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29
Elastic morphing of 2D and 3D objects on a shape manifold
 in Image Analysis and Recognition, ser. Lecture Notes in Computer Science
"... Abstract. We present a new method for morphing 2D and 3D objects. In particular we focus on the problem of smooth interpolation on a shape manifold. The proposed method takes advantage of two recent works on 2D and 3D shape analysis to compute elastic geodesics between any two arbitrary shapes and i ..."
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Abstract. We present a new method for morphing 2D and 3D objects. In particular we focus on the problem of smooth interpolation on a shape manifold. The proposed method takes advantage of two recent works on 2D and 3D shape analysis to compute elastic geodesics between any two arbitrary shapes and interpolations on a Riemannian manifold. Given a finite set of frames of the same (2D or 3D) object from a video sequence, or different expressions of a 3D face, our goal is to interpolate between the given data in a manner that is smooth. Experimental results are presented to demonstrate the effectiveness of our method. 1
LARGE DEFORMATION DIFFEOMORPHIC METRIC MAPPING AND FASTMULTIPOLE BOUNDARY ELEMENT METHOD PROVIDE NEW INSIGHTS FOR BINAURAL ACOUSTICS
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Improving Efficiency of Data Assimilation Procedure for a Biomechanical Heart Model by Representing Surfaces as Currents
 in "FIMH  7th International Conference on Functional Imaging and Modeling of the Heart  2013
"... Abstract. We adapt the formalism of currents to compare data surfaces and surfaces of a mechanical model and we use this discrepancy measure to feed a data assimilation procedure. We apply our methodology to perform parameter estimation in a biomechanical model of the heart using synthetic observa ..."
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Abstract. We adapt the formalism of currents to compare data surfaces and surfaces of a mechanical model and we use this discrepancy measure to feed a data assimilation procedure. We apply our methodology to perform parameter estimation in a biomechanical model of the heart using synthetic observations of the endo and epicardium surfaces of an infarcted left ventricle. We compare this formalism with a more classical signed distance operator between surfaces and we numerically show that we have improved the efficiency of our estimation justifying the use of stateoftheart computational geometry formalism in the data assimilation measurements processing.
CONSTRUCTING REPARAMETERIZATION INVARIANT METRICS ON SPACES OF PLANE CURVES
"... Abstract. Metrics on shape spaces are used to describe deformations that take one shape to another, and to define a distance between shapes. We study a family of metrics on the space of curves, which includes several recently proposed metrics, for which the metrics are characterised by mappings into ..."
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Abstract. Metrics on shape spaces are used to describe deformations that take one shape to another, and to define a distance between shapes. We study a family of metrics on the space of curves, which includes several recently proposed metrics, for which the metrics are characterised by mappings into vector spaces where geodesics can be easily computed. This family consists of Sobolevtype Riemannian metrics of order one on the space Imm(S1, R2) of parameterized plane curves and the quotient space Imm(S1, R2) / Diff(S 1) of unparameterized curves. For the space of open parameterized curves we find an explicit formula for the geodesic distance and show that the sectional curvatures vanish on the space of parameterized open curves and are nonnegative on the space of unparameterized open curves. For one particular metric we provide a numerical algorithm that computes geodesics between unparameterized, closed curves, making use of a constrained formulation that is implemented numerically using the RATTLE algorithm. We illustrate the algorithm with some numerical tests between shapes. 1.
On computing mapping of 3D objects: A survey
, 2014
"... We review the computation of 3D geometric data mapping, which establishes onetoone correspondence between or among spatial/spatiotemporal objects. Effective mapping benefits many scientific and engineering tasks that involve the modeling and processing of correlated geometric or image data. We mo ..."
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We review the computation of 3D geometric data mapping, which establishes onetoone correspondence between or among spatial/spatiotemporal objects. Effective mapping benefits many scientific and engineering tasks that involve the modeling and processing of correlated geometric or image data. We model mapping computation as an optimization problem with certain geometric constraints and go through its general solving pipeline. Different mapping algorithms are discussed and compared according to their formulations of objective functions, constraints, and optimization strategies.
Elastic Morphing of 2D and 3D Objects on a Shape Manifold
, 2009
"... Abstract. We present a new method for morphing 2D and 3D objects. In particular we focus on the problem of smooth interpolation on a shape manifold. The proposed method takes advantage of two recent works on 2D and 3D shape analysis to compute elastic geodesics between any two arbitrary shapes and i ..."
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Abstract. We present a new method for morphing 2D and 3D objects. In particular we focus on the problem of smooth interpolation on a shape manifold. The proposed method takes advantage of two recent works on 2D and 3D shape analysis to compute elastic geodesics between any two arbitrary shapes and interpolations on a Riemannian manifold. Given a finite set of frames of the same (2D or 3D) object from a video sequence, or different expressions of a 3D face, our goal is to interpolate between the given data in a manner that is smooth. Experimental results are presented to demonstrate the effectiveness of our method. 1
SIAM J. Imaging Sci. 5, 1 (2012), 394433. Sectional Curvature in terms of the Cometric, with Applications to the Riemannian Manifolds of Landmarks
"... by FWFproject 21030, DM was supported by NSF grant DMS0704213, and all authors were ..."
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by FWFproject 21030, DM was supported by NSF grant DMS0704213, and all authors were
Variational methods in shape analysis
"... The analysis of shapes as elements in a frequently infinitedimensional space of shapes has attracted increasing attention over the last decade. There are pioneering contributions in the theoretical foundation of shape space as a Riemannian manifold as well as pathbreaking applications to quantitat ..."
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The analysis of shapes as elements in a frequently infinitedimensional space of shapes has attracted increasing attention over the last decade. There are pioneering contributions in the theoretical foundation of shape space as a Riemannian manifold as well as pathbreaking applications to quantitative shape comparison, shape recognition, and shape statistics. The aim of this chapter is to adopt a primarily physical perspective on the space of shapes and to relate this to the prevailing geometric perspective. Indeed, we here consider shapes given as boundary contours of volumetric objects, which consist either of a viscous fluid or an elastic solid. In the first case, shapes are transformed into each other via viscous transport of fluid material, and the flow naturally generates a connecting path in the space of shapes. The viscous dissipation rate—the rate at which energy is converted into heat due to friction—can be defined as a metric on an associated Riemannian manifold. Hence, via the computation of shortest transport paths one defines a distance measure between shapes.
Direct LDDMM of Discrete Currents with Adaptive Finite Elements
, 2011
"... Abstract. We consider Large Deformation Diffeomorphic Metric Mapping of general mcurrents. After stating an optimization algorithm in the function space of admissable morph generating velocity fields, two innovative aspects in this framework are presented and numerically investigated: First, we spa ..."
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Abstract. We consider Large Deformation Diffeomorphic Metric Mapping of general mcurrents. After stating an optimization algorithm in the function space of admissable morph generating velocity fields, two innovative aspects in this framework are presented and numerically investigated: First, we spatially discretize the velocity field with conforming adaptive finite elements and discuss advantages of this new approach. Second, we directly compute the temporal evolution of discrete mcurrent attributes. 1
Registration, Atlas Estimation and Variability Analysis of White Matter Fiber Bundles Modeled as Currents
"... This paper proposes a generic framework for the registration, the template estimation and the variability analysis of white matter fiber bundles extracted from diffusion images. This framework is based on the metric on currents for the comparison of fiber bundles. This metric measures anatomical dif ..."
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This paper proposes a generic framework for the registration, the template estimation and the variability analysis of white matter fiber bundles extracted from diffusion images. This framework is based on the metric on currents for the comparison of fiber bundles. This metric measures anatomical differences between fiber bundles, seen as global homologous structures across subjects. It avoids the need to establish correspondences between points or between individual fibers of different bundles. It can measure differences both in terms of the geometry of the bundles (like its boundaries) and in terms of the density of fibers within the bundle. It is robust to fiber interruptions and reconnections. In addition, a recently introduced sparse approximation algorithm allows us to give an interpretable representation of the fiber bundles and their variations in the framework of currents. First, we used this metric to drive the registration between two sets of homologous fiber bundles of two different subjects. A dense deformation of the underlying white matter is estimated, which is constrained by the bundles seen as global anatomical landmarks. By contrast, the alignment obtained from image registration is driven only by the local gradient of the image. Second, we propose a generative statistical model for the analysis of a