- D. Burago, Y. Burago, S. Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, 2001.
A comprehensive reference on metric geometry.
- M. Spivak, A Comprehensive introduction to differential geometry, Vol 2 (Third edition), Publish or Perish, 1999, pp. 112-148.
An exciting and thorough explanation of Gauss' results, analysis of the properties of the Gaussian curvature from the classical reference on differential geometry. Spivak gives a proof of Theorema Egregium and shows intrinsic expressions of the Gaussian curvature through the perimeter defect (Bertrand-Puiseux formula).
- R. Connelly, "A counterexample to the rigidity conjecture for polyhedra", in Publications Mathématiques de l'IHÉS, Vol. 47, 1997, pp. 333-338.
A counter example to the Euler rigidity conjecture, showing a flexible polyhedron (Connelly sphere).
- S. Dasgupta, "Performance guarantees for hierarchical clustering", in Computational learning theory (J. Kivinen and Robert H. Sloan Eds.), LNAI vol. 2375, Springer, 2002, pp. 351-364.
Properties of farthest point sampling and proof of Hochbaum-Shmoys theorem.
- G. Leibon, D. Letscher, "Delaunay triangulations and Voronoi diagrams for Riemannian manifolds", in Proc. Symp. Computational Geometry, 2000, pp. 341-349.
Sufficient sampling density conditions that guarantee the existence of Delaunay triangulation for Riemannian surfaces; Amenta and Bern result on sufficient sampling density related to local feature size that guarantees the Delaunay triangulation is a valid approximation of the surface.
- G. Taubin, "Estimating the tensor of curvature of a surface from a polyhedral approximation", in Proc. ICCV, 1995.
Seminal paper on estimation of principal curvatures and principal directions.
Shortest path problems
- M. Bernstein, V. de Silva, J. C. Langford, J. B. Tenenbaum, "Graph approximations to geodesics on embedded manifolds", 2000
Conditions that guarantee consistent geodesic approximation in graphs.
Fast marching methods
- J. N. Tsitsiklis, "Efficient algorithms for globally optimal trajectories", 1995
Tsitsikli's formulation of computationally-optimal Eikonal solver motivated by optimal control theory.
- J. A. Sethian, "A fast marching level set method for monotonically advancing fronts", 1996
Sethian's formulation of computationally optimal first-order accurate Eikonal solver on weighted Cartesian grids.
- R. Kimmel, J. A. Sethian, "Computing geodesic paths on manifolds", 1998
Extension of fast marching to arbitrary triangulated meshes, introducing the idea of virtual connections splitting obtuse triangles.
- A. Spira, R. Kimmel, "An efficient solution to the eikonal equation on parametric manifolds", 2004
Formulation of fast marching for (globally) parametric manifolds sampled on a Cartesian grid, and expression of the obtuse triangle unfolding trick entirely in the parametrization coordinates.
- A. M. Bronstein, M. M. Bronstein, R. Kimmel, "Weighted distance maps computation on parametric three-dimensional manifolds", 2007
Extension of the unfolding trick to three-dimensional parametric manifolds.
- F. Memoli, G. Sapiro, "Fast computation of weighted distance functions and geodesics on implicit hyper-surfaces", 2001
Narrow-band version of fast marching for implicit surfaces.
- F. Memoli, G. Sapiro, "On distance functions and geodesic on submanifolds of R^d and point clouds", 2005
More rigorous treatment of the ideas from the 2001 paper and their application to distances on point clouds. Includes probabilistic analysis of sampling conditions.
- O. Weber, Y. Devir, A. M. Bronstein, M. M. Bronstein, R. Kimmel, "Parallel algorithms for approximation of distance maps on parametric surface", 2008
Raster-scan based parallel algorithms for distance computation. Includes extension to multi-chart parametrizations and implementation details on SIMD and GPU architectures.
- L. Yatziv, A. Bartesaghi, G. Sapiro, "A fast O(N) implementation of the fast marching algorithm
Linear-complexity approximation of heap-based fast-marching based on distance map quantization. Idea dates back to Tsitsiklis' 2005 paper.
- A. Elad, R. Kimmel,
"On bending invariant signatures for surfaces", IEEE Trans. PAMI, Vol. 25/10, pp. 1285-1295, 2003.
Intrinsic similarity computation using canonical forms
- M. M. Bronstein, A. M. Bronstein, R. Kimmel, I. Yavneh, "Multigrid multidimensional scaling", Numerical Linear Algebra with Applications (NLAA), Special issue on multigrid methods, Vol. 13/2-3, pp. 149-171, March-April 2006.
Multigrid solver for MDS problems.
- G. Rosman, A. M. Bronstein, M. M. Bronstein, A. Sidi, R. Kimmel, "Fast multidimensional scaling using vector extrapolation", Techn. Report CIS-2008-01, Dept. of Computer Science, Technion, Israel, January 2008
Using vector extrapolation to accelerate convergence in MDS problems.
Non-Euclidean embedding and GMDS
- A. M. Bronstein, M. M. Bronstein, R. Kimmel, "Generalized multidimensional scaling: a framework for isometry-invariant partial surface matching", Proc. National Academy of Sciences (PNAS), Vol. 103/5, pp. 1168-1172, January 2006.
- N. Thorstensen, R. Keriven, "Non-rigid shape matching using geometry and photometry", Proc. ACCV, 2009
Extension of GMDS incorporating photometric information.
- M. Belkin, P. Niyogi, "Laplacian eigenmaps for dimensionality reduction and data representation", Neural Computation, Vol. 15/6, pp. 1373-1396, June 2003.
Local embedding using graph Laplacian.
- M. Reuter, F.-E. Wolter and N. Peinecke, "Laplace-Beltrami spectra as "Shape-DNA" of surfaces and solids", Computer-Aided Design, Vol. 38/4, pp. 342-366, April 2006.
Using Laplace-Beltrami spectrum as intrinsic shape descriptors.
- R. M. Rustamov, "Laplace-Beltrami Eigenfunctions for Deformation Invariant Shape Representation", Proc. SGP 2007.
Using Laplace-Beltrami eigenfunctions as "infinite-dimensional canonical forms.
- M. Wardetzky, S. Mathur, F. Kalberer, E. Grinspun, "Discrete Laplace operators: No free lunch", Proc. SGP 2007.
Proof why it is impossible to construct an ideal discretization of the Laplace-Beltrami operator.
- R. R. Coifman, S. Lafon, A. B. Lee, M. Maggioni, B. Nadler, F. Warner, S. Zucker, "Geometric diffusions as a tool for harmonic analysis and structure definition of data: diffusion maps", Proc. National Academy of Sciences (PNAS), Vol. 102/21, pp. 7426-7431, 2005.
Introduction of the diffusion metric.
- M. Mahmoudi, G. Sapiro, "Three-dimensional point cloud recognition via distributions of geometric distances", Elsevier Journal of Graphical Models, Vol. 71/1, pp. 22-31, 2009.
Signatures of shapes based on distributions of
geodesic and diffusion distances.
- A. M. Bronstein, M. M. Bronstein, R. Kimmel, M. Mahmoudi, G. Sapiro, "A Gromov-Hausdorff framework with diffusion geometry for topologically-robust non-rigid shape matching", IJCV, 2009.
Gromov-Hausdorff framework with the diffusion geometry and its uses for isometry-invariant and topology-insensitive shape similarity and correspondence.
- F. Memoli, "Spectral Gromov-Wasserstein distances for shape matching", Proc. NORDIA, 2009.
Gromov-Wasserstein framework with the diffusion metric.
Isometry invariant similarity and correspondence
- F. Memoli, G. Sapiro, "A theoretical and computational framework for isometry invariant recognition of point cloud data", Found. Comput. Math. Vol. 5/3, pp. 313-347, 2005
Probabilistic approximation of the Gromov-Hausdorff distance.
- A. M. Bronstein, M. M. Bronstein, R. Kimmel, "Efficient computation of isometry-invariant distances between surfaces", SIAM J. Sci. Comp, Vol. 28/5, pp. 1812-1836, 2006.
Computation of the Gromov-Hausdorff distance using GMDS.
- F. Memoli, "On the Use of Gromov-Hausdorff Distances for Shape Comparison", Symposium on Point Based Graphics, 2007
Gromov-Wasserstein distance and relatation to EMD.
- F. Memoli, "Gromov-Hausdorff distances in Euclidean spaces", Proc. NORDIA, 2008
Equivalence between ICP and Gromov-Hausdorff distance.
- A. M. Bronstein, M. M. Bronstein, "Metric approaches to invariant shape similarity", chapter in Handbook of Mathematical Methods in Imaging (O. Scherzer Ed.), Springer
Introduction to invariant shape similarity, overviewing different approach from the metric perspective.
- Y. Lipman, T. Funkhouser, "Moebius voting for surface correspondence", Proc. ACM SIGGRAPH, 2009.
Invariant shape similarity and correspondence of disk-like shapes based on uniformization and parametrization of isometries as disk-to-disk Moebius transformations.
- A. M. Bronstein, M. M. Bronstein, A. M. Bruckstein, R. Kimmel, "Partial similarity of objects, or how to compare a centaur to a horse", IJCV, Vol. 84/2, pp. 163-183, 2009.
Introduction of the Pareto framework for partial shape similarity.
- A. M. Bronstein, M. M. Bronstein, Y. Carmon, R. Kimmel, "Partial similarity of shapes using a statistical significance measure", IPSJ Trans. Computer Vision and Application, Vol. 1, pp. 105-114, 2009.
Statistical weighting for the definition of part significance in isometry-invariant partial shape similarity problems.
- A. M. Bronstein, M. M. Bronstein, "Not only size matters: regularized partial matching of nonrigid shapes", Proc. NORDIA, 2008.
Regularization of parts in isometry-invariant partial shape similarity problems.
- N. J. Mitra, L. Guibas, M. Pauly, "Partial and approximate symmetry detection for 3D geometry", Proc. ACM SIGGRAPH, 2006.
Detection of full and partial extrinsic symmetry of shapes.
- N. J. Mitra, L. Guibas, M. Pauly, "Symmetrization", Proc. ACM SIGGRAPH, 2007.
Extrinsic symmetrization of shapes.
- D. Raviv, A. M. Bronstein, M. M. Bronstein, R. Kimmel, "Symmetries of non-rigid shapes", Proc. Workshop on Non-rigid Registration and Tracking through Learning (NRTL), 2007
Extension of the notion of symmetry to non-rigid shapes.
- D. Raviv, A. M. Bronstein, M. M. Bronstein, R. Kimmel, "Full and partiall symmetries of non-rigid shapes", IJCV, 2009 (submitted).
A more detailed exploration of intrinsic symmetries, including the definition of partial symmetry and approximations of the symmetry group.
- M. Ovsjanikov, J. Sun, L. Guibas, "Global Intrinsic Symmetries of Shapes", Computer Graphics Forum, Vol. 27(5), pp. 1341-1348, 2008.
Global intrinsic symmetries defined through the eignefunction of the Laplace-Beltrami operator.
- J. Sun, M. Ovsjanikov, L. Guibas, "A Concise and provably informative multi-scale signature based on heat diffusion", Proc. Eurographics Symposium on Geometry Processing (SGP), 2009.
Isometry-invariant and topology-insensitive descriptors based on heat kernels.
- M. Ovsjanikov, A. M. Bronstein, M. M. Bronstein, L. Guibas, "ShapeGoogle: a computer vision approach for invariant shape retrieval", Proc. NORDIA, 2009.
Isometry-invariant shape retrieval using bags-of-features and spatially-sensitive bags-of-features based on heat-kernels descriptors.
Expression-invariant face recognition
- A. M. Bronstein, M. M. Bronstein, R. Kimmel, "Three-dimensional face recognition", IJCV, Vol. 64/1, pp. 5-30, August 2005.
Expression-invariant face recognition based on canonical forms.
- A. M. Bronstein, M. M. Bronstein, R. Kimmel, "Calculus of non-rigid surfaces for geometry and texture manipulation", IEEE Trans. Visualization and Computer Graphics, Vol. 13/5, pp. 902-913.
Affine calculus of shapes based on intrinsic correspondence computed using GMDS.
- M. Kilian, N. J. Mitra, H. Pottmann, "Geometric modeling in shape space", Proc. ACM SIGGRAPH, 2007.
As-isometric-as possible morphing of shapes with given correspondence.
Shape reconstruction and inverse problems
- Y. S. Devir, G. Rosman, A. M. Bronstein, M. M. Bronstein, R. Kimmel, "Shape reconstruction with intrinsic priors", Techn. Report CIS-2009-03, Dept. of Computer Science, Technion, Israel, February 2009.
Inverse problems involving non-rigid shapes with intrinsic regularization.
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