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20
Constructing reparametrization invariant metrics on spaces of plane curves
, 2012
"... on spaces of plane curves ..."
A Riemannian view on shape optimization
 Foundations of Computational Mathematics
"... Abstract. Shape optimization based on the shape calculus is numerically mostly performed by means of steepest descent methods. This paper provides a novel framework to analyze shapeNewton optimization methods by exploiting a Riemannian perspective. A Riemannian shape Hessian is defined possessing o ..."
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Cited by 6 (2 self)
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Abstract. Shape optimization based on the shape calculus is numerically mostly performed by means of steepest descent methods. This paper provides a novel framework to analyze shapeNewton optimization methods by exploiting a Riemannian perspective. A Riemannian shape Hessian is defined possessing often sought properties like symmetry and quadratic convergence for Newton optimization methods. AMS subject classifications. 49Q10, 49M15, 53B20 Key words. Shape optimization, Riemannian manifold, Newton method
Rtransforms for Sobolev H2metrics on spaces of plane curves
 Geometry, Imaging and Computing
"... Dedicated to David Mumford on the occasion of his 76th birthday Abstract. We consider spaces of smooth immersed plane curves (modulo translations and/or rotations), equipped with reparameterization invariant weak Riemannian metrics involving second derivatives. This includes the full H2metric witho ..."
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Cited by 4 (4 self)
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Dedicated to David Mumford on the occasion of his 76th birthday Abstract. We consider spaces of smooth immersed plane curves (modulo translations and/or rotations), equipped with reparameterization invariant weak Riemannian metrics involving second derivatives. This includes the full H2metric without zero order terms. We find isometries (called Rtransforms) from some of these spaces into function spaces with simpler weak Riemannian metrics, and we use this to give explicit formulas for geodesics, geodesic distances, and sectional curvatures. We also show how to utilise the isometries to compute geodesics numerically.
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.
RTRANSFORMS FOR SOBOLEV H 2METRICS ON SPACES OF PLANE CURVES
"... Dedicated to David Mumford on the occasion of his 76th birthday Abstract. We consider spaces of smooth immersed plane curves (modulo translations and/or rotations), equipped with reparameterization invariant weak Riemannian metrics involving second derivatives. This includes the full H 2 metric with ..."
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Cited by 2 (2 self)
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Dedicated to David Mumford on the occasion of his 76th birthday Abstract. We consider spaces of smooth immersed plane curves (modulo translations and/or rotations), equipped with reparameterization invariant weak Riemannian metrics involving second derivatives. This includes the full H 2 metric without zero order terms. We find isometries (called Rtransforms) from some of these spaces into function spaces with simpler weak Riemannian metrics, and we use this to give explicit formulas for geodesics, geodesic distances, and sectional curvatures. We demonstrate the value of using Rtransforms by some numerical experiments.
GEODESIC COMPLETENESS FOR SOBOLEV METRICS ON THE SPACE OF IMMERSED PLANE CURVES
, 2013
"... We study properties of Sobolevtype metrics on the space of immersed plane curves. We show that the geodesic equation for Sobolevtype metrics with constant coefficients of order 2 and higher is globally wellposed for smooth initial data as well as for initial data in certain Sobolev spaces. Thus t ..."
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We study properties of Sobolevtype metrics on the space of immersed plane curves. We show that the geodesic equation for Sobolevtype metrics with constant coefficients of order 2 and higher is globally wellposed for smooth initial data as well as for initial data in certain Sobolev spaces. Thus the space of closed plane curves equipped with such a metric is geodesically complete. We find lower bounds for the geodesic distance in terms of curvature and its derivatives. 2010 Mathematics Subject Classification: 58D15 (primary); 35G55, 53A04, 58B20 (secondary) 1.
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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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
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.
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
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