More recently though, an increasing number of data structures have emerged in computational anatomy that not only involve a geometrical shape but some signal attached on this shape, to which we give the general name of functional shapes
. The most basic example is, of course, classical images for which the geometrical support is simply a rectangle on which is given a ’grey level’ signal. In many cases however, the support can have a much more complex geometry like, for instance, the activation maps on surfaces of cortex obtained through fMRI scans. Signals can also include structures that are more sophisticated than simple real values : we could think of a vector field on a surface as well as tensor-valued signal that appear in DTI imaging. Such a diversity both in shape and signal makes it a particularly delicate issue to embed all geometrico-functional objects in one common framework. Despite several attempts to model them directly as currents, important limitations of currents were found in such problems, which we will develop in section 2. As a result, recent approaches have been rather investigating methods where shape and signal are treated separately instead of trying to define an attachment distance between geometrico-functional objects. This is the case for instance in where authors propose a registration algorithm for fMRI data in which is performed an anatomic matching followed by a second one based on the values of the signals. However, all these frameworks have two important drawbacks : they are first very specific to a certain type of dataset and they require an exact one to one correspondence between the two shapes in order to further compare functional values, whereas in many applications inexact matchings are far more appropriate.
The purpose of this paper is to describe and explore a new analytical setting to work on the most general problem of representation and comparison of geometrico-functional structures (compatible with any change of parametrisation of their geometrical supports) treated as elements of an embedding functional vector space, here a Reproducing Kernel Hilbert Space, on which many desirable operations can be performed.
Our new analytical setting shares some common features with the mathematical current setting that will be recalled briefly in section 2 but overcome its main limitations when dealing with functional shapes. The core idea, developed in section 3 is to augment usual currents with an extra component embedding the signal values by a natural tensor product leading to our definition of functional currents. We consider then various actions on functional currents by diffeomorphic transport in section 3 and shows in section 4 that kernel norms can provide a suitable Hilbertian structure on functional currents generalizing greatly what has been done for currents. We also show in what sense this representation and RKHS metric on functional currents is consistent with the idea of comparing functional shapes with respect to deformations between them, which makes it a good approach for defining attachment distances. The two main results on this topic are the control results of propositions 3 and 4. We then illustrate the potential of this new metric setting in section 5 on two different problems. The first illustration is the construction, via a matching pursuit algorithm, of redundancy reduction or compression algorithm of the representation of functional shapes by functional currents with few examples of compression on curves and surfaces with real-valued data. The second illustration is about the potential benefits of functional currents in the field of computational anatomy. In particular, we show a few basic results of diffeomorphic matching between functional shapes with our extension of large deformation diffeomorphic metric mapping (LDDMM) algorithm  to functional currents.
2. Currents in the modelling of shapes
2.1. A brief presentation of currents in computational anatomy
Currents were historically introduced as a generalization of distributions by L. Schwartz and then G. De Rham in . The theory was later on considerably developed and connected to geometric measure theory in great part by H. Federer . In the first place, these results found interesting applications in calculus of variations as well as differential equations. However, the use of currents in the field of computational anatomy is fairly more recent since it was considered for the first time in . In the following, we try to outline the minimum background of theory about currents needed to recall the link between shapes and currents.
First of all, we fix some notations. Let’s call a generic euclidean space of dimension . We will denote by the space of continuous -differential forms on that vanish at infinity. Every element of is then a continuous function such that for all , . Since we have the isomorphism , we can see both as a p-multilinear and alternated form on and as a linear form on the -dimensional space of -vectors in . For all the following, we will use the notation as the evaluation of a differential form at point and on the -vector . On can be defined an euclidean structure induced by the one of , which is such that if and are two simple -vectors, . The norm of a simple -vector is therefore the volume of the element. The space is then equipped with the infinite norm of bounded functions defined on . These notations adopted, we define the space of -currents on as the topological dual , i.e. the space of linear and continuous forms on . Note that in the special case where , the previous definition is exactly the one of usual distributions on that can be also seen as signed measures on . Simplest examples of currents are given by generalization of a Dirac mass : if and , is the current that associates to any its evaluation .
Now, the relationship between shapes and currents lies fundamentally in the fact that every d-dimensional and oriented sub-manifold of of finite volume can be represented by an element of . Indeed, we know from integration theory on manifolds (,) that any d-differential form of can be integrated along , which associates to a -current such that :
for all . The application is also injective. Equation (1) can be rewritten in a more explicit way if admits a parametrization given by a certain smooth immersion with an open subset of . Then,
It is a straightforward computation to check that the last expression is actually independent of the parametrization (as far as the orientation is conserved). In the general case, there always exists a partition of the unit adapted to the local charts of , so that could be expressed as a combination of such terms. The representation is fully geometric in the sense that it only depends on the manifold structure itself and not on the choice of a parametrization. Currents’ approach therefore allows to consider sub-manifolds of given dimension (curves, surfaces,…) as elements of a fixed functional vector space. This also gives a very flexible setting to manipulate shapes since addition, combination or averages become straightforward to define. On the other hand, spaces of currents contain a lot more than sub-manifolds because general currents do not usually derive from sub-manifolds (think for instance of a punctual current ). However, it encompasses in a unified approach a wide variety of geometrical objects as for instance sets of curves and surfaces which can be relevant in some anatomy problems.
In registration issues, a fundamental operation is the transport of objects by a diffeomorphism of the ambient space. If and , we define the transport of by as the classical push-forward operation denoted :
where is the usual pull-back of a differential form defined for all and by :
being the notation we use for the differential of the diffeomorphism at point . With this definition, it’s a straightforward proof to check that , which means that the -current associated to a submanifold transported by is the -current associated to the transported submanifold .
To complete this brief presentation of currents applied to computational anatomy, we still need to explain how the currents’ representation can be practically implemented and how computations can be made on them. This step consists mainly in approximating the integral in (1) into a discrete sum of punctual currents where are points in E and -vectors encode local elements of volume of the manifold . A manifold would be then stored as a list of momenta consisting of points’ coordinates and corresponding -vectors. However, the transition between and its approximation as a discrete current cannot usually be performed in a standard way. Computationally, a mesh on the sub-manifold is needed. Let’s examine the two most frequent cases of curves and surfaces. Let be a continuous curve in E given by a sampling of points . Starting from this approximation of as a polygonal line, we can associate the -current defined by :
with the center of segment and the vector . It can be proved easily that tends toward zero for all -form as , i.e. as the sampling gets more accurate (cf ). Same process can be applied to a triangulated surface S immersed in . We associate to each triangle of the mesh a punctual current with and . Since we have , the previous formal -vector can be identified to the usual wedge product of vectors in , that is the normal vector to the surface whose norm encodes the area of the triangle. Again, it can be shown that this approximated current gets closer and closer to the actual as the mesh is refined. Eventually, the surface is represented as a finite collection of points and normal vectors in the space E.
Finally, the question of building a metric on the space of currents should be addressed. There are several norms traditionally defined on such as the mass norm or the flat norm. However, those are either not easily computable in practice or unfitted to comparison between shapes (see  chap 1.5). A particularly nice framework to avoid both problems is to define a Hilbert space structure on currents through reproducing kernel Hilbert space (RKHS) theory. This approach consists in defining a vector kernel on E () and its associated RKHS . Under some assumptions on the kernel, it can be shown that the space of -currents is continuously embedded in the dual which is also a Hilbert space. Therefore, in applications, we generally consider instead of as our actual space of currents. For more details on the construction of RKHS on currents, we refer to  and . Since, in applications, manifold are represented by sums of punctual currents, it’s sufficient to be able to compute inner products between two punctual currents. RKHS framework precisely gives simple closed expressions of such products. Indeed, one can show that . Computation of distances between shapes then reduces to simple kernel calculus which can be performed efficiently for well-suited kernels either through fast Gauss transform schemes as in  or through convolutions on linearly spaced grids as explained in .
In summary, this succinct presentation was meant to stress two essential advantages of currents in shape representation. The first one being its flexibility due to the vector space structure and the wide range of geometrical objects that are comprehended without ever requiring any parametrization. The second important point is the fact that computations on currents are made very efficient by the use of kernels which makes them appropriate in various applications as simplification, registration or template estimation. All these elements motivate an extension of the framework of currents to incorporate functional shapes, which will be discussed thoroughly in all the following.
2.2. Functional shapes and the limitations of currents
We now consider, as in the previous section, a -dimensional sub-manifold of the -dimensional vector space but in addition, we assume that functional data is attached to every points of through a function defined on and taking its values in a differentiable manifold , the signal space. What we call a functional shape is then a couple of such objects. The natural question that arises is this : can we model such functional shapes in the framework of currents like purely geometrical shapes ? In the following, we are discussing two possible methods to address this question directly with usual currents and explain why both of them are not fully satisfying in the perspective of applications to computational anatomy.
First attempts to include signals supported geometrically in the currents’ representation were investigated in  with the idea of colored currents. This relies basically on the fact already mentioned that the set of -currents contains a wider variety of objects than -dimensional sub-manifolds like rectifiable sets or flat chains (cf ). In particular, weighted sub-manifolds can be considered as currents in the following very natural way : suppose that is a sub-manifold of of dimension and is a weight or equivalently a real signal at each point of such that is continuous, then we can associate to a -current in E :
Although this approach seems to be the most straightforward way to apply currents to functional shapes since we are still defining a -current in , it’s quite obvious that such a representation suffers from several important drawbacks. The first thing is the difficulty to generalize colored currents for signals that are not simply real-valued, particularly if the signal space is not a vector space (think for instance of the case of a signal consisting of directions in the 3D space, where is therefore the sphere ). The second point arises when the previous equation is discretized into Dirac currents, which leads to an expression of the form . We notice an ambiguity appearing between the signal and the volume element since for any , ; separating geometry from signal in the discretized version appears as a fundamental difficulty. In addition, the energy of Dirac terms are proportional to the value of the signal at the corresponding point which induces an asymmetry between low and high-valued signals. In this setting, areas having very small signals become negligible in terms of current, which is both not justified in general and can affect drastically the matching of colored currents. We show a simple illustration of this issue when matching two colored ellipsoids with this approach in figure 2. Finally, we could also mention some additional pitfalls resulting in that colored currents do not separate clearly geometry from signal. Most problematic is the fact that there is no flexibility to treat signals at different scale levels than geometry which can make the approach highly sensitive to noise.
Another possible and interesting way to represent a functional shape by a current is to view it as a shape in the product space . Somehow, it generalizes the idea of seeing a 2D image as a 3D surface. However, at our level of generality, it is not a completely straightforward process. If the signal function is assumed to be , the set inherits a structure of -dimensional manifold of . With a vector space, it results directly from the previous that can be represented as a -current in the product space, that is as an element of . For a general signal manifold though, we would need to extend our definitions of currents to the manifold case, which could be done (cf ) but the definition of kernels on such spaces would then become a much more involved issue in general compared to the vector space case. This difficulty set apart, there still are some important elements to point out. The first one is the increase of dimensionality of the approach because, while we are still considering manifold of dimension , the co-dimension is higher : the space of -vectors characterizing local geometry is now of dimension , with significant consequences from a computational point of view. From a more theoretical angle, we see that, in such an approach, geometrical support and signal play a symmetric role. In this representation, the modelled topology is no more the one of the original shape because we also take into account variations within the signal space. Wether this is a strength or a weakness is not obvious a priori and would highly depend on the kind of applications. What we can state is that this representation is not robust to topological changes of the shape : in practice, the connectivity between all points becomes crucial, what we illustrate on the simplest example of a plane curve carrying a real signal in figure 3. In the field of computational anatomy, the processing of data such as fiber bundles, where connections between points of the fibers are not always reliable, this would be a clear drawback. We shall illustrate these consequences from the point of view of diffeomorphic matching in the last section of the paper.
To sum up this section, we have investigated two direct ways to see a functional shape as a current. The colored currents’ setting, although being very close to the modelling of purely geometrical shapes, is to be discarded mainly because it mixes geometry and signal in an inconsistent way. As for the second idea of immersing the functional shape in a product space, we have explained its limits both from the difficulty of the practical implementation and from the lack of robustness with respect to topology of the geometrical support. These facts constitute our motivation to redefine a proper class of mathematical objects that would preserve the interest of currents while overcoming the previous drawbacks.
3. Definition and basic properties of functional currents
In this section, we propose an extension of the notion of currents to represent functional shapes. The new mathematical objects we introduce, we call ’functional currents’, are not usual currents strictly speaking, contrarily to the methods presented in section 2.2. They would rather derive from the very general concept of double current introduced originally by De Rham in . Here, we adapt it in a different way to fit with the applications we aim at in computational anatomy.
3.1. Functional -forms and functional currents
Like in the previous section, let be a functional shape, with a -dimensional sub-manifold of the -dimensional Euclidean space and a measurable application from to a signal space . In our framework, can be any Riemannian manifold. Most simple examples are provided by surfaces with real signal data like activation maps on cortex in fMRI imaging but the framework that we present here is made general enough to incorporate signals from very different natures : vector fields, tensor fields, grassmannians. We now define the space of functional currents again as the dual of a space of continuous forms :
We call a functional -form on an element of the space which will be denoted by hereafter. We consider the uniform norm on defined by : . A functional -current (or fcurrent in short) is defined as a continuous linear form on for the uniform norm. The space of functional -current will be therefore denoted .
Just as one can establish a correspondence between shapes and currents, to any functional shape we now associate a fcurrent.
Let be a functional shape, with an oriented sub-manifold of dimension and of finite volume and a measurable function from to . For all , can be integrated along . We set :
Then and therefore associates, to any functional shape, a functional current.
To be more explicit, recall that the integral in (4) is simply defined through local parametrization with a given partition of the unit of sub-manifold . If is a parametrization of with an open subset of , then
Note also, although we did not state it explicitly, that the previous proposition could include sub-manifolds with boundary in the exact same way since the boundary is of zero Hausdorff measure on the sub-manifold. Of course, like for regular currents, the previous correspondence between functional shapes and functional currents is not surjective. For instance, a sum of functional currents of the form do not generally derive from a functional shape. In the functional current framework, Dirac masses are naturally generalized by elementary functional currents or Dirac fcurrents for and such that . In the same way as explained in the previous part, one can give a discretized version of functional currents associated to when a mesh is defined on . is then approximated into a sum of punctual currents :
In the particular case of a triangulated surface, the discretized version of the fcurrent can be simply obtained as explained for classical currents by adding the interpolated value of signal at each center point of triangles. From the previous equation, we can observe that functional currents have a very simple interpretation. It consists in attaching values of the signalto the usual representation of as a -current. At this stage, we could also point out an alternative way to define fcurrents by considering them as tensor products of -currents in and 0-current (i.e. measure) in , following for instance .
3.2. Diffeomorphic transport of fcurrents
What about diffeomorphic transport of functional shapes and currents ? This question cannot be addressed as simply as for the classical current setting if we want to remain completely general. The reason is that, depending on the nature of the signal defined on the manifold, there is not a unique way a deformation can act on a functional shape. In the most simple case where the signal values are not directly correlated to geometry (for instance an activation map on a cortical surface), the natural way to deform a functional shape by a diffeomorphism is to transport the geometry of the shape with the values of the signal unchanged. Therefore, the image of would be . But imagine now that is a tangent vector field on . A diffeomorphism , by transporting the geometrical support also has to act on the signal through its differential in order to have a tangent vector field on the image shape. In this case, the image of is where, for all . In other cases, for instance a tensor field defined on a manifold, the expression of the transport would differ again. In all cases though, what we have is a left group action of diffeomorphisms of on the set of considered functional shapes.
Thus, to remain general, suppose that a certain class of functional shapes together with such a group action are fixed, we will note the action of on a functional shape . Then,
We call a deformation model on the space of functional currents an action of the group of diffeomorphisms of on which is such that for any functional shape and any diffeomorphism , if stands for the action on fcurrents, the following property holds :
for all .
Note the difference with (2) : the action of a diffeomorphism on usual currents is always the simple push forward operation which is automatically compatible with the transport of a shape. Here, it is necessary to adapt the definition of the action on fcurrents to be compatible with a given action on functional shapes by satisfying (6).
In practical applications, this is usually not a difficulty. In the first case mentioned above, the action of on a functional current can be derived in a very similar way to the case of usual currents :
It can be easily checked from the previous equations that for all functional shape , we have as we expected under this model. Since we do not want to focus this paper specifically on deformation, the examples of matching that we will give in the last section are under the hypothesis of this model of transport, which is the simplest and will lead to a convenient generalization of matching algorithms on functional currents. We could go a step further and introduce also a contrast change for so that we end up with a new action of on defined by
and the corresponding action on fcurrent given by duality for which we easily check that
Note that it is not significantly more difficult to express and implement the deformation model on functional currents that corresponds to other types of action, as for instance in the case of tangent vector signal we mentioned earlier.
4. A Hilbert space structure on functional currents
In this section, we address the fundamental question of comparing functional currents through an appropriate metric. For this purpose, we adapt the ideas of RKHS presented briefly for currents in the first part of the paper. This approach allows to view functional currents as elements of a Hilbert space of functions, which opens the way to various processing algorithms on functional shapes as will be illustrated in the next section.
4.1. Kernels on fcurrent spaces
As we have seen for currents, the theory of RKHS defines an inner product between currents through a certain kernel function satisfying some regularity and boundary conditions. Following the idea that functional -currents can be considered as well as tensor product of -currents on E and -currents on M, we can generically define a kernel on .
Let be a positive kernel on the geometrical space and a positive kernel on the signal space . We assume that both kernels are continuous, bounded and vanishing at infinity. Then defines a positive kernel from on whose corresponding reproducing Hilbert space is continuously embedded into . Consequently, every functional p-current belongs to .
This relies essentially on classical properties of kernels. From the conditions on both kernels, we know that to and correspond two RKHS and that are respectively embedded into and (cf ). It is a classical result in RKHS theory that defines a positive kernel. Moreover, since is real-valued, we have the following explicit expression of :
To the kernel corresponds a unique RKHS that is the completion of the vector space spanned by all the functions for . Since functions and are both continuous and vanishing at infinity from what we have said, it is also the case for so that is indeed embedded into . There only remains to prove that we have a continuous embedding, which reduces to dominate the uniform norm by .
Let . For all and such that , we have
Since is a RKHS, all are continuous linear forms on . In addition, Riesz representation theorem provides an isometry . Then :
Now, back to equation (10), we have :
Since we assume that and are bounded we deduce that is bounded with respect to , and with . Hence, by taking the supremum in the previous equation, we finally get
which precisely means that the embedding is continuous. By duality, we get that every functional current is an element of . Note that the dual application is not necessarily injective unless is dense in , which is the case in particular if both and are respectively dense in and . ∎
In other words, a quite natural (but not unique) way to build kernels for functional currents is to make the tensor product of kernels defined separately in the geometrical domain (-currents in ) and in the signal domain (-currents in ). As we see, everything eventually relies on the specification of kernels on and .
Kernels on vector spaces have been widely studied in the past and obviously do not arise any additional difficulty in our approach compared to usual current settings. Among others, classical examples of kernels on a vector space taking values in another vector space are provided by radial scalar kernels defined for by where is a function defined on and vanishing at infinity. This family of kernels is the only one that induces a RKHS norm invariant through affine isometries. The most popular is the Gaussian kernel defined by , being a scale parameter that can be interpreted somehow as a range of interactions between points.
The definition of a kernel on a general manifold is often a more involved issue as we already mentioned in subsection 2.2. Generally, the procedure is reversed : the kernel is defined through a compact operator on differential forms of , which can be diagonalized and hopefully provide a closed expression of the kernel on (cf ). The case of the two-dimensional sphere for instance is thoroughly treated in . However, it’s important to note that, in our setting of functional currents, this issue is drastically simplified because we only need to define a real-valued kernel on . This is contrasting with the idea of product space currents of subsection 2.2, which requires the definition of kernels living in the exterior product of the fiber bundle of . For instance, if is a sub-manifold of a certain vector space, obtaining real-valued kernels on becomes straightforward by restriction to of kernels defined on the ambient vector space.
4.2. Convergence and control results on the RKHS norm
We are now going to explore a little more some properties of the RKHS norm on fcurrents and show the theoretical benefits of our approach with respect to the original problem raised by this article.
Suppose, under the same hypotheses as the previous section, that two kernels and are given respectively on space and manifold , providing two RKHS and . By a simple triangular inequality, we get for any , , any , and any ,
Since both kernels and are assumed to be bounded as in Proposition 2, and are uniformly bounded so that eventually
Therefore, the RKHS distance between punctual fcurrents is dominated both with respect to the variation of their geometrical parts and of their functional values. This is the general idea we will formulate in a more precise way with the two following propositions. We denote by the geodesic distance induced on by its Riemannian structure. The next proposition examines the case where the geometrical support is a fixed sub-manifold and shows that the variation of the -norm is then dominated by the norm on .
Let be a d-dimensional sub-manifold of of finite volume and and two measurable functions defined on the sub-manifold taking value in . We assume that is continuously embedded into . Then, there exists a constant such that :
where is the uniform measure on .
We recall the definition . We will first restrict the proof to the case where admits a parametrization given by a function where U is an open subset of . The general result follows by the use of an appropriate partition of the unit on . Denoting for , we get
Now, for and , we have by triangular inequality on :
From (12), . Now, for any , and we have
the last inequality resulting from the continuous embedding . Therefore we get
Moreover, since we assume that the kernel is bounded, we also have . Back to equation (13), we get from the previous derivations the existence of a constant such that :
which precisely proves the stated result. ∎
A straightforward consequence of Proposition 3 and dominated convergence theorem is that if is a sequence of function on that converges pointwisely to a function , then . In other words, pointwise convergence of signal implies convergence in terms of fcurrents.
Following the same kind of reasoning we eventually give a local bound of the RKHS distance between a functional shape and the same shape deformed through small diffeomorphisms both in geometry and signal. As it is now classical, we consider deformations modelled as flows between 0 and 1 of differential equations given through time varying vector fields. In appendix A, we remind the basic definitions about this modelling and a few useful results for the following. Let (resp. ) be a smooth time dependent vector fields on the geometrical space (resp. on the signal space ) and let (resp. ) the solution at time of the flow of the ODE (resp. ). On these spaces of vector fields, we define the norms :
where and .
Let be a sub-manifold of of finite volume and a measurable function. Assume that and are continuously embedded respectively into and . There exists a universal constant such that, if and are sufficiently small (which means that deformations are ’close’ to identity), then :
The full proof of proposition 4 relies mostly on a few controls which are summed up in appendix A. Given again a local parametrization of , , then, similarly to the previous proposition and using same notations, we have :
where for the volume element , is the transported volume element by equal to . From (12) we get
and using and we get
In a similar way, we know that . Moreover :
which concludes the proof since . ∎
This property shows that the RKHS norm is continuous with respect to deformations of the functional shape (both in its geometry and its signal). More specifically, it is not hard to see that for the action given by (8) and (9) and to extend the proof of the previous proposition to a more general situation of a fcurrent having finite “mass norm” where is the proper extension of the previous finite volume condition. Then we get
where is a universal constant.
This result also provides an answer to wether there is a resversed domination in Proposition 3 for two functional shapes that have the same geometrical support. Indeed, consider a particular case where and is a small deformation that leaves globally invariant (). We wish to compare the initial functional shape with the deformed one . By proposition 4, we know that, for any function , the fcurrent’s distance remains small if the deformation is small. It is no longer true if we compute instead , the distance on (). This is easily seen if we choose for the unit circle and consider crenellated signals as in figure 4. Introducing the operator that acts on functional shapes by rotation of an angle , we see indeed that :
whereas, according to Proposition 4