Meric geometry

• 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.



Differential 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).



Discrete geometry

• 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 ", 2006
  Linear-complexity approximation of heap-based fast-marching based on distance map quantization. Idea dates back to Tsitsiklis' 2005 paper.



Multidimensional scaling

• 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.
  Generalized MDS.



• N. Thorstensen, R. Keriven, "Non-rigid shape matching using geometry and photometry", Proc. ACCV, 2009
  Extension of GMDS incorporating photometric information.




Spectral methods

• 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.




Diffusion geometry

• 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.




Partial similarity

• 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.




Symmetry

• 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.




Feature-based similarity

• 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.




Shape synthesis

• 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.

We do our best to maintain this annotated and categorized bibliography as comprehensive as possible.
If, nevertheless, we have omitted an important reference, please let us know at bronstein@ieee.org.