Results 1  10
of
41
Surface matching via currents
 IPMI 2005. LNCS
, 2005
"... Abstract. We present a new method for computing an optimal deformation between two arbitrary surfaces embedded in Euclidean 3dimensional 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 appropriat ..."
Abstract

Cited by 61 (1 self)
 Add to MetaCart
Abstract. We present a new method for computing an optimal deformation between two arbitrary surfaces embedded in Euclidean 3dimensional 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 onetoone 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. 1
Computational anatomy: Shape, growth, and atrophy comparison via diffeomorphisms
 NeuroImage
, 2004
"... Computational anatomy (CA) is the mathematical study of anatomy I a I = I a BG, an orbit under groups of diffeomorphisms (i.e., smooth invertible mappings) g a G of anatomical exemplars Iaa I. The observable images are the output of medical imaging devices. There are three components that CA examine ..."
Abstract

Cited by 50 (2 self)
 Add to MetaCart
Computational anatomy (CA) is the mathematical study of anatomy I a I = I a BG, an orbit under groups of diffeomorphisms (i.e., smooth invertible mappings) g a G of anatomical exemplars Iaa I. The observable images are the output of medical imaging devices. There are three components that CA examines: (i) constructions of the anatomical submanifolds, (ii) comparison of the anatomical manifolds via estimation of the underlying diffeomorphisms g a G defining the shape or geometry of the anatomical manifolds, and (iii) generation of probability laws of anatomical variation P(d) on the images I for inference and disease testing within anatomical models. This paper reviews recent advances in these three areas applied to shape, growth, and atrophy.
Simultaneous nonrigid registration of multiple point sets and atlas construction
 in European Conference on Computer Vision (ECCV), 2006
, 2006
"... Abstract. Estimating a meaningful average or mean shape from a set of shapes represented by unlabeled pointsets is a challenging problem since, usually this involves solving for point correspondence under a nonrigid motion setting. In this paper, we propose a novel and robust algorithm that is cap ..."
Abstract

Cited by 19 (4 self)
 Add to MetaCart
Abstract. Estimating a meaningful average or mean shape from a set of shapes represented by unlabeled pointsets is a challenging problem since, usually this involves solving for point correspondence under a nonrigid motion setting. In this paper, we propose a novel and robust algorithm that is capable of simultaneously computing the mean shape from multiple unlabeled pointsets (represented by finite mixtures) and registering them nonrigidly to this emerging mean shape. This algorithm avoids the correspondence problem by minimizing the JensenShannon (JS) divergence between the point sets represented as finite mixtures. We derive the analytic gradient of the cost function namely, the JSdivergence, in order to efficiently achieve the optimal solution. The cost function is fully symmetric with no bias toward any of the given shapes to be registered and whose mean is being sought. Our algorithm can be especially useful for creating atlases of various shapes present in images as well as for simultaneously (rigidly or nonrigidly) registering 3D range data sets without having to establish any correspondence. We present experimental results on nonrigidly registering 2D as well as 3D real data (point sets). 1
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 ..."
Abstract

Cited by 14 (1 self)
 Add to MetaCart
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
Groupwise point pattern registration using a novel CDFbased JensenShannon divergence
 in: IEEE Computer Vision and Pattern Recognition
"... In this paper, we propose a novel and robust algorithm for the groupwise nonrigid registration of multiple unlabeled pointsets with no bias toward any of the given pointsets. To quantify the divergence between multiple probability distributions each estimated from the given point sets, we develop ..."
Abstract

Cited by 12 (2 self)
 Add to MetaCart
In this paper, we propose a novel and robust algorithm for the groupwise nonrigid registration of multiple unlabeled pointsets with no bias toward any of the given pointsets. To quantify the divergence between multiple probability distributions each estimated from the given point sets, we develop a novel measure based on their cumulative distribution functions that we dub the CDFJS divergence. The measure parallels the well known JensenShannon divergence (defined for probability density functions) but is more regular than the JS divergence since its definition is based on CDFs as opposed to density functions. As a consequence, CDFJS is more immune to noise and statistically more robust than the JS. We derive the analytic gradient of the CDFJS divergence
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 9 (2 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.
A symmetrybased generative model for shape
 In ICCV2007 [59
"... We propose a novel generative language for shape that is based on the shock graph: given a shock graph topology, we explore constraints on the geometry and dynamics of the shock graph branches at each point required to generate a valid shape, i.e., with no selfintersection, cusps, or crossovers. We ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
We propose a novel generative language for shape that is based on the shock graph: given a shock graph topology, we explore constraints on the geometry and dynamics of the shock graph branches at each point required to generate a valid shape, i.e., with no selfintersection, cusps, or crossovers. We model the shape boundary as a piecewise smooth circular arc spline, which is dense in the space of piecewise smooth curves. Using this model we derive an independent set of parameters which generate a variety of shapes and satisfy the reconstruction constraints. We show simple examples of using this generative model as an active deformable shape and for morphing between two shapes. The results illustrate that it is possible to generate any generic shape with relatively few parameters, further reduced if prior knowledge of shape is available. 1.
Modeling planar shape variation via Hamiltonian flows of curves
 Analysis and Statistics of Shapes, Modeling and Simulation in Science, Engineering and Technology, chapter 14. Birkhäuser
, 2005
"... Summary. The application of the theory of deformable templates to the study of the action of a group of diffeomorphisms on deformable objects provides a powerful framework to compute dense onetoone matchings on ddimensional domains. In this paper, we derive the geodesic equations that govern the ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
Summary. The application of the theory of deformable templates to the study of the action of a group of diffeomorphisms on deformable objects provides a powerful framework to compute dense onetoone matchings on ddimensional domains. In this paper, we derive the geodesic equations that govern the time evolution of an optimal matching in the case of the action on 2D curves with various driving matching terms, and provide a Hamiltonian formulation in which the initial momentum is represented by an L 2 vector field on the boundary of the template. Key words: Infinitedimensional Riemannian manifolds, Hamiltonian systems, shape representation and recognition.
Sectional Curvature in terms of the Cometric, with Applications to the Riemannian Manifolds of Landmarks
"... statistics of the infinite dimensional shape manifolds). MM would like to thank Andrea Bertozzi of UCLA for her continuous advice and and support. This paper deals with the computation of sectional curvature for the manifolds of N landmarks (or feature points) in D dimensions, endowed with the Riema ..."
Abstract

Cited by 6 (4 self)
 Add to MetaCart
statistics of the infinite dimensional shape manifolds). MM would like to thank Andrea Bertozzi of UCLA for her continuous advice and and support. This paper deals with the computation of sectional curvature for the manifolds of N landmarks (or feature points) in D dimensions, endowed with the Riemannian metric induced by the group action of diffeomorphisms. The inverse of the metric tensor for these manifolds (i.e. the cometric), when written in coordinates, is such that each of its elements depends on at most 2D of the ND coordinates. This makes the matrices of partial derivatives of the cometric very sparse in nature, thus suggesting solving the highly nontrivial problem of developing a formula that expresses sectional curvature in terms of the cometric and its first and second partial derivatives (we call this Mario’s formula). We apply such formula to the manifolds of landmarks and in particular we fully explore the case of geodesics on which only two points have nonzero momenta and compute the sectional curvatures of 2planes spanned by the tangents to such geodesics. The latter example gives insight to the geometry of the full manifolds of landmarks. 1