-differentialform of Ωd

0 (E) can be integrated along X, which associatesto X a d-current CX such that:

CX(ω) =∫

X

ω (1)

for all ω ∈ Ωd0 (E). The application X → CX is also injec-

tive. Equation (1) can be rewritten in a more explicit way ifX admits a parametrization given by a certain smooth im-mersion F : U → E with U an open subset of R

d . Then,

CX(ω)

=∫

(x1,...,xd )∈UωF(x1,...,xd )

(∂F

∂x1∧ · · · ∧ ∂F

∂xd

)dx1 · · ·dxd .

It is a straightforward computation to check that the last ex-pression is actually independent of the parametrization (aslong as the orientation is conserved). In the general case,there always exists a partition of unity adapted to the lo-cal charts of X, so that CX can be expressed as a combi-nation of such terms. The representation is fully geometricin the sense that it only depends on the manifold structureitself and not on the choice of a parametrization. It there-fore enables us to consider submanifolds of given dimension(curves, surfaces, . . .) as elements of a fixed functional vec-tor space. This also gives a very flexible setting to manipu-late shapes since addition, combination or averages becomestraightforward to define. On the other hand, spaces of cur-rents contain a lot more than submanifolds because generalcurrents do not usually derive from submanifolds (think forinstance of a punctual current δ

ξx ). However, it encompasses,

in a unified framework, a wide variety of geometrical objectssuch as bundles of curves and surfaces which can be relevantin some anatomy problems.

In registration problems, a fundamental operation is thetransport of objects by a diffeomorphism of the ambientspace. If C ∈ Ω

p

0 (E)′ and φ ∈ Diff(E), we define the trans-port of C by φ as the classical push-forward operation de-noted φC:

∀ω ∈ Ωp

0 (E), (φC)(ω) = C(φω

)(2)

where φ∗ω is the usual pull-back of a differential form de-fined for all x ∈ E and ξ = ξ1 ∧ · · · ∧ ξp ∈ ΛpE by:(φω

)x(ξ) = ωφ(x)

(dxφ(ξ1) ∧ · · · ∧ dxφ(ξp)

)(3)

dxφ being the notation we use for the differential of the dif-feomorphism at point x. With this definition, it is straightfor-ward to check that φCX = Cφ(X), which means that trans-porting by push-forward the d-current associated to a sub-manifold yields the d-current associated to the transportedsubmanifold φ(X).

To complete this brief presentation of currents appliedto computational anatomy, we still need to explain how thecurrent representation can be practically implemented andhow computations can be made with them. This step con-sists mainly in approximating the integral in (1) into a dis-crete sum of punctual currents CX ≈ ∑

k=1..N δξkxk

where xk

are points in E and d-vectors ξk encode local elements ofvolume of the manifold X. From a computational point ofview, a mesh on the shape is needed for which each cellwill generate one Dirac. In the case of curves for instance, ifγ : I → E is a continuous curve in E given by a samplingof N points {xk = γ (tk)}k=1..N , we associate the 1-currentcorresponding to the approximation of γ as a polygonal line,that is:

C̃γ =N−1∑

j=1

δτjcj

with cj the center of segment [xjxj+1] and τj the vectorxj+1 − xj . It can be easily shown that |CX(ω) − C̃X(ω)|tends toward zero for all 1-form ω as maxk{|tk+1 − tk|} → 0,i.e. as the sampling gets more accurate (cf. [11]). The sameprocess can be applied to a triangulated surface S immersedin E = R

3, by associating to each triangle, one Dirac en-coding the position of the center and the normal vector (seeillustrations of Fig. 1).

Finally, the question of building a metric on the space ofcurrents should be addressed. As discussed in [6, Chap. 1.5],a particularly convenient framework to build computablemetrics is to define a Hilbert space structure on currentsthrough reproducing kernel Hilbert space (RKHS) theory.This approach consists in defining a vector kernel on E (K :E ×E → L(ΛpE), L(ΛpE) being the set of linear applica-tions of ΛpE into itself) and consider its associated RKHSW . Under some assumptions on the kernel, it can be shownthat the space of p-currents is continuously embedded in thedual W ′ which is also a Hilbert space. Therefore, in appli-cations, we generally consider W ′ instead of Ω

p

0 (E)′ as ouractual space of currents. For more details on the construc-tion of RKHS on currents, we refer to [6] and [11]. Since, inapplications, manifolds are represented by sums of punctualcurrents, it is sufficient to be able to compute inner productsbetween two punctual currents. The RKHS framework givessimple closed form expressions of such products. Indeed,

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Fig. 1 Representation of acurve and a surface in Diraccurrents

one can show that 〈δξ1x1 , δ

ξ2x2〉W ′ = ξT

1 K(x1, x2)ξ2. Compu-tation of distances between shapes then reduces to simplekernel calculus which can be performed efficiently for well-suited kernels either through fast Gauss transform schemesas in [11] or through convolutions on linearly spaced gridsas explained in [6].

In summary, these few theoretical reminders were meantto stress two essential advantages of currents in shape rep-resentation. The first being its flexibility due to the vectorspace structure and the wide range of geometrical objectsthat are represented, without ever requiring any parametriza-tion. The second important point is the fact that computa-tions on currents are made very efficient by the use of ker-nels, which makes them appropriate in various applicationssuch as simplification, registration or template estimation.All these elements suggest an extension of the frameworkof currents to incorporate functional shapes, which will bediscussed in detail below.

2.2 Functional Shapes and the Limitations of Currents

We now consider, as in the previous section, a d-dimensionalsubmanifold X of the n-dimensional vector space E but inaddition, we assume that functional data is attached to everypoint of X through a function f defined on X and takingits values in a differentiable manifold M , the signal space.What we call a functional shape is then a couple (X,f ) ofsuch objects. The natural question that arises is this: can wemodel such functional shapes in the framework of currentsas purely geometrical shapes? In the following, we discusstwo possible methods to address this question directly withusual currents and explain why both of them are not fullysatisfying from the perspective of applications to computa-tional anatomy.

First attempts to include signals supported geometricallythrough the current representation were investigated in [6]with the idea of colored currents. This relies basically onthe fact already mentioned that the set of d-currents con-tains a wider variety of objects than d-dimensional subman-ifolds like rectifiable sets or flat chains (cf. [9]). In partic-ular, weighted submanifolds can be considered as currentsin the following very natural way: suppose that X is a sub-manifold of E of dimension d and f : X → R is a weight

or equivalently a real signal at each point of X such that f

is continuous, then we can associate to (X,f ) a d-currentin E:

T(X,f )(ω) =∫

X

f ω

Although this approach seems to be the most straightforwardway to apply currents to functional shapes, as we are stillusing a d-current in E, it quickly emerges that such a repre-sentation suffers from several important drawbacks. First isthe difficulty to generalize colored currents for signals thatare not simply real-valued, particularly if the signal space isnot a vector space (think for instance of the case of a signalconsisting of directions in the 3D space, where M is there-fore the sphere S

2). The second point arises when the previ-ous equation is discretized into Dirac currents, which leadsto an expression of the form

∑k=1..N f (xk)δ

ξkxk

. We noticean ambiguity appearing between the signal and the volumeelement ξ since for any r = 0, f (xk)δ

ξkxk

= rf (xk)δξk/rxk

; sep-arating geometry from signal in the discretized version ap-pears as a fundamental difficulty. In addition, the energy ofDirac terms are proportional to the value of the signal at thecorresponding point which induces an asymmetry betweenlow and high-valued signals. In this setting, areas havingvery small signals become negligible in terms of current,which not justified in general and can drastically affect thematching of colored currents. We show a simple illustrationof this issue when matching two colored ellipsoids with thisapproach in Fig. 2. Finally an additional limitation in usingcolored currents is the fact that there is no flexibility to treatthe signals at different scale levels than geometry, makingthis approach highly sensitive to noise.

Another possible and interesting way to represent a func-tional shape by a current is to view it as a shape in theproduct space E × M . Somehow, it generalizes the ideaof seeing a 2D image as a 3D surface. However, at ourlevel of generality, it is not a completely straightforwardprocess. If the signal function f is assumed to be C1, theset G := {(p,f (p)) | p ∈ X} inherits a structure of a d-dimensional manifold of E × M . With M a vector space,it results directly from the discussion above that G can berepresented as a d-current in the product space, that is as

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Fig. 2 An example of matching between two ellipsoids provided bythe classical LDDMM algorithm. On the left, the adaptation with thecolored currents’ representation. Values of the signals are two diffusedstains both on the source ellipsoid (inside surface) and the target one(exterior shaded surface). We display in blue trajectories of the points.

The points compounding to zero-valued area of the signal in the sourceshape are not matched to the corresponding points in the target surface.On the right, we show what should be the expected result. It is obtainedthrough the approach of functional currents that shall be presented inthe next parts of the paper (Color figure online)

Fig. 3 Product currents and topology. On the left, we show a discon-nected 2D curve with signal values 0 in blue and 1 in red as well asthe connected curve in dashed line. On the right hand side are thecorresponding curves in the 3-dimensional geometry × signal space.What we want to emphasize here is the fact that no RKHS norm on

product currents would provide a continuity of this representation withrespect to connectivity: the difference between the two curves is themagenta dashed part which represents a pure variation in the signaldomain (Color figure online)

an element of Ωd0 (E × M). For a general signal manifold

though, we would need to extend our definitions of currentsto the manifold case, which could be done (cf. [4]) but thedefinition of kernels on such spaces would then become amuch more involved issue in general compared to the vectorspace case. This difficulty set apart, there still are some im-portant elements to point out. The first one is the increase ofdimensionality of the approach, because, while we are stillconsidering a manifold of dimension d , the co-dimension ishigher: the space of d-vectors characterizing local geome-try Λd(E × M) is now of dimension

(n+dim(M)

d

), with sig-

nificant consequences from a computational point of view.From a more theoretical angle, we see that, in such an ap-proach, geometrical support and signal play a symmetricrole. In this representation, the modeled topology is no morethe one of the original shape because we also take into ac-count variations within the signal space. Whether this is astrength or a weakness is not obvious a priori and would de-

pend on the kind of applications. What we can state is thatthis representation is not robust to topological changes ofthe shape: in practice, the connectivity between all pointsbecomes crucial, which we illustrate on the simplest ex-ample of a plane curve carrying a real signal in Fig. 3. Inthe field of computational anatomy, when processing of datasuch as fiber bundles, where connections between points ofthe fibers are not always reliable, this would be a clear draw-back. We shall show an example of this issue in the last sec-tion of the paper.

To summarize this section, we have investigated two di-rect ways to see a functional shape as a current. The coloredcurrent setting, although being very close to the modellingof purely geometrical shapes, is not acceptable mainly be-cause it mixes geometry and signal in an inconsistent way.As for the second idea of immersing the functional shape ina product space, we have explained its limits both in termsof the difficulty in practical implementation and in terms of

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the lack of robustness with respect to topology of the geo-metrical support. These observations constitute our motiva-tion to redefine a proper class of mathematical objects thatwould preserve the usefulness of currents while overcomingthe previous drawbacks.

3 Definition and Basic Properties of FunctionalCurrents

In this section, we propose an extension of the notion of cur-rents to represent functional shapes. The new mathematicalobjects we introduce, named ‘functional currents’, are notusual currents strictly speaking, contrary to the methods pre-sented in Sect. 2.2. They would rather derive from the verygeneral concept of double current introduced originally byde Rham in [4]. Here, we adapt it in a different way to fit theapplications we aim for in computational anatomy.

3.1 Functional p-Forms and Functional Currents

As in the previous section, let (X,f ) be a functional shape,with X a d-dimensional submanifold of the n-dimensionalEuclidean space E and f a measurable mapping from X

to a signal space M . In our framework, M can be any Rie-mannian manifold. Most simple examples are provided bysurfaces with real signal data like activation maps on cortexin fMRI imaging, but the framework that we present here ismade general enough to incorporate a wide range signals:vector fields, tensor fields, Grassmannians. We now definethe space of functional currents again as the dual of a spaceof continuous forms:

Definition 1 We call a functional p-form on (E,M) an ele-ment of the space C0(E ×M,ΛpE∗) which will be denotedby Ω

p

0 (E,M) hereafter. We consider the uniform normon Ω

p

0 (E,M) defined by: ‖ω‖∞ = sup(x,m)∈E×M ‖ω(x,m)‖.A functional p-current (or fcurrent in short) is defined as acontinuous linear form on Ω

p

0 (E,M) for the uniform norm.The space of functional p-currents will be therefore denotedΩ

p

0 (E,M)′.

It is important, at this point, to distinguish the space offunctional currents Ω

p

0 (E,M)′ from the space of currents inE × M , Ω

p

0 (E × M)′ discussed previously. Functional cur-rents simply augment usual currents with values of signal ateach point. The local geometry is still the one of the geomet-rical shape represented by an element of ΛpE as opposed tothe product current setting that models the geometry of thelifted functional shape in E × M , requiring the higher di-mensional space of p-vectors Λp(E × M). Now, just as onecan establish a correspondence between shapes and currents,to any functional shape we now associate a fcurrent.

Proposition 1 Let (X,f ) be a functional shape, with X

an oriented submanifold of dimension d and of finite vol-ume and f a measurable function from X to M . For allω ∈ Ωd

0 (E,M), x → ω(x,f (x)) can be integrated along X.We set:

C(X,f )(ω) :=∫

X

ω(x,f (x)). (4)

Then C(X,f ) ∈ Ωd0 (E,M)′ and therefore (X,f ) → C(X,f )

associates, to any functional shape, a functional current.

To be more explicit, recall that the integral in (4) is sim-ply defined through local parametrization with a given par-tition of the unity of the submanifold X. If F : U → E is aparametrization of X with U an open subset of R

d , then

C(X,f )(ω)

=∫

(x1,...,xd )∈U

ω(F(x1,...,xd ),f ◦F(x1,...,xd ))

(∂F

∂x1∧ · · ·

∧ ∂F

∂xd

)dx1 · · ·dxd.

Note also, although we did not state it explicitly, that the pre-vious proposition could include submanifolds with bound-ary in the exact same way since the boundary is of zeroHausdorff measure on the submanifold. Of course, like forregular currents, the previous correspondence between func-tional shapes and functional currents is not surjective. Forinstance, a sum of functional currents of the form C(X,f ) donot generally derive from a functional shape. In the func-tional current framework, Dirac masses are naturally gen-eralized by elementary functional currents or Dirac fcur-rents δ

ξ

(x,m) for x ∈ X, m ∈ M and ξ ∈ ΛpE such that

δξ

(x,m)(ω) = ω(x,m)(ξ). In the same way as explained in theprevious section, one can give a discretized version of func-tional currents associated to (X,f ) when a mesh is definedon X. C(X,f ) is then approximated by a sum of punctualcurrents:

C(X,f ) ≈∑

k=1..N

δξk

(xk,mk)(5)

In the particular case of a triangulated surface, the dis-cretized version of the fcurrent can be simply obtained asexplained for classical currents and by adding the ‘interpo-lated value’ of signal at each center point of triangles (or fora general signal manifold, the Frechet mean in M). From theprevious equation, we can observe that functional currentshave a very simple interpretation, which consists in attach-ing values of the signal f to the usual representation of X asa d-current. At this stage, we could also point out an alter-native way to define fcurrents by considering them as tensorproducts of d-currents in E and 0-current (i.e. measure) inM , following for instance [4].

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3.2 Diffeomorphic Transport of Fcurrents

What about diffeomorphic transport of functional shapesand currents? This question cannot be addressed as simplyas in the classical current setting if we want to remain com-pletely general. The reason is that, there is not a unique waya deformation can act on a functional shape, it depends onthe nature of the signal defined on the manifold. In the mostsimple case where the signal values are not directly corre-lated to geometry (for instance an activation map on a cor-tical surface), the natural way to deform a functional shape(X,f ) by a diffeomorphism φ is to transport the geome-try of the shape with the values of the signal unchanged.Therefore, the image of (X,f ) would be (φ(X),f ◦ φ−1).But imagine now that f is a tangent vector field on X.A diffeomorphism φ, by transporting the geometrical sup-port also has to act on the signal through its differential inorder to have a tangent vector field on the image shape. Inthis case, the image of (X,f ) is (φ(X),g) where, for ally ∈ φ(X), g(y) = dφ−1(y)φ(f ◦φ−1(y)). In other cases, forinstance a tensor field defined on a manifold, the expressionof the transport would differ again. In all cases though, whatwe have is a left group action of diffeomorphisms of E onthe set of considered functional shapes.

Thus, to remain general, suppose that a certain class offunctional shapes together with such a group action arefixed, we will note φ.(X,f ) the action of φ ∈ Diff(E) ona functional shape (X,f ). Then,

Definition 2 We call a deformation model on the spaceof functional currents an action of the group of diffeomor-phisms of E on Ωd

0 (E,M)′ which is such that for any func-tional shape (X,f ) and any diffeomorphism φ, if φ∗ standsfor the action on fcurrents, the following property holds:

[φ∗C(X,f )](ω) = Cφ.(X,f )(ω) (6)

for all ω ∈ Ωd0 (E,M).

Note the difference with (2): the action of a diffeomor-phism on usual currents is always the simple push forwardoperation which is automatically compatible with the trans-port of a shape. Here, it is necessary to adapt the definition ofthe action on fcurrents to be compatible with a given actionon functional shapes by satisfying (6).

In practical applications, this is usually not difficult. Inthe first case mentioned above, the action of φ ∈ Diff(E) ona functional current C can be derived in a very similar wayto the case of usual currents:⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

φ∗C(ω).= C(φ∗ω), ∀ω ∈ Ωd

0 (E,M)

where for all ∀x ∈ E, m ∈ M,

ξ = ξ1 ∧ · · · ∧ ξp ∈ ΛpE

(φ∗ω)(x,m)(ξ).= ω(φ(x),m)(dxφ(ξ1) ∧ · · · ∧ dxφ(ξd)).

(7)

It can be easily checked from the previous equations thatfor all functional shapes (X,f ), we have φ∗C(X,f ) =C(φ(X),f ◦φ−1) as we expected under this model. Since we donot want to focus this paper specifically on fcurrents’ trans-port, the examples of matching that we will give in the lastsection are under the hypothesis of this model of transport,which is the simplest and will lead to a convenient gener-alization of matching algorithms on functional currents. Wecould go a step further and also introduce a contrast changeψ → ψ ◦ f for ψ ∈ Diff(M) so that we end up with a newaction of Diff(E) × Diff(M) on Ωd

0 (E,M) defined by((φ,ψ)∗ω

)(x,m)

(ξ)

.= ω(φ(x),ψ(m))

(dxφ(ξ1) ∧ · · · ∧ dxφ(ξd)

)(8)

and the corresponding action on the fcurrent (φ,ψ)∗C(ω).=

C((φ,ψ)∗ω) given by duality. For this we easily check that

(φ,ψ)∗(δξx,m

) = δdxφ(ξ1)∧···∧dxφ(ξd )

φ(x),ψ(m) . (9)

Note that it is not significantly more difficult to express andimplement the deformation model on functional currentsthat corresponds to other types of action, as for instance inthe case of tangent vector signals we mentioned earlier.

4 A Hilbert Space Structure on Functional Currents

In this section, we address the fundamental question of com-paring functional currents through an appropriate metric.For this purpose, we adapt the ideas of RKHS presentedbriefly for currents in the first part of the paper. This ap-proach allows us to view functional currents as elements ofa Hilbert space of functions, which opens the way to var-ious processing algorithms on functional shapes as will beillustrated in the next section.

4.1 Kernels on Fcurrent Spaces

As we have seen for currents, the theory of RKHS definesan inner product between currents through a certain ker-nel function satisfying some regularity and boundary con-ditions. Following the idea that functional p-currents canalso be considered as the tensor product of p-currents onE and 0-currents on M, we can generically define a kernelon E × M .

Proposition 2 Let Kg : E × E → L(ΛpE) be a positivekernel on the geometrical space E and Kf : M × M → R apositive kernel on the signal space M . We assume that bothkernels are continuous, bounded and vanishing at infinity.Then Kg ⊗ Kf defines a positive kernel from E × M onΛpE whose corresponding reproducing Hilbert space W iscontinuously embedded into Ω

p

0 (E,M). Consequently, ev-ery functional p-current belongs to W ′.

J Math Imaging Vis

Proof This relies essentially on classical properties of ker-nels. From the conditions on both kernels, we know that Kg

and Kf correspond to two RKHS’s Wg and Wf that are re-spectively embedded into Ω

p

0 (E) and Ω00 (M) (cf. [11]). It

is a classical result in RKHS theory that K := Kg ⊗ Kf de-fines a positive kernel. Moreover, since Kf is real-valued,we have the following explicit expression of K :

K((x1,m1), (x2,m2)

) = Kf (m1,m2).Kg(x1, x2).

The kernel K corresponds to a unique RKHS W that is thecompletion of the vector space spanned by all the functions{Kf (.,m).Kg(., x)ξ} for x ∈ E, m ∈ M , ξ ∈ ΛpE. Sincefunctions Kf (.,m) and Kg(., x) are both continuous andvanishing at infinity from, this holds for Kf (.,m).Kg(., x)ξ

as well, so that W is indeed embedded into Ωp

0 (E,M). Itonly remains to prove that the embedding is continuous,which reduces to dominate the uniform norm by ‖.‖W .

Let ω ∈ W . For all (x,m) ∈ E × M and ξ ∈ ΛpE suchthat |ξ | = 1, we have

|ω(x,m)(ξ)| = |δξ

(x,m)(ω)|. (10)

Since W is a RKHS, all δξ

(x,m) are continuous linear formson W . In addition, the Riesz representation theorem pro-vides an isometry KW : W ′ → W . Then:

⟨δξ1(x1,m1)

, δξ2(x2,m2)

⟩W ′

= ⟨KW

(δξ1(x1,m1)

),KW

(δξ2(x2,m2)

)⟩W

= ⟨Kf (.,m1)Kg(., x1)ξ1,Kf (.,m2)Kg(., x2)ξ1

⟩W

= Kf (m1,m2).ξT2 Kg(x1, x2)ξ1 (11)

Now, back to (10), we have:

∣∣ω(x,m)(ξ)∣∣ ≤ ∥∥δξ

x,m

∥∥W ′ ‖ω‖W

≤√

Kf (m,m).ξT Kg(x, x)ξ ‖ω‖W

Since we assume that m → Kf (m,m) and x → Kg(x, x)

are bounded we deduce that√

Kf (m,m).ξT Kg(x, x)ξ isbounded with respect to x, m and ξ with |ξ | = 1. Hence,by taking the supremum in the previous equation, we finallyget

‖ω‖∞ ≤ C‖ω‖W

which means that the embedding is continuous. By duality,we get that every functional current is an element of W ′.Note that the dual application is not necessarily injective un-less W is dense in Ω

p

0 (E,M), which is the case in particu-lar if both Wg and Wf are respectively dense in Ω

p

0 (E) andΩ0

0 (M). �

In other words, a quite natural (but not unique) way tobuild kernels for functional currents is to make the tensorproduct of kernels defined separately in the geometrical do-main (p-currents in E) and in the signal domain (0-currentsin M). As we see, everything eventually relies on the specifi-cation of kernels on E and M . Note that non product kernelscould also be used but the product situation corresponds toan independence assumption between shape and functionalinformation with is natural when modeling the residual dif-ference between two functional shapes as noise. Moreover,the use of product kernels leads to simpler and faster com-putational schemes.

Kernels on vector spaces have been widely studied inthe past and obviously do not raise any additional diffi-culty in our approach compared to usual current settings.Among others, classical examples of kernels on a vec-tor space E taking values in another vector space H areprovided by radial scalar kernels defined for x, y ∈ E byK(x,y) = k(|x − y|).IdH where k is a function defined onR+ and vanishing at infinity. This family of kernels is theonly one that induces a RKHS norm invariant for affineisometries. The most popular is the Gaussian kernel defined

by K(x,y) = exp(−|x−y|2σ 2 )IdH , σ being a scale parameter

that can be interpreted as a range of interactions betweenpoints.

The definition of a kernel on a general manifold M isoften a more involved issue as we already mentioned inSect. 2.2. However, it is important to note that, in our set-ting of functional currents, this issue is drastically simplifiedbecause we only need to define real-valued kernels on M .This is contrasting with the idea of product space currents ofSect. 2.2, which requires the definition of kernels living inthe exterior product of the fiber bundle of M . For instance, ifM is a submanifold of a certain vector space, obtaining real-valued kernels on M becomes straightforward by restrictionto M of kernels defined on the ambient vector space.

4.2 Convergence and Control Results on the RKHS Norm

We are now going to explore some properties of the RKHSnorm on fcurrents and show the theoretical benefits of ourapproach with respect to the original problem we set out tohandle in this article.

Suppose, under the same hypotheses as the previous sec-tion, that two kernels Kg and Kf are given respectively onspace E and manifold M , providing two RKHS Wg and Wf .By a simple triangular inequality, we get for any x1, x2 ∈ E,any ξ1, ξ2 ∈ ΛpE and any m1, m2 ∈ M

∥∥δξ2(x2,m2)

− δξ1(x1,m1)

∥∥W ′ ≤ ‖δm1‖W ′

f

∥∥δξ2x2

− δξ1x1

∥∥W ′

g

+ ∥∥δξ2x2

∥∥W ′

g‖δm2 − δm1‖W ′

f. (12)

J Math Imaging Vis

Since both kernels Kf and Kg are assumed to be bounded

as in Proposition 2, ‖δm1‖W ′f

and ‖δξ2x2‖W ′

gare uniformly

bounded so that eventually∥∥δ

ξ2(x2,m2)

− δξ1(x1,m1)

∥∥W ′ ≤ Cte

(∥∥δξ2x2

− δξ1x1

∥∥W ′

g

+ ‖δm2 − δm1‖W ′f

).

Therefore, the RKHS distance between punctual fcurrents isdominated both with respect to the variation of their geomet-rical parts and of their functional values. This is the generalidea we will formulate in a more precise way with the twofollowing propositions. We denote by dM the geodesic dis-tance induced on M by its Riemannian structure. The nextproposition examines the case where the geometrical sup-port is a fixed submanifold X and shows that the variationof the W ′-norm is then dominated by the L1 norm on X.

Proposition 3 Let X be a d-dimensional submanifold of E

of finite volume and f1 : X → M and f2 : X → M two mea-surable functions defined on the submanifold X taking valuein M . We assume that Wf is continuously embedded intoC1

0(M,R). Then, there exists a constant β such that:

‖C(X,f1) − C(X,f2)‖W ′ ≤ β

∫

X

dM

(f1(x), f2(x)

)dσ(x)

where σ is the uniform measure on X.

Proof We recall the definition C(X,f ) = ∫X

ω(x,f (x)). Wewill first restrict the proof to the case where X admits aparametrization given by a function G : U → E where U

is an open subset of Rd . The general result follows by the

use of an appropriate partition of the unit on X. Denotingξ(u)

.= ∂G∂u1

(u) ∧ · · · ∧ ∂G∂ud

(u) for u = (u1, . . . , ud) ∈ U , weget

C(X,f )(ω) =∫

u∈U

ω(G(u),f ◦G(u))

(ξ(u)

)du

Now, for g1.= f1 ◦G and g2

.= f2 ◦G, we have by triangularinequality on ‖.‖W ′ :

‖C(X,f1) − C(X,f2)‖W ′

≤∫

U

∥∥δξ(u)

(G(u),g1(u)) − δξ(u)

(G(u),g2(u))

∥∥W ′du. (13)

From (12), ‖δξ(u)

(G(u),g1(u)) − δξ(u)

(G(u),g2(u))‖W ′ ≤ ‖δξ(u)

G(u)‖W ′g×

‖δg2(u) − δg1(u)‖W ′f

. Now, for any m1, m2 ∈ M and h ∈ Wf

we have∣∣(δm1 − δm2)(h)

∣∣ = |h(m1) − h(m2)|≤ ‖Dh‖∞dM(m1,m2)

≤ Cte‖h‖WfdM(m1,m2)

the last inequality resulting from the continuous embeddingWf ↪→ C1

0(M,R). Therefore we get

‖δg2(u) − δg1(u)‖W ′f

≤ CtedM

(g1(u), g2(u)

).

Moreover, since we assume that the kernel Kg is bounded,

we also have ‖δξ(u)

G(u)‖W ′g

≤ Cte |ξ(u)|. Back to (13), we getfrom the previous derivations the existence of a constantβ > 0 such that:

‖C(X,f1) − C(X,f2)‖W ′ ≤ β

∫

U

dM

(g1(u), g2(u)

)|ξ(u)|du

which precisely proves the stated result. �

A straightforward consequence of Proposition 3 and thedominated convergence theorem is that if fn is a sequenceof functions on X that converges pointwisely to a functionf , then C(X,fn) → C(X,f ). In other words, pointwise con-vergence of the signal implies convergence in terms of fcur-rents.

Following the same kind of reasoning we eventually givea local bound of the RKHS distance between a functionalshape and the same shape deformed through small diffeo-morphisms both in geometry and signal. As it is now com-mon in computational anatomy, we consider deformationsmodeled as flows between 0 and 1 of differential equationsgiven through time varying vector fields. In the Appendix,we recall the basic definitions of this approach and someresults needed for the following. Let u(t, x) (resp. v(t,m))be a smooth time dependent vector fields on the geometricalspace E (resp. on the signal space M) and let φ (resp. ψ ) thesolution at time 1 of the flow of the ODE y′ = u(t, y) (resp.y′ = v(t, y)). On these spaces of vector fields, we define thenorms:

‖u‖χ1 =∫ 1

0

∣∣u(t, .)∣∣1,∞dt, ‖v‖χ0 =

∫ 1

0

∣∣v(t, .)∣∣0,∞dt

where |u(t, .)|1,∞ = supx |u(t, x)| + ∑i supx | ∂u

∂xi(t, x)| and

‖v(t, .)‖0,∞ = supm |v(t,m)|.

Proposition 4 Let X be a submanifold of E of finite vol-ume and f : X → M a measurable function. Assume thatWg and Wf are continuously embedded respectively intoC1

0(E,ΛdE∗) and C10(M,R). There exists a universal con-

stant γ > 0 such that, if ‖u‖χ1 and ‖v‖χ0 are sufficientlysmall (which means that deformations are ‘close’ to iden-tity), then:

‖C(X,f ) −C(φ(X),ψ◦f ◦φ−1)‖W ′ ≤ γ Vol(X)(‖u‖χ1 +‖v‖χ0

)

Proof The full proof of Proposition 4 relies mostly on a fewcontrols which are provided in the Appendix. For the entire

J Math Imaging Vis

proof, we shall use the notation Cte to denote the succes-sive different ‘universal’ constants (i.e not dependant on theshape (X,f ) and the deformations φ and ψ ). Given againa local parametrization of X, G : U → X, then, similarly tothe previous proposition and using same notation, we have:

‖C(X,f ) − C(φ(X),ψ◦f ◦φ−1)‖W ′

≤∫

U

∥∥δ

ξ(u)

(G(u),f ◦G(u)) − δξ̃(u)

(φ◦G(u),ψ◦f ◦G(u))

∥∥

W ′du (14)

where for the volume element ξ(u) = ξ1(u) ∧ · · · ∧ ξd(u),ξ̃ (u) is the transported volume element by φ equal to ξ̃ (u) =dφx(ξ1(u)) ∧ · · · ∧ dφx(ξd(u)). From (12) we get

∥∥δξ(x)

(x,f (x) − δξ̃(x)

(φ(x),ψ◦f (x)))∥∥

W ′

≤ ‖δψ◦f (x)‖W ′f

∥∥δξ̃(x)

φ(x) − δξ(x)x

∥∥W ′

g

+ ∥∥δξ(x)x

∥∥W ′

g‖δψ◦f (x) − δf (x)‖W ′

f.

and using ‖δξ(x)x ‖W ′

g≤ Cte |ξ(x)| and ‖δψ◦f (x)−δf (x)‖W ′

f≤

CtedM(ψ ◦ f (x), f (x)) we get

∥∥δξ(x)x

∥∥W ′

g‖δψ◦f (x) − δf (x)‖W ′

f

≤ Cte∣∣ξ(x)

∣∣dM

(ψ ◦ f (x), f (x)

)

≤ Cte∣∣ξ(x)

∣∣‖v‖χ0 (15)

In a similar way, we know that ‖δψ◦f (x)‖W ′f

≤ Cte. More-over:

∥∥δξ(x)x − δ

ξ̃(x)

φ(x)

∥∥W ′

g

≤ Cte(∣∣ξ(x)

∣∣‖ Id−φ‖∞ + ∣∣ξ̃ (x) − ξ(x)∣∣)

≤ Cte∣∣ξ(x)

∣∣‖u‖χ1

the last inequality being obtained thanks to Theorem 3 andCorollary 1 of the Appendix with s = 0 and t = 1. This leadsto:

‖δψ◦f (x)‖W ′f

∥∥δξ(x)x − δ

ξ̃(x)

φ(x)

∥∥W ′

g≤ Cte

∣∣ξ(x)∣∣‖u‖1,∞. (16)

Plugging (15) and (16) in (14), we finally get:

‖C(X,f ) − C(φ(X),ψ◦f ◦φ−1)‖W ′

≤ Cte(‖u‖χ1 + ‖v‖χ0

)∫

U

∣∣ξ(u)∣∣du

which concludes the proof since∫U

|ξ(u)|du = Vol(X). �

This property shows that the RKHS norm is Lipschitzregular with respect to deformations of the functional shape(both in its geometry and its signal). More specifically, it

Fig. 4 Comparison of the fcurrent’s norm and the Lp norm on a fixedgeometrical support: example of crenellations on the unit circle

is not hard to see that C(φ(X),ψ◦f ◦φ−1) = (φ,ψ)∗C(X,f ) forthe action given by (8) and (9) and to extend the proof of theprevious proposition to a more general situation of a fcurrentC ∈ W ′ having finite “mass norm” M(C) where M(C)

.=supω∈W,‖ω‖∞≤1 C(ω) is the proper extension of the previousfinite volume condition. Then we get∥∥(φ,ψ)∗C − C

∥∥W ′ ≤ γM(C)

(‖u‖χ1 + ‖v‖χ0

). (17)

where γ is a universal constant.This result also provides an answer to whether there is

a reversed domination in Proposition 3 for two functionalshapes that have the same geometrical support. Indeed, con-sider a particular case where ψ = Id and φ is a small de-formation that leaves X globally invariant (φ(X) = X). Wewish to compare the initial functional shape (X,f ) with thedeformed one (φ(X),f ◦ φ−1) = (X,f ◦ φ−1). By Propo-sition 4, we know that, for any function f , the fcurrent’sdistance remains small if the deformation φ is small. It is nolonger true if we compute instead

∫X

|f −f ◦φ−1|p , the Lp

distance on X (0 < p ≤ ∞). This is easily seen if we choosefor X the unit circle S

1 and consider crenelated signals as inFig. 4. Introducing the operator τdθ that acts on functionalshapes by rotation of an angle dθ , we see indeed that:

supf ∈L

p(S1),‖f ‖Lp ≤1

∫

S1

∣∣f − f ◦ τ−1

dθ

∣∣p = 1

whereas, according to Proposition 4

supf ∈L

p(S1),‖f ‖Lp ≤1

‖C(X,f ) − C(X,f ◦τ−1

dθ )‖W ′ = O(dθ).

This eventually gives the answer to the previous question:W ′ norm and Lp norm on a fixed geometrical support arenot equivalent in general. This simple example also revealsthe limits of some previous approaches for functional shapecomparison that rely on estimating correspondence maps be-tween the geometrical supports before comparing signals.

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RKHS norms on fcurrents successfully avoid such issues be-cause shape and function are compared through one singlenorm that has enough spatial regularity to compare func-tional values at points in a certain neighborhood.

To sum up this section, we have proved two importantregularity results of RKHS norms on functional currents:regularity with respect to L1 perturbations of the signalat fixed geometrical support (Proposition 3) and regularitywith respect to diffeomorphic deformations of the geomet-rical support (Proposition 4). This implies that we can con-sider variations of the functional shapes both in their supportand their function with respect to one common Hilbert norm.A last point, that was already mentioned and that shall be il-lustrated on examples, is the robustness of this metric withrespect to disconnections or small topological changes. Allthese theoretical arguments justify its use as a data attach-ment metric for registration algorithms, which is the subjectof Sect. 5.2.

5 Processing Functional Shapes with Fcurrents: TwoExamples

We would like to illustrate now how the concept of func-tional currents introduced above offers a genuine solutionto the simultaneous processing of the geometric and sig-nal information of any functional shape. We have explainedhow functional currents can be equipped with a Hilbertiannorm mixing geometrical and functional content of func-tional shapes and how this norm has nice properties with re-spect to geometrical and functional perturbations. It is moreor less clear that the embedding in this convenient Hilbertsetting opens the way for various processing algorithms thatwill be developed in the near future. Since the purpose ofthis paper is to stay focused on the theoretical exposition offcurrents, we will not try to develop a full range of applica-tions but will briefly present two illustrative applications inorder to shed light on the potential of the proposed frame-work. The first application illustrates the full potential of theHilbertian structure with the design of redundancy reductionor compression algorithms for functional shape representa-tions through matching pursuit schemes on functional cur-rents. The second one, closer to the core engine of computa-tional anatomy, is the design of a large deformation match-ing algorithm for the simultaneous geometric and functionalregistration of functional shapes under diffeomorphic trans-port.

5.1 A Compression Algorithm for Functional CurrentRepresentations

Let us start with the issue of the redundancy of fcurrentrepresentations. If we consider for instance a segment in

2D space with constant signal, the discretization in punc-tual fcurrents given by (5) will provide a representation witha number of elements that corresponds to the initial sam-pling of the curve. Generally, this representation could beclearly reduced since, for such a simple functional shape,only a few terms should capture most of the shape. How-ever, the quality of the approximation needs to be quantifiedin a meaningful way, especially when the functional part isalso involved, through an appropriate norm for which wehave a natural candidate given by the Hilbert structure. Thisissue of redundancy reduction or compression is importantfor instance when making means of currents because with-out further treatment, the number of Dirac currents involvedin the representation of the mean would increase dramati-cally. This is even more important when considering higherorder statistics for the estimation of noise or texture mod-els around a mean functional shape possibly coupled witha deformation model learned from a set of inexact geodesicmatchings, as provided for instance by the matching algo-rithm of Sect. 5.2. In the following, we only provide a gen-eral overview of the algorithm and a few numerical resultsto show the functional current behaviors. The details of nu-merical optimization that may deserve a more in depth studyare beyond the scope of the present paper.

As we have said, the problem of redundancy reductionor compression is deeply simplified thanks to the Hilbertspace structure that has been defined on functional currentsin the previous section. Indeed, classical matching-pursuitalgorithms in general Hilbert spaces have already been stud-ied by Mallat and Zhang in [16] and later adapted to cur-rents in [7]. We can proceed in a similar way for func-tional currents. Consider again a discretized fcurrent C =∑

i=1..N δξi

(xi ,mi)∈ W ′. N , the number of momenta, is auto-

matically given by the mesh on the submanifold (point sam-pling for curves, triangulation for surfaces, . . .). This sub-manifold might have some very regular regions with low ge-ometrical and functional variations, which results in a veryredundant representation by fcurrents due to the fact thatmany adjacent nodes present the same local geometry andsignal. The goal of matching-pursuit is to find a more ap-propriate and reduced representation of C in terms of ele-mentary functional currents. Given a certain threshold ε > 0,we want to find Πn(C) such that C = Πn(C) + Rn(C) and‖Rn(C)‖W ′ � ε. Rn(C) will be called the residual of theapproximation. Somehow, this is linked to the problem offinding the best projection of C on a subspace of W ′. Thisproblem is however too time-consuming computationallyfor usual applications. Instead, matching pursuit is a greedyalgorithm that constructs a family of approximating vectorsstep by step. The result is a suboptimal fcurrent that approx-imates the functional current C with a residual whose en-ergy is below threshold. The algorithm basically proceedsas follows. We need to specify a ‘dictionary’ D of elements

J Math Imaging Vis

in W ′. In our case, we typically consider the set of all ele-mentary functional currents {δξ

(x,m)} with ξ a unit vector in

ΛdE. The first step of matching pursuit algorithm is to find

δξ ′

1(x′

1,m′1)

∈ D that is best correlated to C. In other words, we

try to maximize, with respect to x,m, ξ , the quantity:

⟨C,δ

ξ

(x,m)

⟩W ′ = ξT

(N∑

i=1

K((x,m), (xi,mi)

)ξi

)

(18)

Since ξ is taken among unit vectors, the problem is equiv-alent to maximizing ‖∑

i=1..N K((x,m), (xi,mi))ξi‖ =‖γ (x,m)‖ with respect to (x,m) and take ξ as the unit vec-tor in the direction of γ (x,m). We get a first approximationof C:

C = Π1(C) + R1(C).

The algorithm then applies the same procedure to the resid-

ual R1(C), which provides a second vector δξ ′

2(x′

2,m′2)

∈ D, and

a residual R2(C). The algorithm is stopped when the RKHSnorm of the residual decreases below the given threshold ε.

In most cases, it appears that the compression is bet-ter with the orthogonal version of the previous scheme, inwhich the family of vectors is orthonormalized at each step,in order to force the projection and the residual to be or-thogonal in W ′. The classical algorithm is based on a Gram-Schmidt orthonormalization at each step. In our case, it ispossible to obtain a similar result more efficiently by keep-ing the values of (x′

i ,m′i ) found during previous steps and

simply modify the vectors ξ ′i . This is done by imposing

the following orthogonality condition. Call (ek) the canoni-cal basis of the vector space ΛdE. If C = Πn(C) + Rn(C)

and Πn(C) = ∑i=1..n δ

αni

(x′i ,m

′i )

, we will add the orthogonality

constraint:

δek

(x′i ,m

′i )⊥Rn(C)

⇐⇒ ⟨C,δ

ek

(x′i ,m

′i )

⟩W ∗ = ⟨

Πn(C), δek

(x′i ,m

′i )

⟩W ∗

for all basis vectors ek and for all i ∈ {1, . . . , n}. It is thenstraightforward to check that these conditions are equivalentto the following system of linear equations to find the αn

i :

∀i ∈ {1, . . . , n},n∑

j=1

(K

((x′i ,m

′i

),(x′j ,m

′j

))αn

j

)k= γ

(x′i ,m

′i

)k

(19)

We can show that the norm of the residual Rn(C) mono-tonically decreases to zero as n → ∞. Hence the algorithmconverges and eventually when the residual goes below thegiven threshold at a certain step n, we obtain a compressedrepresentation of C with n orthogonal Dirac fcurrents (with

generally n � N , as we shall see on the coming exam-ples). At each step, the time-consuming part of the algorithmis mainly the computation of sums of kernels, which hasquadratic complexity with respect to the number of Diracsof the original current but can be sped up tremendously bymaking computations on a fixed grid with FFT, as intro-duced for currents in [6]. The same kind of numerical trickcan be performed with fcurrents but we will not elaborate onthat in this paper.

Here are now a few illustrative examples for real valueddata on curves or surfaces. We will always consider kernelson fcurrents that are the tensor product of a Gaussian kernelin R

3 of scale parameter λg with a real Gaussian kernel inthe signal space of scale parameter λf . In Figs. 5 and 6, weemphasize the influence of both kernel sizes on the compres-sion factor as well as on the precision of the functional val-ues of the compressed shape. The bigger the parameter λg ,the coarser the scale of representation is and fewer punctualfcurrents are therefore needed to compress shapes but moresmaller features are lost. In Fig. 7, we focus more preciselyon the compression’s behavior when computing matching-pursuit on a simulated fiber bundle of 2D curves carryingdifferent signals. The scale λg is the same for both figuresbut we show the results of matching-pursuit for two radicallydifferent values of λf . In both cases, matching pursuit pro-vides an accurate approximation of the mean (accordinglyto the kernel norm) with a very limited number of Diracscompared to the original sampling. However, note the im-portant influence of λf : taking a larger value for this param-eter means that the radius of averaging for the signal part ishigher.

In conclusion, these first examples of functional shapeprocessing were meant to highlight that the combination ofthe fcurrent’s representation with the use of RKHS metricsprovides an easy solution to address the issue of redundancyand compression. The method provides important compres-sion factors and enables scale analysis on geometry and sig-nal through the kernel parameters λg and λf .

5.2 A Large Deformation Matching Algorithm forFunctional Shapes

As a second illustrative example, we would like to brieflyhighlight the potential of fcurrent representations in the con-text of computational anatomy and more generally in thecontext of shape spaces. It is clear that many importantanatomical manifolds are coming with interesting data in-dexed by them (for instance cortical thickness in anatomicalMRI or activation maps in fMRI scans among many pos-sibilities) and are perfect examples of functional shapes asdefined in this paper. The statistical analysis of a popula-tion of such functional shapes is however a real challengesince the relevant information in a functional shape may be

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Fig. 5 Matching pursuit on a “painted” bunny with different param-eters λg and λf . Geometrically, the surface has 0.16 × 0.22 × 0.12extension in the 3D space and the signal goes from value zero (blue)to one (red). The original sampling of the fcurrent representation has69451 Diracs and we choose a stopping criterion for the algorithm of

ε = 5%. The resulting Dirac fcurrents δξk

(xk,mk)are here represented as

colored vectors accordingly to the functional values mk . Vectors are allof same length covering an area proportional to the norm of ξk . Noticethat the sampling increases as λg is smaller while the vector’s colorsare more accurate when λf is smaller (Color figure online)

Fig. 6 Close up on two of theprevious results

buried in two sources: the pure geometrical shape defined bythe manifold itself and the signal information spread on thesupport. Moreover, the geometrical and functional parts aremore likely intertwined with each other.

When only pure geometrical shapes are considered, theconcept of shape space equipped with a Riemannian metricoffers proper tools for the local analysis of a population ofshapes seen as a distribution of points in a shape space. Inparticular, the use of Riemannian exponential map arounda template conveys an efficient linearization of the shapespace to describe the differences between shapes. However,observed shapes are contaminated by many errors comingfrom various pre-processing pipelines deriving the extrac-tion of shapes from raw data and the shape space is notsufficient to accommodate all observed shapes. Moreover,

and more fundamentally, shapes in a shape space are idealexemplars of real shapes with controlled complexity, nec-essary to properly address estimation issues from a lim-ited sample. Consequently a discrepancy measure or a noisemodel is needed to link ideal shapes in shape space with ob-served shapes. A coherent solution is provided by the cur-rent framework: indeed observed shapes can be representedas a vector in a Hilbert space of currents in which a Rie-mannian shape space N of ideal shapes is also embedded:N ↪→ W ′ so that a population of observed shapes (Si) canbe represented as a sum Si = ni + ri where ni ∈ N and theresidual noise ri ∈ W ′. Introducing a template n0 and usingthe linearization provided around n0 by the Riemannian ex-ponential map Expn0

we can write for any observed shape S:

S = Expn0(u) + r (20)

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Fig. 7 Matching pursuit on a 2D fiber bundle, each fiber carrying onevalue of signal represented by the color. On top, the initial object con-sisting of 300 fibers. Below, we show two results of matching-pursuit

with the same λg but two different values for λf : λf = 200 for the leftfigure, λf = 20 for the right one (Color figure online)

where (u, r) ∈ Tn0 N × W ′. Note that the (u, r) lie in a vec-tor space and t → nt

.= Expn0(tu) is a geodesic on N . Intro-

ducing the metric ‖‖n0 at n0 and the metric ‖‖W ′ on W ′, wecan estimate an optimal decomposition (20) (u(S), r(S)) ofan observed shape S by the minimization of ‖u‖2

n0+‖r‖2

W ′ .When pure geometrical shapes are no longer involved but

functional shapes instead, the previous setting breaks downwith usual currents but is still valid if W ′ is replaced by aRKHS space of fcurrents. The space N itself can be definedas N = {g · n0 | g ∈ G} i.e. the orbit of a template n0 underthe action of a group of deformations G. The diffeomorphictransport discussed in Sect. 3.2 offers several examples ofsuch action. We will consider the simple situation of func-tional shapes with real valued signals (E = R

d, M = R)where the action is given by (7) even if more complex ac-tions as defined by (8) and (9) could be used. In this set-ting, the Riemannian structure on N is inherited from theoptimization of the kinetic energy

∫ 10 ‖vt‖2

V dt on a time-

dependent Eulerian velocity fields (t, x) → v(t, x) of thetrajectory t → φt · n0 where φt is the flow of the ODE y′ =v(t, y) starting from the identity. The overall framework hasbeen popularized as the large deformation diffeomorphicmapping setting (LDDMM). The space V is a RKHS spaceof vector fields, here given by an isotropic Gaussian kernel,generating a right invariant distance on the group G of dif-feomorphisms generated by flows of kinetic energy. This in-duces, by Riemannian submersion, a Riemannian structureon N (see [19, 25] for a more extended presentation of thisgeometrical setting). In particular, if n0 = C(X,f ) with X isa smooth manifold with finite volume or if n0 = C is a moregeneral element of W ′ such that M(C) < ∞ (for instancea countable family of (Xi, fi)’s with

∑vol(Xi) < ∞) then

the continuity result given by Proposition 4 or (17) gives thecontinuous embedding N ↪→ W ′.

Obviously the RKHS norm plays the role of a data at-tachment distance and could be coupled with other matching

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Fig. 8 Example of registrationof two functional curves (topleft) with binary signal (blue iszero and red is one). On topright, we show the classicalmatching with currents on thepurely geometrical curves. Onbottom left, the same curves arematched with our extension ofLDDMM to functional currents.In both cases, the deformedcurve fits closely to the targetone but note the difference ofthe deformation field for thefunctional current’s approach.Finally, on the right, we showthe result of matching we obtainagain with fcurrents’ LDDMMbut with a big value of λf

compared to the signal, in whichcase the matching is nearlysimilar to the current matching(Color figure online)

approaches (even if we think that the previous setting is par-ticularly attractive for further statistical studies). The readernot familiar with the above geodesic setting could replacethe mapping u → n1(u) = Expn0

(u) by any other mappingu → n1(u).

With distances provided by the RKHS norms on fcur-rents, it is then possible to extend LDDMM algorithm tothe registration of functional shapes. Leaving the technicaldetails of implementation to a future paper, we just presentsome results of the method on simple examples in order toemphasize the various positive features of the approach. Aswe can expect, the resulting matching is driven both by thegeometry of the shapes and by the functional values theycarry according to the scales of both kernels, which we firstshow on the example of Fig. 8. If we compare it now tothe colored currents of Sect. 2.2, we see that since func-tional currents clearly separate signal and geometry, we nolonger have the same drawbacks: in the colored surfacesof Fig. 2, we have shown on the right the matching resultwith the functional current approach. In addition, the func-tional current representation is totally robust both to point-wise outlying signal values and to missing connections be-tween points, which is clear from the definition of the RKHSnorm, because geometrically negligible subsets of the shapehave zero norm. It was not the case for instance with theproduct current idea (cf. Sect. 2.2) since variations of signalalso carry non-zero norm. This has important consequences

when trying to match curves with missing connections aswe show on the example of Fig. 9. In our view this makesfunctional currents better suited for the treatment of fiberbundles carrying signal, like the example given in Fig. 10.A second important point is that having a norm defined bythe tensor product of two kernels Kg and Kf with two in-dependent scales provides great flexibility for the matching,geometrically and functionally. The choice of a bigger pa-rameter λf for instance allows the matching of signal val-ues to be accurate only at a bigger scale, hence our methodcould still achieve matching under noisy or imprecise sig-nals on shapes. The counterpart is of course the presenceof an additional parameter that must be adapted to the data,based upon an a priori on the reliability of the signals wewant to match. Multi-scale approaches can also be builtby adding kernels at different scales in the spirit of [21]or [23].

Finally, in Fig. 11, we show an example of registrationbetween the inflated cortical surfaces of two different sub-jects. Retinotopic information (located in the visual area)was measured on these surfaces, linking the position of apoint in the visual field to the location of the correspondingactivation in the brain. Again, observe the difference of theestimated deformations when the matching is computed us-ing both geometry and function compared to a pure geomet-rical matching. With currents, the retinotopic information isspread by the deformation, making further functional com-

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Fig. 9 LDDMM matching of two planar curves with discontinuoussignals and topological disconnections. Each curve has two points offunctional discontinuity, one of them being also a disconnection of thegeometrical support (point b on the source and b′ on the target). Onthe right figure, the matching is performed by representing the coloredcurve as a current in the product space R

2 ×R as explained in Sect. 2.2.

On the left, with the functional currents’ representation. We see that theresulting deformation is much perturbed by the disconnections in thecase of product currents: the algorithm intends to match connected partof the source shape on a connected part of the target shape although itleads to a very unnatural matching (Color figure online)

Fig. 10 Example of matching on the case of a fiber bundle with signal.On the center figure, the source and target functional shapes. On theleft, the resulting matching with the deformed shape and the deforma-tion grid for the functional currents’ setting. On the right, the resultobtained by matching with currents. Note that even if the geometrical

shapes are well matched in both cases, the two deformations are notthe same. Functional currents elongate the dark blue part to fit withthe target shape’s colors whereas currents, by not taking signal intoaccount, shrinks it (Color figure online)

parison far less relevant compared to the functional currentresult.

6 Conclusion and Outlook

We have presented in this paper a way to formally gen-eralize the notion of currents for the purpose of integrat-ing functional shapes into a coherent and robust represen-tation. Functional currents provide a framework to model

geometrically-supported signals of nearly any type and reg-ularity while preserving the advantages of currents to modelthe geometry in computational anatomy. The second mainpoint of the study is the definition of an appropriate normbased on a RKHS structure, which provides a metric onfunctional shapes. This metric allows the comparison offunctional shapes without requiring any preliminary exactmatching between the geometrical supports. These normsalso allow for useful control properties as stated in Sect. 4.2

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Fig. 11 Registration betweentwo artificially smoothed brainsurfaces with retinotopicactivation maps on the visualcortex. We show the result ofmatching with the functionalcurrents’ setting versus a puregeometrical-based registrationwith currents. Data courtesy ofParietal team from Neurospin,CEA Saclay

and robustness to discontinuities of both support and signal.The resulting Hilbert structure on fcurrents opens the way toa very wide class of applications. Although numerical issuesthat appear when computing with currents were not detailedin this paper, we have presented two examples of processingalgorithms for functional shapes: a matching pursuit schemeto address fcurrent compression and averaging as well as anadaptation of LDDMM algorithm for diffeomorphic regis-tration of two functional shapes. Examples were providedessentially in the simplest cases of curves or surfaces withreal-valued signal but same methods could easily apply todifferent kinds of manifolds, signals and deformation mod-els.

In summary, the objective of this article is to set a pathfor the extension of the scope of traditional computationalanatomy to data structures we have called functional shapes.It is quite likely that this will yield improvements in reg-istration and statistical estimation of deformable templates,which constitutes the future step of our work. In the case ofbrain anatomy for instance, by taking into account the addi-tional information on the cortical surfaces provided by fMRImaps or estimations of cortical thickness.

Acknowledgements This work was made possible thanks to HM-TC (Hippocampus, Memory and Temporal Consciousness) grant fromthe ANR (Agence Nationale de la Recherche).

Appendix: Deformations’ Modelling in the LDDMMFramework

In this appendix, we remind a few intermediate results whichare necessary for the full proof of Proposition 4. Most of

them refer to deformations’ modelling and can be found ei-ther in [11] or [26] (Chap. 12).

Using notations of [26], for p ∈ N, let Cp

0 (Rn,Rn) be the

Banach space of p-times continuously differentiable vectorfields v on R

n such that v, dv, . . . , dpv vanish at infinity,which is equipped with the norm |v|p,∞ = ∑p

i=1 |div|∞.Now, let χp be the set of integrable function from the

segment [0,1] into Cp

0 (Rn,Rn). Any element of χp is a

time-varying vector field we will denote v(t, .), t ∈ [0,1].On χp we define the norm:

‖v‖χp =∫ 1

0

∣∣v(t, .)∣∣p,∞dt

Note that we have χp ⊂ χp−1 ⊂ · · · ⊂ χ0 and that if v ∈ χp ,‖v‖χ0 ≤ · · · ≤ ‖v‖χp .

For any v ∈ χ1, we consider the differential equationdydt

= v(t, y) with initial condition y(s) = x ∈ Rn at time

s ∈ [0,1[. We have:

Theorem 1 For all x ∈ Rn and s ∈ [0,1[, there exists a

unique solution on [0,1] of the differential equation dydt

=v(t, y) such that y(s) = x. We denote by φv

s,t (x) the value attime t of this solution. (t, x) → φv

s,t (x) defined on [0,1] ×R

n is called the flow of the differential equation.

In other words, the flow satisfies the following integralequation:

φvs,t (x) = x +

∫ t

s

v(r,φv

s,r (x))dr (21)

We then have:

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Theorem 2 For all v ∈ χ1 and all s, t ∈ [0,1], φvs,t is a C1-

diffeomorphism of Rn. In the special case where v = 0, φv

s,t

is the identity application.

From (21), using Gronwall inequality, it is easy to showthat:

Theorem 3 For all R > 0 there is a constant C(R) > 0 suchthat, for all v ∈ χ1 with ‖v‖χ1 � R:∥∥φv

s,t − Id∥∥∞ � C(R).‖v‖χ0 � C(R).‖v‖χ1

In a similar way, the same kind of control can be obtainedfor the differential of the flow as stated below:

Theorem 4 For all v ∈ χ1 and s, t ∈ [0,1], φvs,t is a C1

function whose differential satisfies the integral equation:

dxφvs,t = Id +

∫ t

s

dxv(r,φv

s,r (x))dr

Theorem 5 For all R > 0 there is a constant C(R) > 0 suchthat, for all v ∈ χ1 with ‖v‖χ1 � R:

∥∥dφv

s,t − Id∥∥∞ � C(R).‖v‖χ1

This last result, together with the multilinearity of exte-rior product and Jacobian, leads to the following corollary:

Corollary 1 For ‖v‖χ1 small enough, there exists constantsα > 0 and β > 0 such that for all x ∈ E:

∣∣Jacx

(φv

s,t

) − 1∣∣ ≤ α‖v‖χ1

∥∥dxφvs,t (ξ1) ∧ · · · ∧ dxφ

vs,t (ξd) − ξ1 ∧ · · · ∧ ξd

∥∥

≤ β‖v‖χ1 .‖ξ1 ∧ · · · ∧ ξd‖

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Nicolas Charon is currently a PhDstudent at CMLA in ENS Cachan(France). In 2008, he received amasters’ degree in applied math-ematics, specialized in signal pro-cessing and statistical learning. Hisresearch areas are now focused onshape spaces, computational anato-my and geometric measure theory,with applications mainly in the fieldof medical imaging.

J Math Imaging Vis

Alain Trouvé is currently Professorat the Center of Mathematics andTheir Application at ENS Cachan,he did his PhD in Stochastic Opti-mization and Bayesian Image Anal-ysis under the supervision of RobertAzencott. His main research inter-ests are shape and object data analy-sis with a particular emphasis on theuse of Riemannian geometry andinfinite dimensional group actionsdriven by applications in computa-tional anatomy and medical imag-ing.