This Article 
 Bibliographic References 
 Add to: 
Reconstructing Surfaces by Volumetric Regularization Using Radial Basis Functions
October 2002 (vol. 24 no. 10)
pp. 1358-1371

Abstract—We present a new method of surface reconstruction that generates smooth and seamless models from sparse, noisy, nonuniform, and low resolution range data. Data acquisition techniques from computer vision, such as stereo range images and space carving, produce 3D point sets that are imprecise and nonuniform when compared to laser or optical range scanners. Traditional reconstruction algorithms designed for dense and precise data do not produce smooth reconstructions when applied to vision-based data sets. Our method constructs a 3D implicit surface, formulated as a sum of weighted radial basis functions. We achieve three primary advantages over existing algorithms: 1) the implicit functions we construct estimate the surface well in regions where there is little data, 2) the reconstructed surface is insensitive to noise in data acquisition because we can allow the surface to approximate, rather than exactly interpolate, the data, and 3) the reconstructed surface is locally detailed, yet globally smooth, because we use radial basis functions that achieve multiple orders of smoothness.

[1] N. Amenta, M. Bern, and M. Kamvysselis, “A New Voronoi-Based Surface Reconstruction Algorithm,” Proc. SIGGRAPH '98, pp. 415-421, 1998.
[2] C.L. Bajaj, J. Chen, and G. Xu, “Free Form Surface Design with A-Patches,” Proc. Graphics Interface, pp. 174-181, 1994.
[3] C.L. Bajaj, F. Bernardini, and G. Xu, “Automatic Reconstruction of Surfaces and Scalar Fields from 3D Scans,” Proc. SIGGRAPH '95, pp. 109-118, Aug. 1995.
[4] F. Bernardini, J. Mittleman, H. Rushmeier, C. Silva, and G. Taubin, “The Ball-Pivoting Algorithm for Surface Reconstruction,” IEEE Trans. Visualization and Computer Graphics, vol. 5, no. 4, pp. 349-359, Oct.-Dec. 1999.
[5] M.M. Blane, Z. Lei, H. Civil, and D.B. Cooper, The 3L Algorithm for Fitting Implicit Polynomials Curves and Surface to Data IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 22, no. 3, Mar. 2000.
[6] J.F. Blinn,“A generalization of algebraic surface drawing,” ACM Trans. on Graphics, vol.1, no.3, pp. 235-256, 1982.
[7] T.E. Boult and J.R. Kender, “Visual Surface Reconstruction Using Sparse Depth Data,” Computer Vision and Pattern Recognition Proc., pp. 68-76, 1986.
[8] J.W. Bruce and P.J. Giblin, Curves and Singularities, second ed. Cambridge Univ. Press, pp. 59-98, 1992.
[9] J.C. Carr, R.K. Beatson, J.B. Cherrie, T.J. Mitchell, W.R. Fright, B.C. McCallum, and T.R. Evans, “Reconstruction and Representation of 3D Objects with Radial Basis Functions,” Proc. SIGGRAPGH, pp. 67-76, 2001.
[10] F. Chen and D. Suter, “Multiple Order Laplacian Spline - Including Splines with Tension,” Technical Report, MECSE 1996-5, Dept. of Electrical and Computer Systems Eng. Monash Univ., July 1996.
[11] W.B. Culbertson, T. Malzbender, and G.G. Slabaugh, “Generalized Voxel Coloring,” Vision Algorithms: Theory and Practice, vol. 1883, pp. 67-74, Sept. 1999.
[12] B. Curless and M. Levoy, “A Volumetric Method for Building Complex Models from Range Images,” Proc. SIGGRAPH '96, pp. 303-312, 1996.
[13] M. Desbrun, M. Meyer, P. Schröder, and A.H. Barr, “Implicit Fairing of Irregular Meshes Using Diffusion and Curvature Flow,” SIGGRAPH '99 Proc., Computer Graphics Proc., Ann. Conf. Series, Aug. 1999.
[14] H. Dinh and G. Turk, “Reconstructing Surfaces by Volumetric Regularization,” Technical Report, GVU-00-26, College of Computing, Georgia Tech., Dec. 2000.
[15] H. Edelsbrunner and D.P. Mücke, “Three-Dimensional Alpha Shapes,” ACM Trans, Graphics, vol. 13, pp. 43-72, 1994.
[16] L. Fang and D. Gossard, “Multidimensional Curve Fitting to Unorganized Data Points by Nonlinear Minimization,” Computer-Aided Design, vol. 27, no. 1, pp. 48-58, Jan. 1995.
[17] A. Fomenko and T. Kunii, Topological Modeling for Visualization. Springer, 1997.
[18] S.F. Frisken, R.N. Perry, A.P. Rockwood, and T.R. Jones, “Adaptively Sampled Distance Fields: A General Representation of Shape for Computer Graphics,” Proc. SIGGRAPH '00, pp. 249-254, 2000.
[19] F. Girosi, M. Jones, and T. Poggio, “Priors, Stabilizers and Basis Functions: From Regularization to Radial, Tensor and Additive Splines,” A. I. Memo No. 1430, C. B. C. L. Paper No. 75, Massachusetts Inst. of Technology Artificial Intelligence Lab, June 1993.
[20] A. Hilton, A.J. Stoddart, J. Illingworth, and T. Windeatt, “Reliable Surface Reconstruction from Multiple Range Images,” Proc. Fourth European Conf. Computer Vision, 1996.
[21] H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, and W. Stuetzle, “Surface Reconstruction from Unorganized Points,” Proc. SIGGRAPH '92, pp. 71-78, 1992.
[22] D. Keren and C. Gotsman, “Tight Fitting of Convex Polyhedral Shapes,” Int'l J. Shape Modeling, (Special Issue on Reverse Eng. Techniques), pp. 111-126, 1998.
[23] D. Keren and C. Gotsman, “Fitting Curves and Surfaces with Constrained Implicit Polynomials,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 21, no. 1, pp. 31-41, Jan. 1999.
[24] C. Kolb, The Rayshade Homepage. info.html.
[25] K.N. Kutulakos and S.M. Seitz, “A Theory of Shape by Space Carving,” Proc. Seventh Int'l Conf. Computer Vision, vol. I, pp. 307-314, Sept. 1999.
[26] M.S. Lee and G. Medioni, "Inferring Segmented Surface Description from Stereo Data," Proc. Computer Vision and Pattern Recognition, pp. 346-352,Santa Barbara, Calif., June 1998.
[27] W.E. Lorensen and H.E. Cline, “Marching Cubes: A High Resolution 3D Surface Construction Algorithm,” Computer Graphics (SIGGRAPH '87 Proc.), vol. 21, pp. 163-169, 1987.
[28] B. Morse, T. Yoo, P. Rheingans, D. Chen, and K.R. Subramanian, “Interpolating Implicit Surfaces from Scattered Surface Data Using Compactly Supported Radial Basis Functions,” Proc. Int'l Conf. Shape Modeling and Applications, pp. 89-98, May 2001.
[29] S. Muraki,“Volumetric shape description of range data using’blobby model’,” Proc. SIGGRAPH’91 (Las Vegas, Nevada, July 29-Aug. 2, 1991). In Computer Graphics, vol. 25, no. 4, pp. 227-235, 1991.
[30] M. Nielsen, L. Florack, and R. Deriche, “Regularization, Scale-Space, and Edge Detection Filters,” J. Math. Images and Vision, 1997.
[31] G.M. Nielson, “Scattered Data Modeling,” IEEE CG&A, Vol. 13, No. 1, Jan. 1993, pp. 60-70.
[32] A.P. Pentland,“Automatic extraction of deformable part models,” Int’l J. Computer Vision, vol. 4, pp. 107-126, 1990.
[33] V.V. Savchenko, A.A. Pasko, O.G. Okunev, and T.L. Kunii, “Function Representation of Solids Reconstructed from Scattered Surface Points and Contours,” Computer Graphics Forum, vol. 14, no. 4, pp. 181-188, 1995.
[34] S. Sclaroff and A. Pentland, “Generalized Implicit Functions for Computer Graphics,” Proc. SIGGRAPH, pp. 247-250, July 1991.
[35] S.M. Seitz and C.R. Dyer, “Photorealistic Scene Reconstruction by Voxel Coloring,” Int'l J. Computer Vision, vol. 35, no. 2, pp. 151-173, 1999.
[36] G. Slabaugh, B. Culbertson, T. Malzbender, and R. Schafer, “A Survey of Methods for Volumetric Scene Reconstruction from Photographs,” Int'l Workshop Volume Graphics, pp. 51-62, June 2001.
[37] M. Soucy and D. Laurendeau, "A General Surface Approach to the Integration of a Set of Range Views," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 17, no. 4, pp. 344-358, Apr. 1995.
[38] D. Suter and F. Chen, “Left Ventricular Motion Reconstruction Based on Elastic Vector Splines,” IEEE Trans. Medical Imaging, pp. 295-305, 2000.
[39] G. Szeliski and S. Lavalée, “Matching 3D Anatomical Surface with Non-Rigid Deformations Using Octree-Splines,” Int'l J. Computer Vision, vol. 18, no. 2, pp. 171-186, 1996.
[40] C.-K. Tang and G. Medioni, “Inference of Integrated Surface, Curve, and Junction Descriptions from Sparse, Noisy 3D Data,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 20, no. 11, pp. 1206-1223, Nov. 1998.
[41] G. Taubin,“Estimation of planar curves, surfaces, and nonplanar space curves defined by implicit equations with applications to edge and range image segmentation,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 13, no. 11, pp. 1115-1137, Nov. 1991.
[42] G. Taubin, "An improved algorithm for algebraic curve and surface fitting," Proc. Fourth Int'l Conf. Computer Vision, pp. 658-665,Berlin, Germany, May 1993.
[43] G. Taubin, "A Signal Processing Approach to Fair Surface Design," Computer Graphics Proc., Ann. Conf. Series, ACM Siggraph, ACM Press, New York, 1995, pp.351-358.
[44] D. Terzopoulos, "Regularization of Inverse Visual Problems Involving Discontinuities," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 8, no. 4, pp. 413-424, 1986.
[45] D. Terzopoulos, "The Computation of Visible Surface Representations," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 10, no. 4, pp. 417-438, Apr. 1988.
[46] D. Terzopoulos and D. Metaxas, “Dynamic 3D Models with Local and Global Deformations: Deformable Superquadrics,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 13, no. 7, pp. 703-714, July 1991.
[47] G. Turk and M. Levoy, “Zippered Polygon Meshes from Range Images,” Proc. SIGGRAPH '94, pp. 311-318, 1994.
[48] G. Turk and J.F. O'Brien, “Shape Transformation Using Variational Implicit Functions,” Proc. SIGGRAPH, pp. 335-342, 1999.
[49] G. Wahba, “Spline Models for Observational Data,” CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, pp. 11-14 and 45-65, 1990.
[50] G. Yngve and G. Turk, “Robust Creation of Implicit Surfaces from Polygonal Meshes,” ACM Trans. Visualization and Computer Graphics, to appear.
[51] A.L. Yuille and N.M. Grzywacz, “The Motion Coherence Theory,” Proc. Int'l Conf. Computer Vision, pp. 344-353, 1988.

Index Terms:
Regularization, surface fitting, implicit functions, noisy range data.
Huong Quynh Dinh, Greg Turk, Greg Slabaugh, "Reconstructing Surfaces by Volumetric Regularization Using Radial Basis Functions," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 24, no. 10, pp. 1358-1371, Oct. 2002, doi:10.1109/TPAMI.2002.1039207
Usage of this product signifies your acceptance of the Terms of Use.