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The approximation of both geodesic distances and shortest paths on point cloud sampled from an embedded submanifold $\mathcal{M}$ of Euclidean space has been a long-standing challenge in computational geometry.
A note on two problems in connexion with graphs
E. W. Dijkstra et al · 1959
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Curvature measures
H. Federer · 1959
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The discrete geodesic problem
J. S. Mitchell, D. M. Mount, and C. H. Papadimitriou · 1987
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A fast marching level set method for monotonically advancing fronts
J. A. Sethian · 1996
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Computing geodesic paths on manifolds
R. Kimmel and J. A. Sethian · 1998
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The approximation power of moving least-squares
D. Levin · 1998
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An introduction to locally linear embedding
L. K. Saul and S. T. Roweis · 2000
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A global geometric framework for nonlinear dimensionality reduction
J. B. Tenenbaum, V. De Silva, and J. C. Langford · 2000
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Error estimates in sobolev spaces for moving least square approximations
M. G. Armentano · 2001
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H. Wendland · 2001
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Distance functions and geodesics on submanifolds of \ \backslash rˆd and point clouds
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D. Mirzaei · 2015
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A. Moscovich, A. Jaffe, and B. Nadler · 2016
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Approximation of functions over manifolds: A moving least-squares approach
B. Sober, Y. Aizenbud, and D. Levin · 2017
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R. Ravier · 2018
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Eyes on the prize: Improved registration via forward propagation
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R. R. Coifman and S. Lafon · 2006
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P. Cignoni, M. Callieri, M. Corsini, M. Dellepiane, F. Ganovelli, and G. Ranzuglia · 2008
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R. J. Ravier · 2018
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Deep eikonal solvers
M. Lichtenstein, G. Pai, and R. Kimmel · 2019
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Manifold approximation by moving least-squares projection (mmls)
B. Sober and D. Levin · 2019
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