Results 1  10
of
60
An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach
 Applied and Computational Harmonic Analysis, 2007. doi: 10.1016/j.acha.2006.07.004. URL http://www.mat.univie.ac.at/~michor/curveshamiltonian.pdf
"... Abstract. Here shape space is either the manifold of simple closed smooth unparameterized curves in R 2 or is the orbifold of immersions from S 1 to R 2 modulo the group of diffeomorphisms of S 1. We investige several Riemannian metrics on shape space: L 2metrics weighted by expressions in length a ..."
Abstract

Cited by 74 (25 self)
 Add to MetaCart
(Show Context)
Abstract. Here shape space is either the manifold of simple closed smooth unparameterized curves in R 2 or is the orbifold of immersions from S 1 to R 2 modulo the group of diffeomorphisms of S 1. We investige several Riemannian metrics on shape space: L 2metrics weighted by expressions in length and curvature. These include a scale invariant metric and a Wasserstein type metric which is sandwiched between two lengthweighted metrics. Sobolev metrics of order n on curves are described. Here the horizontal projection of a tangent field is given by a pseudodifferential operator. Finally the metric induced from the Sobolev metric on the group of diffeomorphisms on R 2 is treated. Although the quotient metrics are all given by pseudodifferential operators, their inverses are given by convolution with smooth kernels. We are able to prove local existence and uniqueness of solution to the geodesic equation for both kinds of Sobolev metrics. We are interested in all conserved quantities, so the paper starts with the Hamiltonian setting and computes conserved momenta and geodesics in general on the space of immersions. For each metric we compute the geodesic equation on shape space. In the end we sketch in some examples the differences between these metrics.
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 ..."
Abstract

Cited by 56 (18 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 31 (2 self)
 Add to MetaCart
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
Removing shapepreserving transformations in squareroot elastic (sre) framework for shape analysis of curves
 In EMMCVPR’07
, 2007
"... Abstract. This paper illustrates and extends an efficient framework, called the squarerootelastic (SRE) framework, for studying shapes of closed curves, that was first introduced in [2]. This framework combines the strengths of two important ideas elastic shape metric and pathstraightening metho ..."
Abstract

Cited by 30 (9 self)
 Add to MetaCart
(Show Context)
Abstract. This paper illustrates and extends an efficient framework, called the squarerootelastic (SRE) framework, for studying shapes of closed curves, that was first introduced in [2]. This framework combines the strengths of two important ideas elastic shape metric and pathstraightening methods for finding geodesics in shape spaces of curves. The elastic metric allows for optimal matching of features between curves while pathstraightening ensures that the algorithm results in geodesic paths. This paper extends this framework by removing two important shape preserving transformations: rotations and reparameterizations, by forming quotient spaces and constructing geodesics on these quotient spaces. These ideas are demonstrated using experiments involving 2D and 3D curves. 1
An Intrinsic Framework for Analysis of Facial Surfaces
"... A quantitative analysis of shapes of facial surfaces can play an important role in biometric authentication. The main difficulty in comparing shapes of surfaces is the lack of a canonical system to represent all surfaces. This paper overcomes that problem by proposing a specific coordinate system, ..."
Abstract

Cited by 25 (15 self)
 Add to MetaCart
(Show Context)
A quantitative analysis of shapes of facial surfaces can play an important role in biometric authentication. The main difficulty in comparing shapes of surfaces is the lack of a canonical system to represent all surfaces. This paper overcomes that problem by proposing a specific coordinate system, on facial surfaces, that enables comparisons of geometries of faces. In this system, a facial surface is represented as a path on the space of closed curves, called facial curves, where each curve is a level curve of distance function from the tip of the nose. Defining H to be the space of paths on the space of closed curves, the paper studies the differential geometry of H and endows it with a Riemannian. Using numerical techniques, it computes geodesic paths between elements of H that represent individual facial surfaces. This Riemannian analysis of faces is then used to: (i) find an optimal deformation from one face to another, (ii) define and compute an average face for a given set of faces, and (iii) compute distances between faces to quantify differences in their shapes. Experimental results are presented to demonstrate and support these ideas.
Disconnected Skeleton: Shape at its Absolute Scale
, 2007
"... We present a new skeletal representation along with a matching framework to address the deformable shape recognition problem. The disconnectedness arises as a result of excessive regularization that we use to describe a shape at an attainably coarse scale. Our motivation is to rely on the stable pro ..."
Abstract

Cited by 20 (5 self)
 Add to MetaCart
We present a new skeletal representation along with a matching framework to address the deformable shape recognition problem. The disconnectedness arises as a result of excessive regularization that we use to describe a shape at an attainably coarse scale. Our motivation is to rely on the stable properties of the shape instead of inaccurately measured secondary details. The new representation does not suffer from the common instability problems of traditional connected skeletons, and the matching process gives quite successful results on a diverse database of 2D shapes. An important difference of our approach from the conventional use of the skeleton is that we replace the local coordinate frame with a global Euclidean frame supported by additional mechanisms to handle articulations and local boundary deformations. As a result, we can produce descriptions that are sensitive to any combination of changes in scale, position, orientation and articulation, as well as invariant ones.
Ricci flow for 3D shape analysis
 PAMI
"... Abstract—Ricci flow is a powerful curvature flow method, which is invariant to rigid motion, scaling, isometric, and conformal deformations. We present the first application of surface Ricci flow in computer vision. Previous methods based on conformal geometry, which only handle 3D shapes with simpl ..."
Abstract

Cited by 15 (11 self)
 Add to MetaCart
(Show Context)
Abstract—Ricci flow is a powerful curvature flow method, which is invariant to rigid motion, scaling, isometric, and conformal deformations. We present the first application of surface Ricci flow in computer vision. Previous methods based on conformal geometry, which only handle 3D shapes with simple topology, are subsumed by the Ricci flowbased method, which handles surfaces with arbitrary topology. We present a general framework for the computation of Ricci flow, which can design any Riemannian metric by userdefined curvature. The solution to Ricci flow is unique and robust to noise. We provide implementation details for Ricci flow on discrete surfaces of either euclidean or hyperbolic background geometry. Our Ricci flowbased method can convert all 3D problems into 2D domains and offers a general framework for 3D shape analysis. We demonstrate the applicability of this intrinsic shape representation through standard shape analysis problems, such as 3D shape matching and registration, and shape indexing. Surfaces with large nonrigid anisotropic deformations can be registered using Ricci flow with constraints of feature points and curves. We show how conformal equivalence can be used to index shapes in a 3D surface shape space with the use of Teichmüller space coordinates. Experimental results are shown on 3D face data sets with large expression deformations and on dynamic heart data. Index Terms—Ricci flow, shape representation, surface matching and registration. Ç
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. ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
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
Constructing reparametrization invariant metrics on spaces of plane curves
, 2012
"... on spaces of plane curves ..."
(Show Context)
A New Riemannian Setting for Surface Registration
, 2011
"... Abstract. We present a new approach for matching regular surfaces in a Riemannian setting. We use a Sobolev type metric on deformation vector fields which form the tangent bundle to the space of surfaces. In this article we compare our approach with the diffeomorphic matching framework. In the latte ..."
Abstract

Cited by 8 (7 self)
 Add to MetaCart
(Show Context)
Abstract. We present a new approach for matching regular surfaces in a Riemannian setting. We use a Sobolev type metric on deformation vector fields which form the tangent bundle to the space of surfaces. In this article we compare our approach with the diffeomorphic matching framework. In the latter approach a deformation is prescribed on the ambient space, which then drags along an embedded surface. In contrast our metric is defined directly on the deformation vector field and can therefore be called an inner metric. We also show how to discretize the corresponding geodesic equation and compute the gradient of the cost functional using finite elements.