Abstract. We present a new method for computing an optimal deformation between two arbitrary surfaces embedded in Euclidean 3-dimensional space. Our main contribution is in building a norm on the space of surfaces via representation by currents of geometric measure theory. Currents are an appropriate choice for representations because they inherit natural transformation properties from differential forms. We impose a Hilbert space structure on currents, whose norm gives a convenient and practical way to define a matching functional. Using this Hilbert space norm, we also derive and implement a surface matching algorithm under the large deformation framework, guaranteeing that the optimal solution is a one-to-one regular map of the entire ambient space. We detail an implementation of this algorithm for triangular meshes and present results on 3D face and medical image data.

In the paper, we study the problem of optimal matching of two generalized functions (distributions) via a diffeomorphic transformation of the ambient space. In the particular case of discrete distributions (weighted sums of Dirac measures), we provide a new algorithm to compare two arbitrary unlabelled sets of points, and show that it behaves properly in limit of continuous distributions on submanifolds. As a consequence, the algorithm may apply to various matching problems, such as curve or surface matching (via a sub-sampling), or mixings of landmark and curve data. As the solution forbids high energy solutions, it is also robust towards addition of noise and the technique can be used for nonlinear projection of datasets. We present 2D and 3D experiments. . IntroductionMatching embedded geometric structures is of particular importance in many computer vision tasks and medical imaging problems. The setting generally includes two images or volumes, within which points, curves or surfaces have been extracted, and formulates the following problems:1. Detect suitable correspondences between the manifolds extracted from the first image and those extracted from the second.2. Interpolate these correspondences to obtain a dense displacement field between the two images.In most of the approaches developped for matching (like [3, 4, 14, 10, 1, 8, 13, 5]), the considered geometric structures are points and both problems are solved separately: the second is addressed by elastic matching techniques coupled with spline interpolation, and the first one (point to point correspondence) is most of the time solved by hand (labels being provided by experts), in the absence of a reliable matching procedure. As an exception, [16] addresses the problems simultaneously, with the paradigm that the best correspondences should correspond to the smoothest deformations, yielding an automated point matching procedure. To match higher dimensional structures (curves and surfaces), the same approach is used, after discretization and representation by a set of points.However, curve and surface matching cannot be considered as limit problems of point matching (through discretization), because a given point in the discretized first manifold should be matched to some point of the second manifold, but not necessarily belonging to the set of points into which this manifold has been discretized. Our argument in this paper is that all these problems (surface, curve and unlabeled point matching) all are particular instances of a more general class of problems, which is matching measures on R 2 or R 3 , and more generally distributions on these sets (ie. generalized functions). For this purpose, we develop a theory based on the action of diffeomorphisms on distributions (which we present here in the specific case of measures), along the lines of Grenander's deformable templates theory ([12]), and in the large-deformation setting, as developed in [17,9,15,2,11].A significant contribution related to unlabeled point matching has been provided with the Robu...

We present a matching criterion for curves and integrate it into the large deformation diffeomorphic metric mapping (LDDMM) scheme for computing an optimal transformation between two curves embedded in Euclidean space ℝ d . Curves are first represented as vector-valued measures, which incorporate both location and the first order geometric structure of the curves. Then, a Hilbert space structure is imposed on the measures to build the norm for quantifying the closeness between two curves. We describe a discretized version of this, in which discrete sequences of points along the curve are represented by vector-valued functionals. This gives a convenient and practical way to define a matching functional for curves. We derive and implement the curve matching in the large deformation framework and demonstrate mapping results of curves in ℝ 2 and ℝ 3 . Behaviors of the curve mapping are discussed using 2D curves. The applications to shape classification is shown and experiments with 3D curves extracted from brain cortical surfaces are presented.