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51
A metric on shape spaces with explicit geodesics
, 2007
"... 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 ..."
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Cited by 34 (15 self)
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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 (closedopen, 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
H o type Riemannian metrics on the space of planar curves
"... 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 ..."
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Cited by 24 (2 self)
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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
ALMOST LOCAL METRICS ON SHAPE SPACE OF HYPERSURFACES IN nSPACE
"... 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 ..."
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Cited by 17 (14 self)
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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
Large Deformation Diffeomorphic Metric Curve Mapping
 INT J COMPUT VIS
, 2008
"... 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 vectorvalued measures, which incorporate bot ..."
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Cited by 17 (1 self)
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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 vectorvalued 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 vectorvalued 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
SOBOLEV METRICS ON SHAPE SPACE OF SURFACES
"... 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 Riemanni ..."
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Cited by 11 (11 self)
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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:
A computational model of multidimensional shape
 International Journal of Computer Vision, Online First
, 2010
"... We develop a computational model of shape that extends existing Riemannian models of shape of curves to multidimensional objects of general topological type. We construct shape spaces equipped with geodesic metrics that measure how costly it is to interpolate two shapes through elastic deformations. ..."
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Cited by 10 (0 self)
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We develop a computational model of shape that extends existing Riemannian models of shape of curves to multidimensional objects of general topological type. We construct shape spaces equipped with geodesic metrics that measure how costly it is to interpolate two shapes through elastic deformations. The model employs a representation of shape based on the discrete exterior derivative of parametrizations over a finite simplicial complex. We develop algorithms to calculate geodesics and geodesic distances, as well as tools to quantify local shape similarities and contrasts, thus obtaining a localglobal formulation that accounts for regional shape differences and integrates them into a global measure of dissimilarity. The Riemannian shape spaces provide a common framework to treat numerous problems such as the statistical modeling of shapes, the comparison of shapes associated with different individuals and groups, and modeling and simulation of dynamical shapes. We give multiple examples of geodesic interpolations and illustrations of the use of the model in brain mapping, particularly, the analysis of anatomical variation based on neuroimaging data. 1
A New Geometric Metric in the Space of Curves, and Applications to Tracking Deforming Objects by Prediction and Filtering
, 2010
"... 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 cu ..."
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Cited by 10 (0 self)
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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 wellknown manifold. As a consequence, geodesics and gradients of energies defined on the space can be computed using fast closedform 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 finitedimensional group – such as affine motions – or to finitelyparameterized 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 finitedimensional 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 infinitedimensional quantity) as well. We illustrate these ideas using a simple firstorder dynamical model, and show that it can be effective even on data sets where existing methods fail. 1
CURVATURE WEIGHTED METRICS ON SHAPE SPACE OF HYPERSURFACES IN nSPACE
"... 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 ..."
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Cited by 9 (8 self)
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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
The metric geometry of the manifold of Riemannian metrics
"... Abstract. We prove that the L 2 Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finitedimensional manifold induces a metric space structure. As the L 2 metric is a weak Riemannian metric, this fact does not follow from general results. In addition, we prove sev ..."
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Cited by 9 (2 self)
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Abstract. We prove that the L 2 Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finitedimensional manifold induces a metric space structure. As the L 2 metric is a weak Riemannian metric, this fact does not follow from general results. In addition, we prove several results on the exponential mapping and distance function of a weak Riemannian metric on a Hilbert/Fréchet manifold. The statements are analogous to, but weaker than, what is known in the case of a Riemannian metric on a finitedimensional manifold or a strong Riemannian metric on a Hilbert manifold. 1.
SOBOLEV METRICS ON DIFFEOMORPHISM GROUPS AND THE DERIVED GEOMETRY OF SPACES OF SUBMANIFOLDS
"... Abstract. Given a finite dimensional manifold N, the group DiffS(N) of diffeomorphism of N which fall suitably rapidly to the identity, acts on the manifold B(M,N) of submanifolds on N of diffeomorphism type M where M is a compact manifold with dimM < dimN. For a right invariant weak Riemannian m ..."
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Cited by 8 (6 self)
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Abstract. Given a finite dimensional manifold N, the group DiffS(N) of diffeomorphism of N which fall suitably rapidly to the identity, acts on the manifold B(M,N) of submanifolds on N of diffeomorphism type M where M is a compact manifold with dimM < dimN. For a right invariant weak Riemannian metric on DiffS(N) induced by a quite general operator L: XS(N) → Γ(T ∗ N⊗vol(N)), we consider the induced weak Riemannian metric on B(M,N) and we compute its geodesics and sectional curvature. For that we derive a covariant formula for curvature in finite and infinite dimensions, we show how it makes O’Neill’s formula very transparent, and we use it finally to compute sectional curvature on B(M,N). 1.