Kd-Tree Based OLS in Implicit Surface Reconstruction with Radial Basis Function

In this paper, we propose a new method for surface reconstruction from scattered point set based on least square radial basis function (LSRBF) and orthogonal least square forward selection procedure. Firstly, the traditional RBF formulation is rewritten into least square formula. A implicit surface can be represented with fewer centers. Then, the orthogonal least square procedure is utilized to select significant centers from original point data set. The RBF coefficients can be solved from the triangular matrix from OLS selection through backward substitution method. So, this scheme can offer a fast surface reconstruction tool and can overcome the numerical ill-conditioning of coefficient matrix and over-fitting problem. Some examples are presented to show the effectiveness of our algorithm in 2D and 3D cases.

[1]  Kalpathi R. Subramanian,et al.  Interpolating implicit surfaces from scattered surface data using compactly supported radial basis functions , 2001, Proceedings International Conference on Shape Modeling and Applications.

[2]  Patrick Reuter,et al.  Efficient Reconstruction of Large Scattered Geometric Datasets using the Partition of Unity and Radial Basis Functions , 2004, WSCG.

[3]  I. Babuska,et al.  The Partition of Unity Method , 1997 .

[4]  James F. O'Brien,et al.  Variational Implicit Surfaces , 1999 .

[5]  Sheng Chen,et al.  Orthogonal least squares methods and their application to non-linear system identification , 1989 .

[6]  Chandrajit L. Bajaj,et al.  Sampling and reconstructing manifolds using alpha-shapes , 1997, CCCG.

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

[8]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

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

[10]  Patrick Reuter,et al.  Multi-scale reconstruction of implicit surfaces with attributes from large unorganized point sets , 2004, Proceedings Shape Modeling Applications, 2004..

[11]  Marshall W. Bern,et al.  A new Voronoi-based surface reconstruction algorithm , 1998, SIGGRAPH.

[12]  Leif Kobbelt,et al.  A Stream Algorithm for the Decimation of Massive Meshes , 2003, Graphics Interface.

[13]  Paolo Cignoni,et al.  Metro: Measuring Error on Simplified Surfaces , 1998, Comput. Graph. Forum.

[14]  Hans-Peter Seidel,et al.  Multi-level partition of unity implicits , 2005, SIGGRAPH Courses.

[15]  Patrick Reuter,et al.  Multiresolution Reconstruction of Implicit Surfaces with Attributes from Large Unorganized Point Sets , 2004 .

[16]  Richard K. Beatson,et al.  Reconstruction and representation of 3D objects with radial basis functions , 2001, SIGGRAPH.

[17]  James F. O'Brien,et al.  Modelling with implicit surfaces that interpolate , 2002, TOGS.

[18]  Jooyoung Park,et al.  Universal Approximation Using Radial-Basis-Function Networks , 1991, Neural Computation.

[19]  Sunghee Choi,et al.  The power crust , 2001, SMA '01.

[20]  S. Osher,et al.  Fast surface reconstruction using the level set method , 2001, Proceedings IEEE Workshop on Variational and Level Set Methods in Computer Vision.

[21]  James F. O'Brien,et al.  Interpolating and approximating implicit surfaces from polygon soup , 2005, SIGGRAPH 2005.

[22]  Shang-Liang Chen,et al.  Orthogonal least squares learning algorithm for radial basis function networks , 1991, IEEE Trans. Neural Networks.