Scattered data fitting with simplex splines in two and three dimensional spaces

We present a method for scattered data fitting in 2D and 3D euclidean spaces that uses simplex splines. The fitting process includes establishing the domain of the definition of a set of 2D or 3D data with the ahull method, segmentation for generating a proper initial triangulation, knot generation, least squares fitting with simplex splines, and adaptive domain subdivision. This technique has applications in scientific visualization and databases, computer vision, GIS, and CAD/CAM.

[1]  David C. Gossard,et al.  Reconstruction of smooth parametric surfaces from unorganized data points , 1992 .

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

[3]  C. Micchelli,et al.  Blossoming begets B -spline bases built better by B -patches , 1992 .

[4]  C. Micchelli,et al.  On the Linear Independence of Multivariate B-Splines, I. Triangulations of Simploids , 1982 .

[5]  D. Shepard.  A two-dimensional interpolation function for irregularly-spaced data , 1968, ACM National Conference.

[6]  Herbert Edelsbrunner,et al.  Three-dimensional alpha shapes , 1992, VVS.

[7]  Robert C. Beardsley,et al.  CTD observations off northern California during the Shelf Mixed Layer Experiment, SMILE, November 1988 , 1989 .

[8]  Gerald Farin,et al.  Triangular Bernstein-Bézier patches , 1986, Comput. Aided Geom. Des..

[9]  Rae A. Earnshaw,et al.  Computer Graphics: Developments in Virtual Environments , 1995, Computer Graphics.

[10]  H. Seidel,et al.  Simplex splines support surprisingly strong symmetric structures and subdivision , 1994 .

[11]  Nicholas M. Patrikalakis,et al.  Topologically reliable approximation of composite Bézier curves , 1996, Comput. Aided Geom. Des..

[12]  Martial Hebert,et al.  Energy functions for regularization algorithms , 1991, Optics & Photonics.

[13]  Hans-Peter Seidel,et al.  Fitting Triangular B‐Splines to Functional Scattered Data , 1996, Comput. Graph. Forum.

[14]  C. Micchelli,et al.  On multivariate -splines , 1989 .

[15]  Hong Qin,et al.  Dynamic manipulation of triangular B-splines , 1995, Symposium on Solid Modeling and Applications.

[16]  Thomas A. Grandine,et al.  The computational cost of simplex spline functions , 1987 .

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

[18]  I. K Crain,et al.  Treatment of non-equispaced two-dimensional data with a digital computer , 1967 .

[19]  Hans-Peter Seidel,et al.  Modeling with triangular B-splines , 1993, IEEE Computer Graphics and Applications.

[20]  Robert E. Barnhill,et al.  Representation and Approximation of Surfaces , 1977 .

[21]  G. P. Cressman.  AN OPERATIONAL OBJECTIVE ANALYSIS SYSTEM , 1959 .

[22]  G. Nielson.  A method for interpolating scattered data based upon a minimum norm network , 1983 .

[23]  R. L. Hardy.  Multiquadric equations of topography and other irregular surfaces , 1971 .

[24]  Hans-Peter Seidel,et al.  An implementation of triangular B-spline surfaces over arbitrary triangulations , 1993, Comput. Aided Geom. Des..

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

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

[27]  Malcolm A. Sabin,et al.  Piecewise Quadratic Approximations on Triangles , 1977, TOMS.

[28]  George Celniker,et al.  Deformable curve and surface finite-elements for free-form shape design , 1991, SIGGRAPH.

[29]  Hans Hagen,et al.  Research issues in data modeling for scientific visualization , 1994, IEEE Computer Graphics and Applications.

[30]  Tony DeRose,et al.  Piecewise smooth surface reconstruction , 1994, SIGGRAPH.

[31]  C. R. Traas,et al.  Practice of Bivariate Quadratic Simplicial Splines , 1990 .

[32]  Nicholas M. Patrikalakis,et al.  Reliable Interrogation of 3D Non-linear Geophysical Databases , 1995, Computer Graphics.

[33]  L. Schumaker.  Fitting surfaces to scattered data , 1976 .

[34]  R. Franke.  Scattered data interpolation: tests of some methods , 1982 .

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

[36]  Hans-Peter Seidel,et al.  Control Points for Multivariate B‐Spline Surfaces over Arbitrary Triangulations , 1991, Comput. Graph. Forum.

[37]  Demetri Terzopoulos,et al.  Sampling and reconstruction with adaptive meshes , 1991, Proceedings. 1991 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[38]  P. Dierckx.  An algorithm for surface-fitting with spline functions , 1981 .

[39]  Marian Neamtu,et al.  Approximation and geometric modeling with simplex B-splines associated with irregular triangles , 1991, Comput. Aided Geom. Des..

[40]  Tony DeRose,et al.  Generalized B-spline surfaces of arbitrary topology , 1990, SIGGRAPH.

[41]  Jingfang Zhou,et al.  Interval simplex splines for scientific databases , 1995 .

[42]  Hong Qin,et al.  Dynamic NURBS with geometric constraints for interactive sculpting , 1994, TOGS.