Chapter 5: Smooth Surface Reconstruction Using Doo-Sabin Subdivision Surfaces

A new technique for the reconstruction of a smooth surface from a set of 3D data points is presented. The reconstructed surface is represented by an everywhere C1-continuous subdivision surface which interpolates all the given data points. The new technique consists of two major steps. First, an efficient surface reconstruction method is applied to produce a polyhedral approximation to the given data set M. A Doo-Sabin subdivision surface that smoothly passes through all the points in the given data set M is then constructed. The Doo-Sabin subdivision surface is constructed by iteratively modifying the vertices of the polyhedral approximation until a new control mesh Mmacr, whose Doo-Sabin subdivision surface interpolates M, is reached. This iterative process converges for meshes of any size and any topology. Therefore the surface reconstruction processes well-defined. The new technique has the advantages of both a local method and a global method, and the surface reconstruction process can reproduce special features such as edges and corners faithfully.

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

[2]  Juan Manuel Peña,et al.  Progressive iterative approximation and bases with the fastest convergence rates , 2007, Comput. Aided Geom. Des..

[3]  Peter Schröder,et al.  Fitting subdivision surfaces , 2001, Proceedings Visualization, 2001. VIS '01..

[4]  Thomas A. Funkhouser,et al.  The Princeton Shape Benchmark (Figures 1 and 2) , 2004, Shape Modeling International Conference.

[5]  N. Dyn,et al.  A butterfly subdivision scheme for surface interpolation with tension control , 1990, TOGS.

[6]  G. Farin.  Curves and Surfaces for Cagd: A Practical Guide , 2001 .

[7]  Malcolm A. Sabin,et al.  Behaviour of recursive division surfaces near extraordinary points , 1998 .

[8]  Ahmad H. Nasri,et al.  Surface interpolation on irregular networks with normal conditions , 1991, Comput. Aided Geom. Des..

[9]  H. Bao,et al.  Totally positive bases and progressive iteration approximation , 2005 .

[10]  Thomas A. Funkhouser,et al.  The Princeton Shape Benchmark , 2004, Proceedings Shape Modeling Applications, 2004..

[11]  David Cohen-Steiner,et al.  A greedy Delaunay-based surface reconstruction algorithm , 2004, The Visual Computer.

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

[13]  Jianmin Zheng,et al.  Interpolation over arbitrary topology meshes using a two-phase subdivision scheme , 2006, IEEE Transactions on Visualization and Computer Graphics.

[14]  Charles T. Loop,et al.  Smooth Subdivision Surfaces Based on Triangles , 1987 .

[15]  E. Catmull,et al.  Recursively generated B-spline surfaces on arbitrary topological meshes , 1978 .

[16]  Jianmin Zheng,et al.  Making Doo-Sabin surface interpolation always work over irregular meshes , 2005, The Visual Computer.

[17]  Tony DeRose,et al.  Efficient, fair interpolation using Catmull-Clark surfaces , 1993, SIGGRAPH.

[18]  Leif Kobbelt,et al.  Interpolatory Subdivision on Open Quadrilateral Nets with Arbitrary Topology , 1996, Comput. Graph. Forum.

[19]  C. D. Boor.  How does Agee ’ s smoothing method work ? , .

[20]  Guojin Wang,et al.  Constructing iterative non-uniform B-spline curve and surface to fit data points , 2004, Science in China Series : Information Sciences.

[21]  Deming Li.  The Star Chromatic Numbers of Some Planar Graphs Derived from Wheels , 2002 .

[22]  Fuhua Cheng,et al.  Similarity based interpolation using Catmull–Clark subdivision surfaces , 2006, The Visual Computer.

[23]  J. Magnus,et al.  Matrix Differential Calculus with Applications in Statistics and Econometrics , 1991 .

[24]  Peter Schröder,et al.  Interpolating Subdivision for meshes with arbitrary topology , 1996, SIGGRAPH.