B-spline surface fitting by iterative geometric interpolation/approximation algorithms

Recently, the use of B-spline curves/surfaces to fit point clouds by iteratively repositioning the B-spline's control points on the basis of geometrical rules has gained in popularity because of its simplicity, scalability, and generality. We distinguish between two types of fitting, interpolation and approximation. Interpolation generates a B-spline surface that passes through the data points, whereas approximation generates a B-spline surface that passes near the data points, minimizing the deviation of the surface from the data points. For surface interpolation, the data points are assumed to be in grids, whereas for surface approximation the data points are assumed to be randomly distributed. In this paper, an iterative geometric interpolation method, as well as an approximation method, which is based on the framework of the iterative geometric interpolation algorithm, is discussed. These two iterative methods are compared with standard fitting methods using some complex examples, and the advantages and shortcomings of our algorithms are discussed. Furthermore, we introduce two methods to accelerate the iterative geometric interpolation algorithm, as well as a method to impose geometric constraints, such as reflectional symmetry, on the iterative geometric interpolation process, and a novel fairing method for non-uniform complex data points. Complex examples are provided to demonstrate the effectiveness of the proposed algorithms.

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

[2]  Weiyin Ma,et al.  Parameterization of randomly measured points for least squares fitting of B-spline curves and surfaces , 1995, Comput. Aided Des..

[3]  David H. Douglas,et al.  ALGORITHMS FOR THE REDUCTION OF THE NUMBER OF POINTS REQUIRED TO REPRESENT A DIGITIZED LINE OR ITS CARICATURE , 1973 .

[4]  Shi-Min Hu,et al.  Direct manipulation of FFD: efficient explicit solutions and decomposible multiple point constraints , 2001, Vis. Comput..

[5]  Hongwei Lin,et al.  An extended iterative format for the progressive-iteration approximation , 2011, Comput. Graph..

[6]  Yinglin,et al.  RECONSTRUCTION OF SYMMETRIC B-SPLINE CURVES AND SURFACES , 2007 .

[7]  Ahmad H. Nasri,et al.  Generating B-spline curves with points, normals and curvature constraints: a constructive approach , 2010, The Visual Computer.

[8]  Gerald Farin,et al.  Curves and surfaces for computer aided geometric design , 1990 .

[9]  Shigefumi Tamura,et al.  Point-tangent/point-normal B-spline curve interpolation by geometric algorithms , 2009, Comput. Aided Des..

[10]  Fengtao Fan,et al.  Subdivision Based Interpolation with Shape Control , 2008 .

[11]  William H. Press,et al.  Numerical recipes in C , 2002 .

[12]  Wolfgang Böhm,et al.  A survey of curve and surface methods in CAGD , 1984, Comput. Aided Geom. Des..

[13]  Nicholas M. Patrikalakis,et al.  Shape Interrogation for Computer Aided Design and Manufacturing , 2002, Springer Berlin Heidelberg.

[14]  Ulrich Dietz.  Fair surface reconstruction from point clouds , 1998 .

[15]  Gábor Renner,et al.  Advanced surface fitting techniques , 2002, Comput. Aided Geom. Des..

[16]  Hongwei Lin,et al.  Uniform B-Spline Curve Interpolation with Prescribed Tangent and Curvature Vectors , 2012, IEEE Transactions on Visualization and Computer Graphics.

[17]  Lizheng Lu,et al.  Weighted progressive iteration approximation and convergence analysis , 2010, Comput. Aided Geom. Des..

[18]  Masayuki Morioka,et al.  Loop Subdivision Surface Fitting by Geometric Algorithms , 2008 .

[19]  Timothy A. Davis,et al.  Direct methods for sparse linear systems , 2006, Fundamentals of algorithms.

[20]  David Bommes,et al.  Efficient Linear System Solvers for Mesh Processing , 2005, IMA Conference on the Mathematics of Surfaces.

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

[22]  Phillip N. Azariadis.  Parameterization of clouds of unorganized points using dynamic base surfaces , 2004, Comput. Aided Des..

[23]  Takashi Maekawa,et al.  Interpolation by geometric algorithm , 2007, Comput. Aided Des..

[24]  Shi-Min Hu,et al.  A second order algorithm for orthogonal projection onto curves and surfaces , 2005, Comput. Aided Geom. Des..

[25]  Hongwei Lin.  The convergence of the geometric interpolation algorithm , 2010, Comput. Aided Des..

[26]  L. Piegl,et al.  The NURBS Book , 1995, Monographs in Visual Communications.

[27]  Josef Hoschek,et al.  Fundamentals of computer aided geometric design , 1996 .

[28]  Bert Jüttler,et al.  Techniques for fair- and shape-preserving surface fitting with tensor-product-B-splines , 1999 .

[29]  Martin Reimers,et al.  Meshless parameterization and surface reconstruction , 2001, Comput. Aided Geom. Des..

[30]  Fujio Yamaguchi,et al.  Curves and Surfaces in Computer Aided Geometric Design , 1988, Springer Berlin Heidelberg.

[31]  Marc Alexa,et al.  Computing and Rendering Point Set Surfaces , 2003, IEEE Trans. Vis. Comput. Graph..