Sampling and Reconstruction of Surfaces and Higher Dimensional Manifolds

Abstract We present new sampling theorems for surfaces and higher dimensional manifolds. The core of the proofs resides in triangulation results for manifolds with boundary, not necessarily bounded. The method is based upon geometric considerations that are further augmented for 2-dimensional manifolds (i.e surfaces). In addition, we show how to apply the main results to obtain a new, geometric proof of the classical Shannon sampling theorem, and also to image analysis.

[1]  Takis Sakkalis,et al.  Approximating Curves via Alpha Shapes , 1999, Graph. Model. Image Process..

[2]  J. R. Higgins,et al.  The Sampling Theorem and Several Equivalent Results in Analysis , 2000 .

[3]  Tamal K. Dey,et al.  Manifold reconstruction from point samples , 2005, SODA '05.

[4]  J. Munkres.  Obstructions to the smoothing of piecewise-differentiable homeomorphisms , 1959 .

[5]  S. S. Cairns,et al.  A simple triangulation method for smooth manifolds , 1961 .

[6]  A. Papoulis,et al.  Generalized sampling expansion , 1977 .

[7]  Alexander M. Bronstein,et al.  Isometric Embedding of Facial Surfaces into , 2005, Scale-Space.

[8]  Jack Snoeyink,et al.  Theory and practice of sampling and reconstruction for manifolds with boundaries , 2001 .

[9]  S T Roweis,et al.  Nonlinear dimensionality reduction by locally linear embedding. , 2000, Science.

[10]  W. Thurston,et al.  Three-Dimensional Geometry and Topology, Volume 1 , 1997, The Mathematical Gazette.

[11]  Herbert Edelsbrunner,et al.  Geometry and Topology for Mesh Generation , 2001, Cambridge monographs on applied and computational mathematics.

[12]  D. Burago,et al.  A Course in Metric Geometry , 2001 .

[13]  Herbert Edelsbrunner,et al.  Geometry and Topology for Mesh Generation , 2001, Cambridge monographs on applied and computational mathematics.

[14]  Peter W. Hallinan.  A low-dimensional representation of human faces for arbitrary lighting conditions , 1994, 1994 Proceedings of IEEE Conference on Computer Vision and Pattern Recognition.

[15]  H. Feichtinger,et al.  Exact iterative reconstruction algorithm for multivariate irregularly sampled functions in spline-like spaces: The $L^p$-theory , 1998 .

[16]  M. Gromov,et al.  Partial Differential Relations , 1986 .

[17]  Yehoshua Y. Zeevi,et al.  Local versus Global in Quasi-Conformal Mapping for Medical Imaging , 2008, Journal of Mathematical Imaging and Vision.

[18]  Alexander M. Bronstein,et al.  Three-Dimensional Face Recognition , 2005, International Journal of Computer Vision.

[19]  S. S. Cairns.  On the Triangulation of Regular Loci , 1934 .

[20]  David Eppstein,et al.  Dihedral bounds for mesh generation in high dimensions , 1995, SODA '95.

[21]  M. Gromov,et al.  Embeddings and immersions in Riemannian geometry , 1970 .

[22]  J. Pach,et al.  Combinatorial geometry , 1995, Wiley-Interscience series in discrete mathematics and optimization.

[23]  Yehoshua Y. Zeevi,et al.  Two-dimensional sampling and representation of folded surfaces embedded in higher dimensional manifolds , 2006, 2006 14th European Signal Processing Conference.

[24]  Robert Schrader,et al.  On the curvature of piecewise flat spaces , 1984 .

[25]  J. Nash.  The imbedding problem for Riemannian manifolds , 1956 .

[26]  E. Moise.  Geometric Topology in Dimensions 2 and 3 , 1977 .

[27]  Akram Aldroubi,et al.  Nonuniform Sampling and Reconstruction in Shift-Invariant Spaces , 2001, SIAM Rev..

[28]  Robert J. Marks,et al.  Differintegral interpolation from a bandlimited signal's samples , 1981 .

[29]  J. Munkres,et al.  Elementary Differential Topology. , 1967 .

[30]  N. Aronszajn.  Theory of Reproducing Kernels. , 1950 .

[31]  Tamal K. Dey,et al.  Sampling and meshing a surface with guaranteed topology and geometry , 2004, SCG '04.

[32]  S. S. Cairns.  Polyhedral Approximations to Regular Loci , 1936 .

[33]  Yehoshua Y. Zeevi,et al.  Nonuniform sampling and antialiasing in image representation , 1993, IEEE Trans. Signal Process..

[34]  M. Berger.  A Panoramic View of Riemannian Geometry , 2003 .

[35]  J. Zerubia,et al.  A Generalized Sampling Theory without bandlimiting constraints , 1998 .

[36]  Tamal K. Dey,et al.  Anisotropic surface meshing , 2006, SODA '06.

[37]  A. Zapadinskaya.  MODULUS OF CONTINUITY FOR QUASIREGULAR MAPPINGS WITH FINITE DISTORTION EXTENSION , 2008 .

[38]  H. Sebastian Seung,et al.  The Manifold Ways of Perception , 2000, Science.

[39]  János Pach,et al.  Combinatorial Geometry , 2012 .

[40]  Emil Saucan.  Note on a Theorem of Munkres , 2004 .

[41]  W. Kühnel,et al.  Smooth approximation of polyhedral surfaces regarding curvatures , 1982 .

[42]  M. Spivak.  A comprehensive introduction to differential geometry , 1979 .

[43]  W. Thurston,et al.  Three-Dimensional Geometry and Topology, Volume 1 , 1997, The Mathematical Gazette.

[44]  B. Andrews,et al.  Notes on the isometric embedding problem and the Nash-Moser implicit function theorem , 2002 .

[45]  S. Smale,et al.  Shannon sampling and function reconstruction from point values , 2004 .

[46]  Ron Kimmel,et al.  Images as Embedded Maps and Minimal Surfaces: Movies, Color, Texture, and Volumetric Medical Images , 2000, International Journal of Computer Vision.

[47]  Manfredo P. do Carmo,et al.  Differential geometry of curves and surfaces , 1976 .

[48]  Emil Saucan.  The Existence of Quasimeromorphic Mappings in Dimension 3 , 2003 .

[49]  Xiang-Yang Li,et al.  Generating well-shaped Delaunay meshed in 3D , 2001, SODA '01.

[50]  M. M. Dodson,et al.  The Whittaker–Kotel’nikov– Shannon Theorem, Spectral Translates and Plancherel’s Formula , 2004 .

[51]  Xiang-Yang Li.  Generating Well-Shaped d-dimensional Delaunay Meshes , 2001, COCOON.

[52]  W. Thurston,et al.  Three-Dimensional Geometry and Topology, Volume 1: Volume 1 , 1997 .

[53]  I. Holopainen.  Riemannian Geometry , 1927, Nature.