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Constructing reparametrization invariant metrics on spaces of plane curves
, 2012
"... on spaces of plane curves ..."
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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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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.
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.
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.