Abstract
Alexa [1] and Ivrissimtzis et al. [2] have proposed a classification mechanism for bivariate subdivision schemes. Alexa considers triangular primal schemes, Ivrissimtzis et al. generalise this both to quadrilateral and hexagonal meshes and to dual and mixed schemes. I summarise this classification and then proceed to analyse it in order to determine which classes of subdivision scheme are likely to contain useful members. My aim is to ascertain whether there are any potentially useful classes which have not yet been investigated or whether we can say, with reasonable confidence, that all of the useful classes have already been considered.
I apply heuristics related to the mappings of element types (vertices, face centres, and mid-edges) to one another, to the preservation of symmetries, to the alignment of meshes at different subdivision levels, and to the size of the overall subdivision mask. My conclusion is that there are only a small number of useful classes and that most of these have already been investigated in terms of linear, stationary subdivision schemes. There is some space for further work, particularly in the investigation of whether there are useful ternary linear, stationary subdivision schemes, but it appears that future advances are more likely to lie elsewhere.
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Alexa, M.: Refinement operators for triangle meshes. Computer Aided Geometric Design 19, 169–172 (2002)
Ivrissimtzis, I.P., Dodgson, N.A., Sabin, M.A.: A generative classification of mesh refinement rules with lattice transformations. Computer Aided Geometric Design 22, 99–109 (2004)
Sloan, I.H., Joe, S.: Lattice methods for multiple integration. Oxford Science Publications, Oxford (1994)
Warren, J., Weimer, H.: Subdivision Methods for Geometric Design. Morgan Kaufmann, San Francisco (2001)
Martin, G.E.: Transformation Geometry. Springer, New York (1982)
Catmull, E., Clark, J.: Recursively generated B-spline surfaces on arbitrary topological meshes. Computer Aided Design 10, 350–355 (1978)
Doo, D., Sabin, M.: Behaviour of recursive division surfaces near extraordinary points. Computer-Aided Design 10, 356–360 (1978)
Loop, C.T.: Smooth subdivision surfaces based on triangles. Master’s thesis, University of Utah, Department of Mathematics (1987)
Peters, J., Reif, U.: The simplest subdivision scheme for smoothing polyhedra. ACM Transactions on Graphics 16, 420–431 (1997)
Peters, J., Shiue, L.J.: Combining 4- and 3-direction subdivision. ACM Transactions on Graphics 23, 980–1003 (2004)
Velho, L.: Quasi 4-8 subdivision. Computer Aided Geometric Design 18, 299–379 (2001)
Velho, L., Zorin, D.: 4-8 subdivision. Computer Aided Geometric Design 18, 397–427 (2001)
Kobbelt, L.: \(\sqrt{3}\)-Subdivision. In: SIGGRAPH 2000 Conference Proceedings, pp. 103–112 (2000)
Claes, J., Beets, K., van Reeth, F.: A corner-cutting scheme for hexagonal subdivision surfaces. In: Proceedings of Shape Modelling International, pp. 13–20 (2002)
Dyn, N., Levin, D., Gregory, J.A.: A butterfly subdivision scheme for surface interpolation with tension control. ACM Transactions on Graphics 9, 160–169 (1990)
Kobbelt, L.: Interpolatory subdivision on open quadrilateral nets with arbitrary topology. Computer Graphics Forum 15, 409–420 (1996)
Loop, C.T.: Smooth ternary subdivision of triangular meshes. In: Cohen, A., Merrien, J.L., Schumaker, L.L. (eds.) Curve and Surface Fitting: Saint-Malo 2002, pp. 295–302. Nashboro Press (2003)
Dodgson, N.A., Sabin, M.A., Barthe, L., Hassan, M.F.: Towards a ternary interpolating subdivision scheme for the triangular mesh. University of Cambridge Computer Laboratory Technical Report No. 539 (2002)
Sabin, M.A.: Eigenanalysis and artifacts of subdivision curves and surfaces. In: Iske, A., Quak, E., Floater, M.S. (eds.) Tutorials on Multiresolution in Geometric Modelling, pp. 69–92. Springer, Heidelberg (2002)
Zorin, D., Schröder, P.: A unified framework for primal/dual quadrilateral subdivision schemes. Computer Aided Geometric Design 18, 429–454 (2001)
Oswald, P., Schröder, P.: Composite primal/dual \(\sqrt3\)-subdivision schemes. Computer Aided Geometric Design 20, 135–164 (2003)
Ivrissimtzis, I.P., Dodgson, N.A., Hassan, M.F., Sabin, M.A.: On the geometry of recursive subdivision. International Journal of Shape Modeling 8, 23–42 (2002)
Ivrissimtzis, I.P., Dodgson, N.A., Sabin, M.A.: Recursive subdivision and hypergeometric functions. In: Proceedings of SMI2002: International Conference on Shape Modelling and Applications, pp. 145–153. IEEE Press, Los Alamitos (2002)
Labsik, U., Greiner, G.: Interpolatory \(\sqrt3\)-subdivision. Computer Graphics Forum 19, 131–138 (2000)
Hassan, M.F., Ivrissimtzis, I.P., Dodgson, N.A., Sabin, M.A.: An interpolating 4-point C 2 ternary stationary subdivision scheme. Computer Aided Geometric Design 19, 1–18 (2002)
Han, B.: Classification and construction of bivariate subdivision schemes. In: Cohen, A., Merrien, J.L., Schumaker, L.L. (eds.) Curve and Surface Fitting: Saint-Malo 2002, pp. 187–197. Nashboro Press (2003)
Ostromoukhov, V., Donohue, C., Jodoin, P.M.: Fast hierarchical importance sampling with blue noise properties. In: ACM Transactions on Graphics, vol. 23, pp. 488–495 (2004); Proc. SIGGRAPH 2004 (2004)
Dodgson, N.A., Ivrissimtzis, I.P., Sabin, M.A.: Characteristics of dual \(\sqrt3\) subdivision schemes. In: Cohen, A., Merrien, J.L., Schumaker, L.L. (eds.) Curve and Surface Fitting: Saint-Malo 2002, pp. 119–128. Nashboro Press (2003)
Niven, I.M.: Irrational Numbers. In: Carus Mathematical Monographs, Wiley, New York (1956)
Ivrissimtzis, I.P., Sabin, M.A., Dodgson, N.A.: \(\sqrt5\) subdivision. In: Dodgson, N.A., Floater, M.S., Sabin, M.A. (eds.) Advances in Multiresolution for Geometric Modelling, pp. 285–299. Springer, Heidelberg (2005)
Maillot, J., Stam, J.: A unified subdivision scheme for polygonal modeling. Computer Graphics Forum 20, 471–479 (2001)
Dyn, N., Levin, D., Simoens, J.: Face value subdivision schemes on triangulations by repeated averaging. In: Cohen, A., Merrien, J.L., Schumaker, L.L. (eds.) Curve and Surface Fitting: Saint-Malo 2002, pp. 129–138. Nashboro Press (2003)
Morin, G., Warren, J., Weimer, H.: A subdivision scheme for surfaces of revolution. Computer Aided Geometric Design 18, 483–502 (2001)
Stam, J., Loop, C.: Quad/triangle subdivision. Computer Graphics Forum 22, 79–85 (2003)
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Dodgson, N.A. (2005). An Heuristic Analysis of the Classification of Bivariate Subdivision Schemes. In: Martin, R., Bez, H., Sabin, M. (eds) Mathematics of Surfaces XI. Lecture Notes in Computer Science, vol 3604. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11537908_10
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DOI: https://doi.org/10.1007/11537908_10
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