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11
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 ..."
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Cited by 8 (7 self)
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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.
Variational time discretization of geodesic calculus
, 2012
"... Abstract. We analyze a variational time discretization of geodesic calculus on finite and certain classes of infinitedimensional Riemannian manifolds. We investigate the fundamental properties of discrete geodesics, the associated discrete logarithm, discrete exponential maps, and discrete paralle ..."
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Cited by 3 (2 self)
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Abstract. We analyze a variational time discretization of geodesic calculus on finite and certain classes of infinitedimensional Riemannian manifolds. We investigate the fundamental properties of discrete geodesics, the associated discrete logarithm, discrete exponential maps, and discrete parallel transport, and we prove convergence to their continuous counterparts. The presented analysis is based on the direct methods in the calculus of variation, on Γconvergence, and on weighted finite element error estimation. The convergence results of the discrete geodesic calculus are experimentally confirmed for a basic model on a twodimensional Riemannian manifold. This provides a theoretical basis for the application to shape spaces in computer vision, for which we present one specific example.
Discrete geodesic regression in shape space
"... Abstract. A new approach for the effective computation of geodesic regression curves in shape spaces is presented. Here, one asks for a geodesic curve on the shape manifold that minimizes a sum of dissimilarity measures between given two or threedimensional input shapes and corresponding shapes al ..."
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Cited by 1 (0 self)
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Abstract. A new approach for the effective computation of geodesic regression curves in shape spaces is presented. Here, one asks for a geodesic curve on the shape manifold that minimizes a sum of dissimilarity measures between given two or threedimensional input shapes and corresponding shapes along the regression curve. The proposed method is based on a variational time discretization of geodesics. Curves in shape space are represented as deformations of suitable reference shapes, which renders the computation of a discrete geodesic as a PDE constrained optimization for a family of deformations. The PDE constraint is deduced from the discretization of the covariant derivative of the velocity in the tangential direction along a geodesic. Finite elements are used for the spatial discretization, and a hierarchical minimization strategy together with a Lagrangian multiplier type gradient descent scheme is implemented. The method is applied to the analysis of root growth in botany and the morphological changes of brain structures due to aging. 1
DISCRETE GEODESIC CALCULUS IN SHAPE SPACE
"... Abstract. Based on a local approximation of the Riemannian distance on a manifold by a computationally cheap dissimilarity measure, a time discrete geodesic calculus is developed, and applications to shape space are explored. The dissimilarity measure is derived from a deformation energy whose Hessi ..."
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Abstract. Based on a local approximation of the Riemannian distance on a manifold by a computationally cheap dissimilarity measure, a time discrete geodesic calculus is developed, and applications to shape space are explored. The dissimilarity measure is derived from a deformation energy whose Hessian reproduces the underlying Riemannian metric, and it is used to define length and energy of discrete paths in shape space. The notion of discrete geodesics defined as energy minimizing paths gives rise to a discrete logarithmic map, a variational definition of a discrete exponential map, and a time discrete parallel transport. This new concept is applied to a shape space in which shapes are considered as boundary contours of physical objects consisting of viscous material. The flexibility and computational efficiency of the approach is demonstrated for topology preserving shape morphing, the representation of paths in shape space via local shape variations as path generators, shape extrapolation via discrete geodesic flow, and the transfer of geometric features. Key words. Shape space, geodesic paths, exponential map, logarithm, parallel transport
Mathematical Foundations of Computational Anatomy Geometrical and Statistical Methods for Biological Shape Variability Modeling
"... Computational anatomy is an emerging discipline at the interface of geometry, statistics and image analysis which aims at modeling and analyzing the biological shape of tissues and organs. The goal is to estimate representative organ anatomies across diseases, populations, species or ages, to model ..."
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Computational anatomy is an emerging discipline at the interface of geometry, statistics and image analysis which aims at modeling and analyzing the biological shape of tissues and organs. The goal is to estimate representative organ anatomies across diseases, populations, species or ages, to model the organ development across time (growth or aging), to establish their variability, and to correlate this variability information with other functional, genetic or structural information. The Mathematical Foundations of Computational Anatomy (MFCA) workshop aims at fostering the interactions between the mathematical community around shapes and the MICCAI community in view of computational anatomy applications. It targets more particularly researchers investigating the combination of statistical and geometrical aspects in the modeling of the variability of biological shapes. The workshop is a forum for the exchange of the theoretical ideas and aims at being a source of inspiration for new methodological developments in computational anatomy. A special emphasis is put on theoretical
DISCRETE GEODESIC CALCULUS IN THE SPACE OF VISCOUS FLUIDIC OBJECTS
"... Abstract. Based on a local approximation of the Riemannian distance on a manifold by a computationally cheap dissimilarity measure, a time discrete geodesic calculus is developed, and applications to shape space are explored. The dissimilarity measure is derived from a deformation energy whose Hessi ..."
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Abstract. Based on a local approximation of the Riemannian distance on a manifold by a computationally cheap dissimilarity measure, a time discrete geodesic calculus is developed, and applications to shape space are explored. The dissimilarity measure is derived from a deformation energy whose Hessian reproduces the underlying Riemannian metric, and it is used to define length and energy of discrete paths in shape space. The notion of discrete geodesics defined as energy minimizing paths gives rise to a discrete logarithmic map, a variational definition of a discrete exponential map, and a time discrete parallel transport. This new concept is applied to a shape space in which shapes are considered as boundary contours of physical objects consisting of viscous material. The flexibility and computational efficiency of the approach is demonstrated for topology preserving shape morphing, the representation of paths in shape space via local shape variations as path generators, shape extrapolation via discrete geodesic flow, and the transfer of geometric features. Key words. Shape space, geodesic paths, exponential map, logarithm, parallel transport
TimeDiscrete Geodesics in the Space of Shells
"... Figure 1: Discrete geodesic computed from two input poses (leftmost and rightmost hand). Building on concepts from continuum mechanics, we offer a computational model for geodesics in the space of thin shells, with a metric that reflects viscous dissipation required to physically deform a thin shell ..."
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Figure 1: Discrete geodesic computed from two input poses (leftmost and rightmost hand). Building on concepts from continuum mechanics, we offer a computational model for geodesics in the space of thin shells, with a metric that reflects viscous dissipation required to physically deform a thin shell. Different from previous work, we incorporate bending contributions into our deformation energy on top of membrane distortion terms in order to obtain a physically sound notion of distance between shells, which does not require additional smoothing. Our bending energy formulation depends on the socalled relative Weingarten map, for which we provide a discrete analogue based on principles of discrete differential geometry. Our computational results emphasize the strong impact of physical parameters on the evolution of a shell shape along a geodesic path. Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Computational geometry and object modeling—Physically based modeling 1.
Mathematical Foundations of Computational Anatomy Geometrical and Statistical Methods for Biological Shape Variability Modeling
, 2013
"... Computational anatomy is an emerging discipline at the interface of geometry, statistics and image analysis which aims at modeling and analyzing the biological shape of tissues and organs. The goal is to estimate representative organ anatomies across diseases, populations, species or ages, to model ..."
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Computational anatomy is an emerging discipline at the interface of geometry, statistics and image analysis which aims at modeling and analyzing the biological shape of tissues and organs. The goal is to estimate representative organ anatomies across diseases, populations, species or ages, to model the organ development across time (growth or aging), to establish their variability, and to correlate this variability information with other functional, genetic or structural information. The Mathematical Foundations of Computational Anatomy (MFCA) workshop aims at fostering the interactions between the mathematical community around shapes and the MICCAI community in view of computational anatomy applications. It targets more particularly researchers investigating the combination of statistical and geometrical aspects in the modeling of the variability of biological shapes. The workshop is a forum for the exchange of the theoretical ideas and aims at being a source of inspiration for new methodological developments in computational anatomy. A special emphasis is put on theoretical
Contents
, 2011
"... The HahnBanach Theorem is one of the most fundamental results in functional analysis. We present a fully formal proof of two versions of the theorem, one for general linear spaces and another for normed spaces. This development is based on simplytyped classical settheory, as provided by ..."
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The HahnBanach Theorem is one of the most fundamental results in functional analysis. We present a fully formal proof of two versions of the theorem, one for general linear spaces and another for normed spaces. This development is based on simplytyped classical settheory, as provided by
Geometric Monitoring of Heterogeneous Streams (Long Version, with Proofs of the Theorems)
"... Interest in stream monitoring is shifting toward the distributed case. In many applications the data is high volume, dynamic, and distributed, making it infeasible to collect the distinct streams to a central node for processing. Often, the monitoring problem consists of determining whether the val ..."
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Interest in stream monitoring is shifting toward the distributed case. In many applications the data is high volume, dynamic, and distributed, making it infeasible to collect the distinct streams to a central node for processing. Often, the monitoring problem consists of determining whether the value of a global function, defined on the union of all streams, crossed a certain threshold. We wish to reduce communication by transforming the global monitoring to the testing of local constraints, checked independently at the nodes. Geometric monitoring (GM) proved useful for constructing such local constraints for general functions. Alas, in GM the constraints at all nodes share an identical structure and are thus unsuitable for handling heterogeneous streams. Therefore, we propose a general approach for monitoring heterogeneous streams (HGM), which defines constraints tailored to fit the data distributions at the nodes. While we prove that optimally selecting the constraints is NPhard, we provide a practical solution, which reduces the running time by hierarchically clustering nodes with similar data distributions and then solving simpler optimization problems. We also present a method for efficiently recovering from local violations at the nodes. Experiments yield an improvement of over an order of magnitude in communication relative to GM. index terms: heterogeneous data streams, distributed streams, geometric monitoring, data modeling, safe zones. 1 1