This Article 
 Bibliographic References 
 Add to: 
Interpolation over Arbitrary Topology Meshes Using a Two-Phase Subdivision Scheme
May/June 2006 (vol. 12 no. 3)
pp. 301-310

Abstract—The construction of a smooth surface interpolating a mesh of arbitrary topological type is an important problem in many graphics applications. This paper presents a two-phase process, based on a topological modification of the control mesh and a subsequent Catmull-Clark subdivision, to construct a smooth surface that interpolates some or all of the vertices of a mesh with arbitrary topology. It is also possible to constrain the surface to have specified tangent planes at an arbitrary subset of the vertices to be interpolated. The method has the following features: 1) It is guaranteed to always work and the computation is numerically stable, 2) there is no need to solve a system of linear equations and the whole computation complexity is O(K) where K is the number of the vertices, and 3) each vertex can be associated with a scalar shape handle for local shape control. These features make interpolation using Catmull-Clark surfaces simple and, thus, make the new method itself suitable for interactive free-form shape design.

[1] P. Brunet, “Including Shape Handles in Recursive Subdivision Surfaces,” Computer Aided Geometric Design, vol. 5, no. 1, pp. 41-50, June 1988.
[2] E. Catmull and J. Clark, “Recursively Generated B-Spline Surfaces on Arbitrary Topological Meshes,” Computer-Aided Design, vol. 10, pp. 350-355, Sept. 1978.
[3] T. DeRose, M. Kass, and T. Truong, “Subdivision Surfaces in Character Animation,” Proc. SIGGRAPH '98}, pp. 85-94, 1998.
[4] D. Doo and M. Sabin, “Behaviour of Recursive Division Surfaces Near Extraordinary Points,” Computer-Aided Design, vol. 10, pp. 356-360, Sept. 1978.
[5] N. Dyn, D. Levin, and J.A. Gregory, “A Butterfly Subdivision Scheme for Surface Interpolation with Tension Control,” ACM Trans. Graphics, vol. 9, no. 2, pp. 160-169, Apr. 1990.
[6] P. Fjällström, “Smoothing of Polyhedral Models,” Proc. Second Ann. Symp. Computational Geometry, pp. 226-235, 1986.
[7] G. Golub and C. van Loan, Matrix Computations, second ed. Baltimore: The John Hopkins University Press, 1989.
[8] M. Halstead, M. Kass, and T. DeRose, “Efficient, Fair Interpolation Using Catmull-Clark Surfaces,” Proc. Computer Graphics (SIGGRAPH '93), vol. 27, pp. 35-44, Aug. 1993.
[9] H. Hoppe, T. DeRose, T. Duchamp, M. Halstead, H. Jin, J. McDonald, J. Schweitzer, and W. Stuetzle, “Piecewise Smooth Surface Reconstruction,” Proc. Conf. Computer Graphics (SIGGRAPH '94), pp. 295-302. 1994.
[10] L. Kobbelt, “Interpolatory Subdivision on Open Quadrilateral Nets with Arbitrary Topology,” Computer Graphics Forum, vol. 15, no. 3, pp. 409-420, 485, Sept. 1996.
[11] A. Levin, “Interpolating Nets of Curves by Smooth Subdivision Surfaces,” Proc. Conf. Computer Graphics (SIGGRAPH '99), pp. 57-64, Aug. 1999.
[12] N. Litke, A. Levin, and P. Schröder, “Fitting Subdivision Surfaces,” Proc. Visualization 2001, pp. 319-324, 2001.
[13] C. Loop, “Smooth Subdivision Surfaces Based on Triangles,” master's thesis, Dept. of Math., Univ. of Utah, 1987.
[14] A. Nasri, “Polyhedral Subdivision Methods for Free-Form Surfaces,” ACM Trans. Graphics, vol. 6, no. 1, pp. 29-73, Jan. 1987.
[15] A. Nasri, “Surface Interpolation on Irregular Networks with Normal Conditions,” Computer Aided Geometric Design, vol. 8, no. 1, pp. 89-96, Feb. 1991.
[16] N.M. Ntoko, “A Generalisation of the Timmer Blending Functions forBezier-Type Cubic Curves,” Proc. Eurographics '85, C.E. Vandoni, ed., pp. 311-314, 1985.
[17] J. Peters, “Smooth Free-Form Surfaces over Irregular Meshes Generalizing Quadratic Splines,” Computer Aided Geometric Design, vol. 10, pp. 347-361, 1993.
[18] J. Peters, “Constructing ${C}^1$ Surfaces of Arbitrary Topology Using Biquadratic Bicubic Splines,” Designing Fair Curves and Surfaces, N. Sapidis, ed., pp. 1-19. SIAM Publications, 1994.
[19] J. Peters, “${C}^1$ Surface Spline,” SIAM J. Numerical Analysis, vol. 32, 1995.
[20] M. Sabin, “Cubic Recursive Division with Bounded Curvature,” Curves and Surfaces, P.J. Laurent, A.L. Méhauté, and L.L. Schumaker, eds., pp. 411-414. Boston: Academic Press, 1991.
[21] P. Shilane, P. Min, M. Kazhdan, and T. Funkhouser, “The Princeton Shape Benchmark,” Shape Modeling Int'l, 2004.
[22] J. Stam, “Exact Evaluation of Catmull-Clark Subdivision Surfaces at Arbitrary Parameter Values,” Proc. SIGGRAPH '98, pp. 395-404, 1998.
[23] H.G. Timmer, “Alternative Representation for Parametric Cubic Curves and Surfaces,” Computer-Aided Design, vol. 12, pp. 25-28, 1980.
[24] J. Warren, J.D. Warren, and H. Weimer, Subdivision Methods for Geometric Design: A Constructive Approach. Morgan Kaufmann, 2001.
[25] D. Zorin and P. Schröder, “Subdivision for Modeling and Animation,” Course Notes SIGGRAPH '00, 2000.
[26] D. Zorin, P. Schröder, and W. Sweldens, “Interpolating Subdivision for Meshes with Arbitrary Topology,” Computer Graphics, Ann. Conf. Series, vol. 30, pp. 189-192, 1996.

Index Terms:
Computer graphics; computational geometry and object modeling; curve, surface, solid, and object representations; computer-aided engineering; computer-aided design.
Jianmin Zheng, Yiyu Cai, "Interpolation over Arbitrary Topology Meshes Using a Two-Phase Subdivision Scheme," IEEE Transactions on Visualization and Computer Graphics, vol. 12, no. 3, pp. 301-310, May-June 2006, doi:10.1109/TVCG.2006.49
Usage of this product signifies your acceptance of the Terms of Use.