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ALMOST LOCAL METRICS ON SHAPE SPACE OF HYPERSURFACES IN nSPACE
"... Abstract. This paper extends parts of the results from [12] for plane curves to the case of hypersurfaces in Rn. Let M be a compact connected oriented n − 1 dimensional manifold without boundary like S2 or the torus S1 × S1. Then shape space is either the manifold of submanifolds of Rn of type M, or ..."
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Cited by 23 (17 self)
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Abstract. This paper extends parts of the results from [12] for plane curves to the case of hypersurfaces in Rn. Let M be a compact connected oriented n − 1 dimensional manifold without boundary like S2 or the torus S1 × S1. Then shape space is either the manifold of submanifolds of Rn of type M, or the orbifold of immersions from M to Rn modulo the group of diffeomorphisms of M. We investigate almost local Riemannian metrics on shape space: These are induced by metrics of the following form on the space of immersions: Z Gf(h, k) = Φ(Vol(M), Tr(L))〈h, k 〉 · vol(f
SOBOLEV METRICS ON SHAPE SPACE, II: WEIGHTED SOBOLEV METRICS AND ALMOST LOCAL METRICS
"... Abstract. In continuation of [5] we discuss metrics of the form G P ∫ p ∑ ( ) ( f (h,k) = Φi Vol(f) g (Pi)fh,k M i=0) vol(f ∗ g) on the space of immersions Imm(M,N) and on shape space Bi(M,N) = Imm(M,N)/Diff(M). Here (N,g) is a complete Riemannian manifold, M is a compact manifold, f: M → N is an ..."
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Cited by 8 (5 self)
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Abstract. In continuation of [5] we discuss metrics of the form G P ∫ p ∑ ( ) ( f (h,k) = Φi Vol(f) g (Pi)fh,k M i=0) vol(f ∗ g) on the space of immersions Imm(M,N) and on shape space Bi(M,N) = Imm(M,N)/Diff(M). Here (N,g) is a complete Riemannian manifold, M is a compact manifold, f: M → N is an immersion, h and k are tangent vectors to f in the space of immersions, f∗g is the induced Riemannian metric on M, vol(f∗g) is the induced volume density on M, Vol(f) = ∫ M vol(f∗g), Φi are positive realvalued functions, and (Pi)f are operators like some power of the Laplacian ∆f∗g. We derive the geodesic equations for these metrics and show that they are sometimes wellposed with the geodesic exponential mapping a local diffeomorphism. The new aspect here are the weights Φi(Vol(f)) which we use to construct scale invariant metrics and order 0 metrics with positive geodesic distance. We treat several concrete special cases in detail. 1.
SOBOLEV METRICS ON THE MANIFOLD OF ALL RIEMANNIAN METRICS
"... Abstract. On the manifold M(M) of all Riemannian metrics on a compact manifold M one can consider the natural L 2metric as described first by [11]. In this paper we consider variants of this metric which in general are of higher order. We derive the geodesic equations, we show that they are wellpo ..."
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Cited by 4 (2 self)
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Abstract. On the manifold M(M) of all Riemannian metrics on a compact manifold M one can consider the natural L 2metric as described first by [11]. In this paper we consider variants of this metric which in general are of higher order. We derive the geodesic equations, we show that they are wellposed under some conditions and induce a locally diffeomorphic geodesic exponential mapping. We give a condition when Ricci flow is a gradient flow for one of these metrics. 1.
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.
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.
Sobolev metrics on the Riemannian manifold of all Riemannian metrics
, 2010
"... Abstract. On the manifold M(M) of all Riemannian metrics on a compact manifold M one can consider the natural L 2metric as decribed first by [10]. In this paper we consider variants of this metric which in general are of higher order. We derive the geodesic equations, we show that they are wellpos ..."
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Cited by 2 (1 self)
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Abstract. On the manifold M(M) of all Riemannian metrics on a compact manifold M one can consider the natural L 2metric as decribed first by [10]. In this paper we consider variants of this metric which in general are of higher order. We derive the geodesic equations, we show that they are wellposed under some conditions and induce a locally diffeomorphic geodesic exponential mapping. We give a condition when Ricci flow is a gradient flow for one of this metrics. 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.
curves
"... On the geometry and the deformation of shapes represented by piecewise continuous Bézier ..."
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On the geometry and the deformation of shapes represented by piecewise continuous Bézier
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, 2013
"... On the geometry and the deformation of shapes represented by piecewise continuous Bézier curves with application to shape optimization ..."
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On the geometry and the deformation of shapes represented by piecewise continuous Bézier curves with application to shape optimization
SIAM J. Imaging Sci. 5 (2012), pp. 244310. ALMOST LOCAL METRICS ON SHAPE SPACE OF HYPERSURFACES IN nSPACE
"... Abstract. This paper extends parts of the results from [P.W.Michor and D. Mumford, Appl. Comput. Harmon. Anal., 23 (2007), pp. 74–113] for plane curves to the case of hypersurfaces in Rn. Let M be a compact connected oriented n − 1 dimensional manifold without boundary like the sphere or the torus. ..."
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Abstract. This paper extends parts of the results from [P.W.Michor and D. Mumford, Appl. Comput. Harmon. Anal., 23 (2007), pp. 74–113] for plane curves to the case of hypersurfaces in Rn. Let M be a compact connected oriented n − 1 dimensional manifold without boundary like the sphere or the torus. Then shape space is either the manifold of submanifolds of Rn of type M, or the orbifold of immersions from M to Rn modulo the group of diffeomorphisms of M. We investigate almost local Riemannian metrics on shape space. These are induced by metrics of the following form on the space of immersions: Gf (h, k) = Φ(Vol(f), Tr(L))¯g(h, k) vol(f