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Fast and stable evaluation of box-splines via the BB-form

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Abstract

To repeatedly evaluate linear combinations of box-splines in a fast and stable way, in particular along knot planes, the box-spline is converted to and tabulated as piecewise polynomial in BB-form (Bernstein–Bézier-form). We show that the BB-coefficients can be derived and stored as integers plus a rational scale factor and derive a hash table for efficiently accessing the polynomial pieces. This pre-processing, the resulting evaluation algorithm and use in a widely-used ray-tracing package are illustrated for splines based on two trivariate box-splines: the seven-directional box-spline on the Cartesian lattice and the six-directional box-spline on the face-centered cubic lattice.

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References

  1. Casciola, G., Franchini, E., Romani, L.: The mixed directional difference-summation algorithm for generating the Bézier net of a trivariate four-direction box-spline. Numer. Algorithms 43(1), 1017–1398 (2006)

    Article  MathSciNet  Google Scholar 

  2. Cavaretta, A.S., Micchelli, C.A., Dahmen, W.: Stationary Subdivision. American Mathematical Society, Boston, MA, USA (1991)

    Google Scholar 

  3. Chui, C.K., Lai, M.-J.: Algorithms for generating B-nets and graphically displaying spline surfaces on three- and four-directional meshes. Comput. Aided Geom. Des. 8(6), 479–493 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  4. Chung, K.C., Yao, T.H.: On lattices admitting unique lagrange interpolations. SIAM J. Numer. Anal. 14(4), 735–743 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  5. Condat, L., Van De Ville, D.: Three-directional box-splines: characterization and efficient evaluation. IEEE Signal Process. Lett. 13(7), 417–420 (2006)

    Article  Google Scholar 

  6. Conway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices and Groups, third edn. Springer-Verlag New York, Inc., New York, NY, USA (1998)

    Google Scholar 

  7. de Boor, C.: B-form Basics, Geometric Modeling, pp. 131–148. SIAM, Philadelphia, PA (1987)

    Google Scholar 

  8. de Boor, C.: On the evaluation of box splines. Numer. Algor. 5(1–4), 5–23 (1993)

    Article  MATH  Google Scholar 

  9. de Boor, C. Höllig, K., Riemenschneider, S.: Box Splines. Springer-Verlag New York, Inc., New York, NY, USA (1993)

    MATH  Google Scholar 

  10. Entezari, A.: Optimal sampling lattices and trivariate box splines. Ph.D. thesis, Simon Fraser University, Vancouver, Canada. http://www.cs.sfu.ca/~torsten/Publications/Thesis/entezari.pdf (July 2007)

  11. Entezari, A., Möller, T.: Extensions of the Zwart-Powell box spline for volumetric data reconstruction on the Cartesian lattice. IEEE Trans. Vis. Comput. Graph. 12(5), 1337–1344 (2006)

    Article  Google Scholar 

  12. Fuchs, H., Kedem, Z.M., Naylor, B.F.: On visible surface generation by a priori tree structures. In: SIGGRAPH ‘80: Proceedings of the 7th Annual Conference on Computer Graphics and Interactive Techniques (New York, NY, USA), pp. 124–133. ACM Press (1980)

  13. Dæhlen, M.: On the Evaluation of Box Splines. Mathematical Methods in Computer Aided Geometric Design, pp. 167–179 (San Diego, CA, USA). Academic Press Professional, Inc. (1989)

  14. Jetter, K., Stöckler, J.: Algorithms for cardinal interpolation using box splines and radial basis functions. Numer. Math. 60(1), 97–114 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  15. Kim, M.: tribox: Matlab packages for evaluation of 6- and 7-directional trivariate box-splines. http://www.cise.ufl.edu/research/SurfLab/tribox

  16. Kobbelt, L.: Stable evaluation of box-splines. Numer. Algor. 14(4), 377–382 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  17. Lai, M.-J.: Fortran subroutines for B-nets of box splines on three- and four-directional meshes. Numer. Algor. 2(1), 33–38 (1992)

    Article  MATH  Google Scholar 

  18. MathWorks: Matlab 7 Function Reference: volume 2 (F–O) http://www.mathworks.com/access/helpdesk/help/pdf_doc/matlab/refbook2.pdf (2006)

  19. MathWorks: Matlab 7 Function Reference: volume 3 (P–Z) http://www.mathworks.com/access/helpdesk/help/pdf_doc/matlab/refbook3.pdf (2006)

  20. McCool, M.D.: Accelerated evaluation of box splines via a parallel inverse FFT. Comput. Graph. Forum 15(1), 35–45 (1996)

    Article  MathSciNet  Google Scholar 

  21. Naylor, B.F.: Binary space partitioning trees. In: Mehta, D.P., Sahni, S. (eds) Handbook of Data Structures and Applications. Chapman & Hall (2005)

  22. Nicolaides, R.A.: On a class of finite elements generated by lagrange interpolation. SIAM J. Numer. Anal. 9(3), 435–445 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  23. Persistence of Vision Pty. Ltd.: POV-Ray: the Persistence of Vision Raytracer. http://www.povray.org

  24. Peters, J.: C 2 surfaces built from zero sets of the 7-direction box spline. In: Mullineux, G. (ed.) IMA Conference on the Mathematics of Surfaces, pp. 463–474. Clarendon Press (1994)

  25. Peters, J., Wittman, M.: Box-spline based CSG blends. In: Proceedings of the Fourth ACM Symposium on Solid Modeling and Applications, pp. 195–205. SIGGRAPH, ACM Press (1997)

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Authors and Affiliations

  1. Department of Computer and Information Science and Engineering, University of Florida, Gainesville, FL, 32611-6120, USA

    Minho Kim & Jörg Peters

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  1. Minho Kim
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  2. Jörg Peters
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Correspondence to Minho Kim.

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Kim, M., Peters, J. Fast and stable evaluation of box-splines via the BB-form. Numer Algor 50, 381–399 (2009). https://doi.org/10.1007/s11075-008-9231-6

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