Publication 2000 Issue No. 6 - June Abstract - Ordering and Parameterizing Scattered 3D Data for B-Spline Surface Approximation
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Ordering and Parameterizing Scattered 3D Data for B-Spline Surface Approximation
June 2000 (vol. 22 no. 6)
pp. 642-648
 ASCII Text x Fernand S. Cohen, Walid Ibrahim, Chuchart Pintavirooj, "Ordering and Parameterizing Scattered 3D Data for B-Spline Surface Approximation," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 22, no. 6, pp. 642-648, June, 2000.
 BibTex x @article{ 10.1109/34.862203,author = {Fernand S. Cohen and Walid Ibrahim and Chuchart Pintavirooj},title = {Ordering and Parameterizing Scattered 3D Data for B-Spline Surface Approximation},journal ={IEEE Transactions on Pattern Analysis and Machine Intelligence},volume = {22},number = {6},issn = {0162-8828},year = {2000},pages = {642-648},doi = {http://doi.ieeecomputersociety.org/10.1109/34.862203},publisher = {IEEE Computer Society},address = {Los Alamitos, CA, USA},}
 RefWorks Procite/RefMan/Endnote x TY - JOURJO - IEEE Transactions on Pattern Analysis and Machine IntelligenceTI - Ordering and Parameterizing Scattered 3D Data for B-Spline Surface ApproximationIS - 6SN - 0162-8828SP642EP648EPD - 642-648A1 - Fernand S. Cohen, A1 - Walid Ibrahim, A1 - Chuchart Pintavirooj, PY - 2000KW - Surface fittingKW - B-SplineKW - Gaussian mapKW - geodesicsKW - NURBS.VL - 22JA - IEEE Transactions on Pattern Analysis and Machine IntelligenceER -

Abstract—Surface representation is intrinsic to many applications in medical imaging, computer vision, and computer graphics. We present a method that is based on surface modeling by B-Spline. The B-Spline constructs a smooth surface that best fits a set of scattered unordered 3D range data points obtained from either a structured light system (a range finder), or from point coordinates on the external contours of a set of surface sections, as for example in histological coronal brain sections. B-Spline stands as of one the most efficient surface representations. It possesses many properties such as boundedness, continuity, local shape controllability, and invariance to affine transformations that makes it very suitable and attractive for surface representation. Despite its attractive properties, however, B-Spline has not been widely applied for representing a 3D scattered nonordered data set. This may be due to the problem in finding an ordering and a choice for the topological parameters of the B-Spline that lead to a physically meaningful surface parameterization based on the scattered data set. The parameters needed for the B-Spline surface construction, as well as finding the ordering of the data points, are calculated based on the geodesics of the surface extended Gaussian map. The set of control points is analytically calculated by solving a minimum mean square error problem for best surface fitting. For a noise immune modeling, we elect to use an approximating rather than an interpolating B-Spline. We also examine ways of making the B-Spline fitting technique robust to local deformation and noise.

[1] R.H. Bartels, J.C. Beatty, and B.A. Barsky, An Introduction to Splines for Use in Computer Graphics and Geometric Modeling, Morgan Kaufmann, Los Altos, Calif., 1987.
[2] L. Piegel, “On NURBS: A Survey,” IEEE Trans. Computer Graphics and Applications, vol. 11, no. 5, 1991.
[3] C. De Boor, A Practical Guide to Splines. New York: Springer-Verlag, 1978.
[4] F. Yamaguchi, Curves and Surfaces in Computer Aided Geometric Design. Berlin: Springer-Verlag, 1988.
[5] F.S. Cohen and J.-Y. Wang, “Modeling Image Curve Using Invariant 3-D Object Curve Models—A Path to 3D Recognition and Shape Estimation from Image Contours,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 16, no. 1, pp. 1-12, Jan. 1994.
[6] J.-Y. Wang and F.S. Cohen, “3D Object Recognition and Shape Estimation from Image Contours Using B-Splines, Shape Invariant Matching, and Neural Network,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 16, no. 1, pp. 13-21, Jan. 1994.
[7] F.S. Cohen, Z. Huang, and Z. Yang, “Invariant Matching and Identification of Curves Using B-Splines Curve Representation,” IEEE Trans. Image Processing, vol. 4, no. 1, Jan. 1995.
[8] Z. Huang and F.S. Cohen, “Affine-Invariant B-Spline Moments for Curve Matching,” IEEE Trans. Image Processing, vol. 5, no. 10, pp. 1,473-1,480, Oct. 1996.
[9] L. Piegl and W. Tiller, The NURBS Book.New York: Springer-Verlag, 1995.
[10] M. Milroy, C. Bradley, G. Vickers, and D. Weir, “${\rm{G}}^1$Continuity of the B-Spline Surface Patches in Reverse Engineering,” Compter Aided Design, vol. 27, pp. 471-478, 1995.
[11] W. Ma and J. Kruth, “Parameterization of Randomly Measured Points for Least Squares Fitting of B-Splines and Surfaces,” Computer Aided Design, vol. 27, pp. 663-675, 1995.
[12] V. Krishnamurthy and M. Levoy, “Fitting Smooth Surfaces to Dense Polygon Meshes,” Proc. SIGGRAPH '96, pp. 313-324, Aug. 1996.
[13] E. Andersson, R. Andersson, M. Boman, T. Elmroth, B. Dahlberg, and B. Johansson, “Automatic Construction of Surfaces with Prescribed Shape,” Computer Aided Design, vol. 20, pp. 317-324, 1988.
[14] M. Eck and H. Hoppe, “Automatic Reconstruction of B-Spline Surfaces of Arbitrary Topological Type,” Proc. SIGGRAPH '96, pp. 325-334, Aug. 1996.
[15] M.P. Do Carmo, Differential Geometry of Curves and Surfaces. Englewood Cliffs, N.J.: Prentice Hall, 1976.
[16] R.S. Millman and G.D. Parher, Elements of Differential Geometry. Englewood Cliffs, N.J.: Prentice Hall, 1977.

Index Terms:
Surface fitting, B-Spline, Gaussian map, geodesics, NURBS.
Citation:
Fernand S. Cohen, Walid Ibrahim, Chuchart Pintavirooj, "Ordering and Parameterizing Scattered 3D Data for B-Spline Surface Approximation," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 22, no. 6, pp. 642-648, June 2000, doi:10.1109/34.862203