Visibility of noisy point cloud data

We present a robust algorithm for estimating visibility from a given viewpoint for a point set containing concavities, non-uniformly spaced samples, and possibly corrupted with noise. Instead of performing an explicit surface reconstruction for the points set, visibility is computed based on a construction involving convex hull in a dual space, an idea inspired by the work of Katz et al. [26]. We derive theoretical bounds on the behavior of the method in the presence of noise and concavities, and use the derivations to develop a robust visibility estimation algorithm. In addition, computing visibility from a set of adaptively placed viewpoints allows us to generate locally consistent partial reconstructions. Using a graph based approximation algorithm we couple such reconstructions to extract globally consistent reconstructions. We test our method on a variety of 2D and 3D point sets of varying complexity and noise content.

[1]  Tony DeRose,et al.  Surface reconstruction from unorganized points , 1992, SIGGRAPH.

[2]  Michael Hoffmann,et al.  The Euclidean degree-4 minimum spanning tree problem is NP-hard , 2009, SCG '09.

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

[4]  Ronen Basri,et al.  Direct visibility of point sets , 2007, ACM Trans. Graph..

[5]  Leonidas J. Guibas,et al.  Uncertainty and Variability in Point Cloud Surface Data , 2004, PBG.

[6]  Nina Amenta,et al.  Defining point-set surfaces , 2004, ACM Trans. Graph..

[7]  Peter Wonka,et al.  Visibility in Computer Graphics , 2003 .

[8]  Heinrich Müller,et al.  Improved Laplacian Smoothing of Noisy Surface Meshes , 1999, Comput. Graph. Forum.

[9]  Markus H. Gross,et al.  Point-based multiscale surface representation , 2006, TOGS.

[10]  David Levin,et al.  The approximation power of moving least-squares , 1998, Math. Comput..

[11]  Tamal K. Dey,et al.  Provable surface reconstruction from noisy samples , 2006, Comput. Geom..

[12]  Raymond E. Miller,et al.  Complexity of Computer Computations , 1972 .

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

[14]  Leonidas J. Guibas,et al.  Dynamic geometry registration , 2007, Symposium on Geometry Processing.

[15]  Tamal K. Dey,et al.  An Adaptive MLS Surface for Reconstruction with Guarantees , 2022 .

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

[17]  Arthur Appel,et al.  Some techniques for shading machine renderings of solids , 1968, AFIPS Spring Joint Computing Conference.

[18]  Sunil Arya,et al.  Approximate nearest neighbor queries in fixed dimensions , 1993, SODA '93.

[19]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[20]  Markus Gross,et al.  Point-Based Graphics , 2007 .

[21]  Tamal K. Dey,et al.  Tight cocone: a water-tight surface reconstructor , 2003, SM '03.

[22]  Luiz Velho,et al.  Surface reconstruction from noisy point clouds , 2005, SGP '05.

[23]  Daniel Cohen-Or,et al.  Consolidation of unorganized point clouds for surface reconstruction , 2009, ACM Trans. Graph..

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

[25]  Matthew J. Sottile,et al.  Curve and surface reconstruction: algorithms with mathematical analysis by Tamal K. Dey Cambridge University Press , 2010, SIGA.

[26]  Steve Marschner,et al.  Filling holes in complex surfaces using volumetric diffusion , 2002, Proceedings. First International Symposium on 3D Data Processing Visualization and Transmission.

[27]  Sunghee Choi,et al.  The power crust, unions of balls, and the medial axis transform , 2001, Comput. Geom..

[28]  Tamal K. Dey,et al.  Detecting undersampling in surface reconstruction , 2001, SCG '01.

[29]  Marc Alexa,et al.  Point set surfaces , 2001, Proceedings Visualization, 2001. VIS '01..

[30]  Leonidas J. Guibas,et al.  Estimating surface normals in noisy point cloud data , 2004, Int. J. Comput. Geom. Appl..

[31]  Edwin Earl Catmull,et al.  A subdivision algorithm for computer display of curved surfaces. , 1974 .