An Optimal Transport Approach to Robust Reconstruction and Simplification of 2d Shapes

We propose a robust 2D shape reconstruction and simplification algorithm which takes as input a defect‐laden point set with noise and outliers. We introduce an optimal‐transport driven approach where the input point set, considered as a sum of Dirac measures, is approximated by a simplicial complex considered as a sum of uniform measures on 0‐ and 1‐simplices. A fine‐to‐coarse scheme is devised to construct the resulting simplicial complex through greedy decimation of a Delaunay triangulation of the input point set. Our method performs well on a variety of examples ranging from line drawings to grayscale images, with or without noise, features, and boundaries.

[1]  Stanley S. Ipson,et al.  A novel triangulation procedure for thinning hand-written text , 2001, Pattern Recognit. Lett..

[2]  Pierre Alliez,et al.  Computational geometry algorithms library , 2008, SIGGRAPH '08.

[3]  Yuqing Song,et al.  Boundary fitting for 2D curve reconstruction , 2010, The Visual Computer.

[4]  Michael Ian Shamos,et al.  Computational geometry: an introduction , 1985 .

[5]  Pierre Alliez,et al.  Signing the Unsigned: Robust Surface Reconstruction from Raw Pointsets , 2010, Comput. Graph. Forum.

[6]  Pierre Alliez,et al.  Eurographics Symposium on Geometry Processing (2007) Voronoi-based Variational Reconstruction of Unoriented Point Sets , 2022 .

[7]  Hans-Peter Seidel,et al.  MovieReshape: tracking and reshaping of humans in videos , 2010, SIGGRAPH 2010.

[8]  Bo Li,et al.  Optimal transportation meshfree approximation schemes for fluid and plastic flows , 2010 .

[9]  In-Kwon Lee,et al.  Curve reconstruction from unorganized points , 2000, Comput. Aided Geom. Des..

[10]  Carlos A. Vanegas,et al.  Ellipse-based principal component analysis for self-intersecting curve reconstruction from noisy point sets , 2010, The Visual Computer.

[11]  Frédéric Chazal,et al.  Geometric Inference for Measures based on Distance Functions , 2011 .

[12]  Stefan Funke,et al.  Reconstructing a collection of curves with corners and endpoints , 2001, SODA '01.

[13]  Michael Balzer,et al.  Capacity-constrained point distributions: a variant of Lloyd's method , 2009, ACM Trans. Graph..

[14]  Tamal K. Dey,et al.  A simple provable algorithm for curve reconstruction , 1999, SODA '99.

[15]  Asish Mukhopadhyay,et al.  Curve reconstruction in the presence of noise , 2007, Computer Graphics, Imaging and Visualisation (CGIV 2007).

[16]  D. Cohen-Or,et al.  Robust moving least-squares fitting with sharp features , 2005, ACM Trans. Graph..

[17]  Nabil H. Mustafa,et al.  Near-Linear Time Approximation Algorithms for Curve Simplification , 2005, Algorithmica.

[18]  Jack Snoeyink,et al.  A One-Step Crust and Skeleton Extraction Algorithm , 2001, Algorithmica.

[19]  Dominique Attali,et al.  r-regular shape reconstruction from unorganized points , 1997, SCG '97.

[20]  David Eppstein,et al.  The Crust and the beta-Skeleton: Combinatorial Curve Reconstruction , 1998, Graph. Model. Image Process..

[21]  Niloy J. Mitra,et al.  Visibility of noisy point cloud data , 2010, Comput. Graph..

[22]  Stefan Funke,et al.  Curve reconstruction from noisy samples , 2003, SCG '03.

[23]  C. Villani Topics in Optimal Transportation , 2003 .

[24]  Kurt Mehlhorn,et al.  Curve reconstruction: Connecting dots with good reason , 2000, Comput. Geom..

[25]  Yuan Zhou,et al.  Quadric-based simplification in any dimension , 2005, TOGS.

[26]  Michael M. Kazhdan,et al.  Poisson surface reconstruction , 2006, SGP '06.

[27]  Leif Kobbelt,et al.  Fast Mesh Decimation by Multiple-Choice Techniques , 2002, VMV.

[28]  Mathieu Desbrun,et al.  HOT: Hodge-optimized triangulations , 2011, ACM Trans. Graph..

[29]  I. Daubechies,et al.  Conformal Wasserstein distances: Comparing surfaces in polynomial time , 2011, 1103.4408.

[30]  Facundo Mémoli,et al.  Gromov–Wasserstein Distances and the Metric Approach to Object Matching , 2011, Found. Comput. Math..

[31]  Cohen-OrDaniel,et al.  ℓ1-Sparse reconstruction of sharp point set surfaces , 2010 .

[32]  Xiaofeng Mi,et al.  Abstraction of 2D shapes in terms of parts , 2009, NPAR '09.

[33]  Tamal K. Dey,et al.  Fast Reconstruction of Curves with Sharp Corners , 2002, Int. J. Comput. Geom. Appl..

[34]  Frédéric Chazal,et al.  Geometric Inference for Probability Measures , 2011, Found. Comput. Math..

[35]  Daniel Cohen-Or,et al.  ℓ1-Sparse reconstruction of sharp point set surfaces , 2010, TOGS.

[36]  Kurt Mehlhorn,et al.  Curve reconstruction: connecting dots with good reason , 1999, SCG '99.

[37]  Christopher Dyken,et al.  Simultaneous curve simplification , 2009, J. Geogr. Syst..

[38]  Julie Delon,et al.  Fast Transport Optimization for Monge Costs on the Circle , 2009, SIAM J. Appl. Math..

[39]  John Hershberger,et al.  An O(nlogn) implementation of the Douglas-Peucker algorithm for line simplification , 1994, SCG '94.

[40]  Mathieu Desbrun,et al.  HOT: Hodge-optimized triangulations , 2011, SIGGRAPH 2011.

[41]  David G. Kirkpatrick,et al.  On the shape of a set of points in the plane , 1983, IEEE Trans. Inf. Theory.

[42]  James F. O'Brien,et al.  Spectral surface reconstruction from noisy point clouds , 2004, SGP '04.

[43]  Leonidas J. Guibas,et al.  The Earth Mover's Distance as a Metric for Image Retrieval , 2000, International Journal of Computer Vision.

[44]  Mathieu Desbrun,et al.  Exoskeleton: Curve network abstraction for 3D shapes , 2011, Comput. Graph..

[45]  Herbert Edelsbrunner,et al.  Topology preserving edge contraction , 1998 .

[46]  Valentin Polishchuk,et al.  Robust curve reconstruction with k-order alpha-shapes , 2008, Shape Modeling International.