Advertisement

Recent Progress in Subdivision: a Survey

  • Malcolm Sabin
Part of the Mathematics and Visualization book series (MATHVISUAL)

Summary

After briefly establishing the traditional concepts in subdivision surfaces, we survey the way in which the literature on this topic has burgeoned in the last five or six years, picking out new trends, ideas and issues which are becoming important.

Keywords

Subdivision Scheme Subdivision Surface Subdivision Algorithm Interpolatory Subdivision Scheme Extraordinary Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    G. de Rham. Un peu de mathématique à propos d'une courbe plane Elemente der Mathematik 2, 73–76, 89–97, 1947.Google Scholar
  2. 2.
    G. Chaikin. An algorithm for high speed curve generation Computer Graphics & Image Processing 3, 346–349, 1974.Google Scholar
  3. 3.
    A. R. Forrest. Notes on Chaikin's algorithm. University of East Anglia Computational Geometry Project Memo, CGP74/1, 1974.Google Scholar
  4. 4.
    R. Riesenfeld. On Chaikin's algorithm. Computer Graphics & Image Processing 4, 304–310, 1975.Google Scholar
  5. 5.
    D. Doo. A subdivision algorithm for smoothing down irregularly shaped polyhedrons. Proc. Int'l Conf. on Interactive Techniques in Computer Aided Design, IEEE Computer Soc., 157–165, 1978.Google Scholar
  6. 6.
    E. Catmull and J. Clark. Recursively generated B-spline surfaces on arbitrary topological meshes. Computer Aided Design 10, 350–355, 1978.CrossRefGoogle Scholar
  7. 7.
    D. Doo and M. Sabin. Behaviour of recursive division surfaces near extraordinary points. Computer Aided Design 10, 356–360, 1978.CrossRefGoogle Scholar
  8. 8.
    L. L. Schumaker. Spline Functions, John Wiley, New York, 1981.Google Scholar
  9. 9.
    A. Ball and D. Storry. Recursively generated B-spline surfaces. Proc. CAD84, 112–119, 1984.Google Scholar
  10. 10.
    W. Dahmen and C. Micchelli. Subdivision algorithms for the generation of box-spline surfaces. Computer Aided Geometric Design 1, 115–129, 1984.CrossRefGoogle Scholar
  11. 11.
    D. Storry. B-spline surfaces over an irregular topology by recursive subdivision. Ph.D. Thesis, Loughborough University, 1984.Google Scholar
  12. 12.
    A. Ball and D. Storry. A matrix approach to the analysis of recursively generated B-spline surfaces. Computer Aided Design 18(8), 437–442, 1986.CrossRefGoogle Scholar
  13. 13.
    C. Loop. Smooth subdivision surfaces based on triangles. Master's thesis, Department of Mathematics, University of Utah, 1987.Google Scholar
  14. 14.
    N. Dyn, D. Levin, and J. A. Gregory. A four-point interpolatory subdivision scheme for curve design. Computer Aided Geometric Design 4, 257–268, 1987.MathSciNetCrossRefGoogle Scholar
  15. 15.
    A. Ball and D. Storry. Conditions for tangent plane continuity over recursively generated B-spline surfaces. ACMTransactions on Graphics 7(2), 83–108, 1988.CrossRefGoogle Scholar
  16. 16.
    G. Deslauriers and S. Dubuc. Symmetric iterative interpolation processes Constructive Approximation 5, 49–68, 1989.MathSciNetCrossRefGoogle Scholar
  17. 17.
    N. Dyn, J. Gregory, and D. Levin. A butterfly subdivision scheme for surface interpolation with tension control. ACM Transactions on Graphics 9, 160–169, 1990.CrossRefGoogle Scholar
  18. 18.
    M. Sabin. Cubic recursive division with bounded curvature. Curves and Surfaces, L. L. Schumaker, J.-P. Laurent, and A. Le Méhauté (eds.), Academic Press, 411–414, 1991.Google Scholar
  19. 19.
    A. Cavaretta, W. Dahmen, and C. Micchelli. Stationary subdivision Memoirs of the AMS, vol 453, 1991.Google Scholar
  20. 20.
    N. Dyn, J. Gregory, and D. Levin. Analysis of uniform binary subdivision schemes for curve design Constructive Approximation 7, 127–147, 1991.MathSciNetCrossRefGoogle Scholar
  21. 21.
    T. DeRose, M. Lounsbery, and J. Warren. Multiresolution analysis for surfaces of arbitrary topological type. ACM Transactions on Graphics 11, 34–73, 1992.Google Scholar
  22. 22.
    D. Moore. Simplicial mesh generation with applications. Ph.D. thesis, Cornell University, 1992.Google Scholar
  23. 23.
    N. Dyn and D. Levin. Stationary and non-stationary binary subdivision schemes. Mathematical Methods in Computer Aided Geometric Design II, T. Lyche, and L. L. Schumaker (eds.), Academic Press, 209–216, 1992.Google Scholar
  24. 24.
    R. Qu and J. Gregory. A subdivision algorithm for non-uniform B-splines Approximation Theory, Spline Functions and Applications, Singh (ed.), 423–436, 1992.Google Scholar
  25. 25.
    J.-L. Merrien. A family of Hermite interpolants by bisection algorithms. Numerical Algorithms 2, 187–200, 1992.MATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    N. Dyn. Subdivision schemes in computer aided geometric design. Advances in Numerical Analysis — Volume II, Wavelets, Subdivision Algorithms and Radial Basis Functions, W. Light (ed) Clarendon Press, Oxford, 36–104, 1992.Google Scholar
  27. 27.
    D. L. Donoho. Smooth wavelet decompositions with blocky coefficient kernels. Recent Advances in Wavelet Analysis, L. L. Schumaker and G. Webb (eds.), Academic Press, Boston, 259–308, 1993.Google Scholar
  28. 28.
    H. Hoppe, T. DeRose, T. Duchamp, M. Halstead, H. Jin, J. McDonald, J. Schweitzer, and W. Stützle. Piecewise smooth surface reconstruction Proc. ACM SIGGRAPH '94, 295–302, 1994.Google Scholar
  29. 29.
    J.-L. Merrien. Dyadic Hermite interpolation on a triangulation. Numerical Algorithms 7, 391–410, 1994.MATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    N. Dyn and D. Levin. Analysis of asymptotically equivalent binary subdivision schemes. Journal of Math.Anal.Appl. 193, 594–621, 1995.MathSciNetCrossRefGoogle Scholar
  31. 31.
    N. Dyn, J. Gregory, and D. Levin. Piecewise uniform subdivision schemes. Mathematical Methods for Curves and Surfaces [33], 111–119, 1995.Google Scholar
  32. 32.
    J. Warren. Binary subdivision schemes for functions of irregular knot sequences. Mathematical Methods for Curves and Surfaces [33], 543–562, 1995.Google Scholar
  33. 33.
    M. Daehlen, T. Lyche, and L. Schumaker (eds.). Mathematical Methods for Curves and Surfaces, Vanderbilt University Press, ISBN 8265-1268-2, 1995.Google Scholar
  34. 34.
    U. Reif. A unified approach to subdivision algorithms near extraordinary vertices. Computer Aided Geometric Design 12,2, 153–174, 1995.MATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    U. Reif. A degree estimate for subdivision surfaces of higher regularity. Proc AMS 124(7), 2167–2174, 1996.MATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    J. Gregory and R. Qu. Non-uniform corner cutting. Computer Aided Geometric Design 13, 763–772, 1996.MathSciNetCrossRefGoogle Scholar
  37. 37.
    L. Kobbelt. Interpolatory subdivision on open quadrilateral nets with arbitrary topology. Computer Graphics Forum 15, 409–420, 1996.CrossRefGoogle Scholar
  38. 38.
    D. Zorin, P. Schröder, and W. Sweldens. Interpolatory subdivision for meshes with arbitrary topology. Proc. ACM SIGGRAPH '96, 189–192, 1996.Google Scholar
  39. 39.
    R. MacCracken and K. Joy. Free-form deformations with lattices of arbitrary topology Proc. ACM SIGGRAPH '96, 181–188, 1996.Google Scholar
  40. 40.
    F. Holt. Toward a curvature continuous stationary subdivision algorithm. ZAMM 1996 S1 (Proc GAMM), 423–424, 1996.Google Scholar
  41. 41.
    J. Peters and U. Reif. The simplest subdivision scheme for smoothing polyhedra. ACM Transactions on Graphics 16, 420–431, 1997.CrossRefGoogle Scholar
  42. 42.
    R. van Damme. Bivariate Hermite subdivision. Computer Aided Geometric Design 14, 847–875, 1997.MATHMathSciNetCrossRefGoogle Scholar
  43. 43.
    J. Warren. Sparse filter banks for binary subdivision schemes. Proc. Mathematics of Surfaces VII [45], 427–438, 1997.Google Scholar
  44. 44.
    J. Peters and M. Wittman. Smooth blending of basic surfaces using trivariate box splines. Proc. Mathematics of Surfaces VII [45], 409–426, 1997.Google Scholar
  45. 45.
    T. Goodman and R. Martin (eds.. Proc. Mathematics of Surfaces VII, Information Geometers, ISBN 1-874728-12-7, 1997.Google Scholar
  46. 46.
    H. Prautzsch and G. Umlauf. Improved triangular subdivision schemes. Proc. Computer Graphics International, 626–632, 1998.Google Scholar
  47. 47.
    H. Prautzsch and G. Umlauf. A G2 subdivision algorithm. Computing Supplements 13, Springer Verlag, 217–224, 1998.MathSciNetGoogle Scholar
  48. 48.
    H. Prautzsch. Smoothness of subdivision surfaces at extraordinary points. Adv. Comput. Math. 9, 377–389, 1998.MATHMathSciNetCrossRefGoogle Scholar
  49. 49.
    T. DeRose et al. Texture mapping and other uses of scalar fields on subdivision surfaces in computer graphics and animation US Patent 6,037,949, 1998.Google Scholar
  50. 50.
    T. DeRose, M. Kass, and T. Truong. Subdivision surfaces in character animation. Proc. ACM SIGGRAPH '98, 85–94, 1998.Google Scholar
  51. 51.
    T. Sederberg, D. Sewell, and M. Sabin. Non-uniform recursive subdivision surfaces. Proc. ACM SIGGRAPH '98, 387–394, 1998.Google Scholar
  52. 52.
    J. Stam. Exact Evaluation of Catmull-Clark subdivision surfaces at arbitrary parameter values. Proc. ACM SIGGRAPH '98, 395–404, 1998.Google Scholar
  53. 53.
    I. Guskov. Multivariate subdivision schemes and divided differences. Princeton University preprint, http://www.cs.caltech.edu/ĩvguskov/two.ps.gz, 1998.Google Scholar
  54. 54.
    J. Peters and U. Reif. Analysis of algorithms generalizing B-spline subdivision SIAM J. Numerical Analysis 35, 728–748, 1998.MathSciNetCrossRefGoogle Scholar
  55. 55.
    A. Levin. Combined subdivision schemes. Ph.D. thesis, Tel-Aviv University, 1999.Google Scholar
  56. 56.
    A. Levin. Combined subdivision schemes for the design of surfaces satisfying boundary conditions. Computer Aided Geometric Design 16, 345–354, 1999.MATHMathSciNetCrossRefGoogle Scholar
  57. 57.
    F. Cirak, M. Ortiz, and P. Schröder. Subdivision surfaces: A new paradigm for thin-shell finite element analysis. Technical report http://www.multires.caltech.edu/pubs/, 1999.Google Scholar
  58. 58.
    K. Qin and H. Wang. Eigenanalysis and continuity of non-uniform Doo-Sabin surfaces. Proc. Pacific Graphics 99, 179–186, 1999.Google Scholar
  59. 59.
    H. Weimer and J. Warren. Subdivision schemes for fluid flow. Proc. ACM SIGGRAPH '99, 111–120, 1999.Google Scholar
  60. 60.
    F. Samavati and R. Bartels. Multiresolution curve and surface representation: reversing subdivision rules by least-squares data fitting. Computer Graphics Forum 18, 97–119, 1999.CrossRefGoogle Scholar
  61. 61.
    H. Suzuki, S. Takeuchi, and T. Kanai. Subdivision surface fitting to a range of points. Pacific Graphics 99, 158–167, 1999.Google Scholar
  62. 62.
    N. Dyn. Using Laurent polynomial representation for the analysis of non-uniform binary subdivision schemes. Adv. Comput. Math. 11, 41–54, 1999.MathSciNetCrossRefGoogle Scholar
  63. 63.
    N. Dyn and D. Levin. Analysis of Hermite interpolatory subdivision schemes. Spline Functions and the Theory of Wavelets, S. Dubuc (ed.), AMS series CRM Proceedings and Lecture Notes 18, 105–113, 1999.Google Scholar
  64. 64.
    N. Dyn and T. Lyche. Hermite subdivision scheme for the evaluation of the Powell-Sabin 12-split element. Approximation Theory IX, C. Chui and L. L. Schumaker (eds.), Vanderbilt University Press, 1–6, 1999.Google Scholar
  65. 65.
    I. Guskov, K. Vidimce, W. Sweldens, and P. Schröder. Normal meshes. Proc. ACM SIGGRAPH 2000, 95–102, 2000.Google Scholar
  66. 66.
    L. Kobbelt. \(\sqrt 3 \) subdivision Proc. ACM SIGGRAPH 2000, 103–112, 2000.Google Scholar
  67. 67.
    H. Biermann, A. Levin, and D. Zorin. Piecewise smooth subdivision surfaces with normal control Proc. ACM SIGGRAPH 2000, 113–120, 2000.Google Scholar
  68. 68.
    A. Khodakovsky, P. Schröder, and W. Sweldens. Progressive geometry compression. Proc. ACM SIGGRAPH 2000, 271–278, 2000.Google Scholar
  69. 69.
    J. Peters and G. Umlauf. Gaussian and mean curvature of subdivision surfaces. The Mathematics of Surfaces IX, Cipolla and Martin (eds.), Springer, ISBN 1-85233-358-8, pages 59–69, 2000.Google Scholar
  70. 70.
    H. Prautzsch and G. Umlauf. A G1 and a G2 subdivision scheme for triangular nets. Journal of Shape Modeling 6, 21–35, 2000.Google Scholar
  71. 71.
    N. Dyn and E. Farkhi. Spline subdivision schemes for convex compact sets. Journal of Computational and Applied Mathematics 119, 133–144, 2000.MathSciNetCrossRefGoogle Scholar
  72. 72.
    A. Cohen, N. Dyn, K. Kaber, and M. Postel. Multiresolution schemes on triangles for scalar conservation laws. Journal of Computational Physics 161, 264–286, 2000.MathSciNetCrossRefGoogle Scholar
  73. 73.
    J. Peters. Patching Catmull-Clark meshes. Proc. ACM SIGGRAPH 2000, pp255–258, 2000.Google Scholar
  74. 74.
    U. Reif and P. Schröder. Curvature integrability of subdivision surfaces. Adv. Comp. Math. 12, 1–18, 2000.Google Scholar
  75. 75.
    W. Ma and N. Zhao. Catmull-Clark surface fitting for reverse engineering applications. Proc. Geometric Modeling and Processing 2000, IEEE, 274–283, 2000.Google Scholar
  76. 76.
    A. Nasri, K. van Overfeld, and B. Wyvill. A recursive subdivision algorithm for piecewise circular spline. Computer Graphics Forum 20(1), 35–45, 2001.CrossRefGoogle Scholar
  77. 77.
    L. Velho. Quasi 4–8 subdivision. Computer Aided Geometric Design 18(4), 345–358, 2001.MATHMathSciNetCrossRefGoogle Scholar
  78. 78.
    L. Velho and D. Zorin. 4–8 subdivision. Computer Aided Geometric Design 18(5), 397–427, 2001.MathSciNetCrossRefGoogle Scholar
  79. 79.
    G. Morin, J. Warren, and H. Weimer. A subdivision scheme for surfaces of revolution. Computer Aided Geometric Design 18(5), 483–502, 2001.MathSciNetCrossRefGoogle Scholar
  80. 80.
    S. Skaria, E. Akleman, and F. Parke. Modeling subdivision control meshes for creating cartoon faces. Proc. Shape Modeling and Applications 2001, 216–225, 2001.Google Scholar
  81. 81.
    C. Loop. Triangle mesh subdivision with bounded curvature and the convex hull property. Technical report MSR-TR-2001-24, Microsoft Research, 2001.Google Scholar
  82. 82.
    N. Dyn and E. Farkhi. Spline subdivision schemes for compact sets with metric averages. Trends in Approximation Theory, K. Kopotun, T. Lyche and M. Neamtu (eds.), Vanderbilt University Press. Nashville, TN, 93–102, 2001.Google Scholar
  83. 83.
    D. Zorin and P. Schröder. A unified framework for primal/dual quadrilateral subdivision schemes. Computer Aided Geometric Design 18, 429–454, 2001.MathSciNetCrossRefGoogle Scholar
  84. 84.
    R. Bartels and F. Samavati. Reversing subdivision rules: local linear conditions and observations on inner products. J. Comp and Appl.Math., 119,(1–2), 29–67, 2001.MathSciNetGoogle Scholar
  85. 85.
    F. Samavati and R. Bartels. Reversing Subdivision using local linear conditions: generating multiresolutions on regular triangular meshes. preprint, http://www.cgl.uwaterloo.ca/~rhbartel/Papers/TriMesh.pdf, 2001.Google Scholar
  86. 86.
    N. Dyn and E. Farkhi. Spline subdivision schemes for compact sets — a survey. Serdica Math. J. Vol.28(4), 349–360, 2002.MathSciNetGoogle Scholar
  87. 87.
    M. Alexa. Refinement operators for triangle meshes. Computer Aided Geometric Design 19(4), 169–172, 2002.MATHMathSciNetCrossRefGoogle Scholar
  88. 88.
    M. Hassan, I. P. Ivrissimtzis, N. A. Dodgson, and M. A. Sabin. An interpolating 4-point C2 ternary stationary subdivision scheme. Computer Aided Geometric Design 19, 1–18, 2002.MathSciNetCrossRefGoogle Scholar
  89. 89.
    L. Barthe, B. More, N. A. Dodgson, and M. A. Sabin. Triquadratic reconstruction for interactive modelling of potential fields. Proc. Shape Modeling and Applications 2002, 145–153, 2002.Google Scholar
  90. 90.
    J. Warren and H. Weimer. Subdivision Methods for Geometric design., Morgan Kaufmann, 2002.Google Scholar
  91. 91.
    F. Cirak, M. Scott, E. Antonsson, M. Ortiz, and P. Schröder. Integrated modeling, finite element analysis and engineering design for thin-shell structures using subdivision. Computer Aided Design 34, 137–148, 2002.CrossRefGoogle Scholar
  92. 92.
    W. Ma, X. Ma, S-K. Tso, and Z. Pan. Subdivision surface fitting from a dense triangle mesh. Proc. Geometric Modeling and Processing 2002, 94–103, 2002.Google Scholar
  93. 93.
    M. Sabin. Interrogation of subdivision surfaces. Handbook of Computer Aided Design, Chap. 12, 327–341, 2002.Google Scholar
  94. 94.
    N. Dyn and D. Levin. Subdivision schemes in geometric modelling. Acta Numerica, 73–144, 2002.Google Scholar
  95. 95.
    C. Bajaj, S. Schaefer, J. Warren, and G. Xu. A subdivision scheme for hexahedral meshes. The Visual Computer 18, 343–356, 2002.CrossRefGoogle Scholar
  96. 96.
    G. Taubin. Detecting and reconstructing subdivision connectivity. The Visual Computer 18, 357–367, 2002.CrossRefGoogle Scholar
  97. 97.
    B. Jüttler, U. Schwanecke. Analysis and design of Hermite subdivision schemes The Visual Computer 18, 326–342, 2002.CrossRefGoogle Scholar
  98. 98.
    D. Zorin, D. Kristjansson. Evaluation of piecewise smooth subdivision surfaces. The Visual Computer 18, 299–315, 2002.CrossRefGoogle Scholar
  99. 99.
    C. Loop. Bounded curvature triangle mesh subdivision with the convex hull property. The Visual Computer 18, 316–325, 2002.CrossRefGoogle Scholar
  100. 100.
    M. Sabin. Subdivision of box-splines. Tutorials on Multiresolution in Geometric Modelling [104], 3–23, 2002.Google Scholar
  101. 101.
    N. Dyn. Interpolatory subdivision schemes. Tutorials on Multiresolution in Geometric Modelling [104], 25–50, 2002.Google Scholar
  102. 102.
    N. Dyn. Analysis of convergence and smoothness by the formalism of Laurent polynomials. Tutorials on Multiresolution in Geometric Modelling [104], 51–68, 2002.Google Scholar
  103. 103.
    M. Sabin. Eigenanalysis and artifacts of subdivision curves and surfaces. Tutorials on Multiresolution in Geometric Modelling [104], 69–92, 2002.Google Scholar
  104. 104.
    A. Iske, E. Quak, and M. S. Floater (eds. Tutorials on Multiresolution in Geometric Modelling., Springer, ISBN 3-540-43639-1, 2002.Google Scholar
  105. 105.
    N. A. Dodgson, M. A. Sabin, L. Barthe, and M. F. Hassan. Towards a ternary interpolating scheme for the triangular mesh. Technical Report UCAM-CLTR-539, Computer Laboratory, University of Cambridge, 2002.Google Scholar
  106. 106.
    I. P. Ivrissimtzis, N. A. Dodgson, and M. A. Sabin. A generative classification of mesh refinement rules with lattice transformations. Preprint of [128], Technical Report UCAM-CL-TR-542, Computer Laboratory, University of Cambridge, 2002.Google Scholar
  107. 107.
    Y. Chang, K. McDonnell, and H. Qin. A new solid subdivision scheme based on box splines. Proc. of the Seventh ACM Symposium on Solid Modeling and Applications, 226–233, 2002.Google Scholar
  108. 108.
    F. Samavati, N. Mahdavi-Amiri, and R. Bartels. Multiresolution surfaces having arbitrary topologies by a reverse Doo subdivision method. Computer Graphics Forum 21, 121–136, 2002.CrossRefGoogle Scholar
  109. 109.
    A. Nasri and M. Sabin. A taxonomy of interpolation constraints on recursive subdivision surfaces. The Visual Computer 18, 382–403, 2002.CrossRefGoogle Scholar
  110. 110.
    P. Prusinkiewicz, F. Samavati, C. Smith, and R. Karwowski. L-system description of subdivision curves. International Journal of Shape Modeling 9, 41–59, 2003.Google Scholar
  111. 111.
    N. A. Dodgson, I. P. Ivrissimtzis, and M. A. Sabin. Characteristics of dual triangular \(\sqrt 3 \) subdivision. Curve and Surface Fitting: Saint-Malo 2002 [117], 119–128, 2003.Google Scholar
  112. 112.
    N. Dyn, D. Levin, and J. Simoens. Face value subdivision schemes on triangulations by repeated averaging. Curve and Surface Fitting: Saint-Malo 2002 [117], 129–138, 2003.Google Scholar
  113. 113.
    B. Han. Classification and construction of bivariate subdivision schemes. Curve and Surface Fitting: Saint-Malo 2002 [117], 187–198, 2003.Google Scholar
  114. 114.
    M. F. Hassan and N. A. Dodgson. Ternary and Three-point univariate subdivision schemes. Curve and Surface Fitting: Saint-Malo 2002 [117], 199–208, 2003.Google Scholar
  115. 115.
    C. Loop. Smooth ternary subdivision of triangle meshes. Curve and Surface Fitting: Saint-Malo 2002 [117], 295–302, 2003.Google Scholar
  116. 116.
    M. Sabin and L. Barthe. Artifacts in recursive subdivision surfaces. Curve and Surface Fitting: Saint-Malo 2002 [117], 353–362, 2003.Google Scholar
  117. 117.
    A. Cohen, J-L. Merrien, and L. L. Schumaker (eds.). Curve and Surface Fitting: Saint-Malo 2002, Nashboro Press, Brentwood, TN, ISBN 0-9728482-1-5, 2003.Google Scholar
  118. 118.
    J. Stam and C. Loop. Quad/triangle subdivision. Computer Graphics Forum 22(1), 79–85, 2003.CrossRefGoogle Scholar
  119. 119.
    V. Surazhsky and C. Gotsman. Explicit surface remeshing. Eurographics symposium on geometry processing [120], 17–27, 2003.Google Scholar
  120. 120.
    L. Kobbelt, P. Schröder, and H. Hoppe (eds. Eurographics symposium on geometry processing., Eurographics Association, 2003.Google Scholar
  121. 121.
    A. Levin. Polynomial generation and quasi-interpolation in stationary non-uniform subdivision Computer Aided Geometric Design 20, 41–60, 2003.MATHMathSciNetCrossRefGoogle Scholar
  122. 122.
    P. Oswald and P. Schröder. Composite primal/dual sqrt(3) subdivision schemes Computer Aided Geometric Design 20, 135–164, 2003.MathSciNetCrossRefGoogle Scholar
  123. 123.
    A. Levin and D. Levin. Analysis of quasi-uniform subdivision. Applied and Computational Harmonic Analysis 15, 18–32, 2003.MathSciNetCrossRefGoogle Scholar
  124. 124.
    B. Han. Computing the smoothness exponent of a symmetric multivariate refinable function. SIAM Journal on Matrix Analysis and its Applications, 24, 693–714, 2003.MATHCrossRefGoogle Scholar
  125. 125.
    N. Dyn, D. Levin, and A. Luzzatto. Non-stationary interpolatory subdivision schemes reproducing spaces of exponential polynomials. Found. Comput. Math, 187–206, 2003.Google Scholar
  126. 126.
    M. Sabin and A. Bejancu. Boundary conditions for the 3-direction box-spline Maths of Surfaces X, Springer, 2003.Google Scholar
  127. 127.
    A. Ron. Private communication, 2003.Google Scholar
  128. 128.
    I. P. Ivrissimtzis, N. A. Dodgson, and M. A. Sabin. A generative classification of mesh refinement rules with lattice transformations. Computer Aided Geometric Design vol 21(1), 99–109, 2004.MathSciNetCrossRefGoogle Scholar
  129. 129.
    I. P. Ivrissimtzis, N. A. Dodgson, and M. A. Sabin. \(\sqrt 5 \) Subdivision. Advances in Multiresolution for Geometric Modelling [135], pages 285–299 (this book), 2004.Google Scholar
  130. 130.
    M. F. Hassan and N. A. Dodgson. Reverse Subdivision. Advances in Multiresolution for Geometric Modelling [135], pages 271–283 (this book), 2004.Google Scholar
  131. 131.
    N. Alkalai and N. Dyn. Optimizing 3D triangulations for improving the initial triangulation for the butterfly subdivision scheme. Advances in Multiresolution for Geometric Modelling [135], pages 231–244 (this book), 2004.Google Scholar
  132. 132.
    N. Dyn, D. Levin, and M. Marinov. Geometrical interpolation shape-preserving 4-point schemes. Advances in Multiresolution for Geometric Modelling [135], pages 301–315 (this book), 2004.Google Scholar
  133. 133.
    L. Barthe, C. Gérot, M. A. Sabin, and L. Kobbelt. Simple computation of the eigencomponents of a subdivision matrix in the frequency domain. Advances in Multiresolution for Geometric Modelling [135], pages 245–257 (this book), 2004.Google Scholar
  134. 134.
    C. Gérot, L. Barthe, N. A. Dodgson, and M. A. Sabin. Subdivision as a sequence of sampled Cp surfaces. Advances in Multiresolution for Geometric Modelling [135], pages 259–270 (this book), 2004.Google Scholar
  135. 135.
    N. A. Dodgson, M. S. Floater, and M. A. Sabin (eds.). Advances in Multiresolution for Geometric Modelling, Springer-Verlag (this book), 2004.Google Scholar
  136. 136.
    A. Cohen, N. Dyn, and B. Matei. Quasilinear subdivision schemes with applications to ENO interpolation. Applied and Computational Harmonic Analysis, to appear.Google Scholar
  137. 137.
    I. P. Ivrissimtzis, N. A. Dodgson, and M. A. Sabin. The support of recursive subdivision surfaces. ACM Transactions on Graphics, to appear.Google Scholar
  138. 138.
    B. Han, T. Yu, and Yong-Gang Xue. Non-interpolatory Hermite subdivision schemes. Preprint.Google Scholar
  139. 139.
    J. Peters and L-J. Shiue. 4–3 Directionally Ripple-free Subdivision. Submitted to ACM Transactions on Graphics.Google Scholar
  140. 140.
    N. Dyn and E. Farkhi. Convexification rates in Minkowski averaging processes. In preparation.Google Scholar
  141. 141.
    M. Sabin. A circle-preserving interpolatory subdivision scheme. In preparation.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Malcolm Sabin
    • 1
    • 2
  1. 1.Computer LaboratoryUniversity of CambridgeUK
  2. 2.Numerical Geometry Ltd.CambridgeUK

Personalised recommendations