Abstract. This paper studies a specific metric on plane curves that has the property of being isometric to classical manifold (sphere, complex projective, Stiefel, Grassmann) modulo change of parametrization, each of these classical manifolds being associated to specific qualifications of the space of curves (closed-open, modulo rotation etc...) Using these isometries, we are able to explicitely describe the geodesics, first in the parametric case, then by modding out the paremetrization and considering horizontal vectors. We also compute the sectional curvature for these spaces, and show, in particular, that the space of closed curves modulo rotation and change of parameter has positive curvature. Experimental results that explicitly compute minimizing geodesics between two closed curves are finally provided

An H 2 type metric on the space of planar curves is proposed and equation of the geodesic is derived. A numerical example is given to illustrate the differneces between H 1 and H 2 metrics. 1

Abstract. This paper extends parts of the results from [12] for plane curves to the case of hypersurfaces in Rn. Let M be a compact connected oriented n − 1 dimensional manifold without boundary like S2 or the torus S1 × S1. Then shape space is either the manifold of submanifolds of Rn of type M, or the orbifold of immersions from M to Rn modulo the group of diffeomorphisms of M. We investigate almost local Riemannian metrics on shape space: These are induced by metrics of the following form on the space of immersions: Z Gf(h, k) = Φ(Vol(M), Tr(L))〈h, k 〉 · vol(f

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

Abstract. This paper extends parts of the results from [14] for plane curves to the case of surfaces in Rn. Let M be a compact connected oriented manifold of dimension less than n without boundary. Then shape space is either the manifold of submanifolds of Rn of type M, or the orbifold of immersions from M to Rn modulo the group of diffeomorphisms of M. We investigate the Sobolev Riemannian metrics on shape space: These are induced by metrics of the following form on the space of immersions:

Abstract. Let M be a compact connected oriented n−1 dimensional manifold without boundary. In this work, shape space is the orbifold of unparametrized immersions from M to Rn. The results of [1], where mean curvature weighted metrics were studied, suggest to incorporate Gauß curvature weights in the definition of the metric. This leads us to study metrics on shape space that are induced by metrics on the space of immersions of the form Gf (h, k) = Φ.¯g(h, k) vol(f

Abstract. Let M and N be connected manifolds without boundary with dim(M) < dim(N), and let M compact. Then shape space in this work is either the manifold of submanifolds of N that are diffeomorphic to M, or the orbifold of unparametrized immersions of M in N. We investigate the Sobolev Riemannian metrics on shape space: These are induced by metrics of the following form on the space of immersions:

Abstract. We present a new framework for multidimensional shape analysis. The proposed framework represents solid objects as points on an infinite-dimensional Riemannian manifold and distances between objects as minimal length geodesic paths. Intershape distance forms the foundation for shape-based statistical analysis. The proposed method incorporates a metric that naturally prevents self-intersections of object boundaries and thus produces a well-defined interior and exterior along every geodesic path. This paper presents an implementation of the geodesic computations for 2D shapes and gives examples of geodesic paths that demonstrate the advantages of enforcing well-defined boundaries. This compares favorably with equivalent paths under a linear L 2 metric, which do not prevent self-intersection of the boundary, and thus do not produce valid solid objects. 1

We present and study a family of metrics on the space of compact subsets of � N (that we call “shapes”). These metrics are “geometric”, that is, they are independent of rotation and translation; and these metrics enjoy many interesting properties, as, for example, the existence of minimal geodesics. We view our space of shapes as a subset of Banach (or Hilbert) manifolds: so we can define a “tangent manifold ” to shapes, and (in a very weak form) talk of a “Riemannian Geometry” of shapes. Some of the metrics that we propose are topologically equivalent to the Hausdorff metric; but at the same time, they are more “regular”, since we can hope for a local uniqueness of minimal geodesics. We also study general properties of the metrics obtained by isometrically identifying a generic metric space with a subset of a Banach space and we obtain a rigidity result.

We define a novel metric on the space of closed planar curves. According to this metric centroid translations, scale changes and deformations are orthogonal, and the metric is also invariant with respect to reparameterizations of the curve. The Riemannian structure that is induced on the space of curves is a smooth Riemannian manifold, which is isometric to a classical well-known manifold. As a consequence, geodesics and gradients of energies defined on the space can be computed using fast closed-form formulas, and this has obvious benefits in numerical applications. The obtained Riemannian manifold of curves is apt to address complex problems in computer vision; one such example is the tracking of highly deforming objects. Previous works have assumed that the object deformation is smooth, which is realistic for the tracking problem, but most have restricted the deformation to belong to a finite-dimensional group – such as affine motions – or to finitely-parameterized models. This is too restrictive for highly deforming objects such as the contour of a beating heart. We adopt the smoothness assumption implicit in previous work, but we lift the restriction to finite-dimensional motions/deformations. We define a dynamical model in this Riemannian manifold of curves, and use it to perform filtering and prediction to infer and extrapolate not just the pose (a finitely parameterized quantity) of an object, but its deformation (an infinite-dimensional quantity) as well. We illustrate these ideas using a simple first-order dynamical model, and show that it can be effective even on data sets where existing methods fail. 1