Fast and stable evaluation of splines and their derivatives generated by the seven-direction quartic box-spline

Abstract

In this paper, we propose a fast and stable evaluation method for the analytic derivatives of splines generated by the 7-direction quartic box-spline. We can maintain the spline structure by determining the derivative functions that can be represented as finite differences of box-splines defined by the sub-directions. Thus, the evaluation overhead can be reduced. We demonstrate that the first and second derivative functions are composed of only three cubic and six quadratic polynomial formulas, respectively, owing to their symmetries. Moreover, for each derivative order, all of the required functions possess change-of-variables relation with each other. Therefore, additional formulas are not required. As a result, for a given point, we only need to evaluate one quartic, three cubic, and six quadratic polynomial formulas to evaluate its spline value, gradient, and Hessian, respectively. This reduction in cases is especially advantageous for graphics processing unit (GPU) kernels, where conditional statements significantly degrade performance. To verify our technique, we implemented a real-time curvature-based GPU isosurface raycaster. Compared with other implementations, our method (i) achieves superior accuracy, (ii) is more than four times faster, and (iii) requires less than 15% of memory.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

References

  1. 1.

    Blu, T., Unser, M.: Quantitative fourier analysis of approximation techniques. I. interpolators and projectors. IEEE Trans. Signal Process. 47(10), 2783–2795 (1999). https://doi.org/10.1109/78.790659

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Bogner, S., Rüde, U., Harting, J.: Curvature estimation from a volume-of-fluid indicator function for the simulation of surface tension and wetting with a free-surface lattice Boltzmann method. Phys. Rev. E 93(4), 043,302 (2016). https://doi.org/10.1103/PhysRevE.93.043302

    MathSciNet  Article  Google Scholar 

  3. 3.

    Coeurjolly, D., Lachaud, J.O., Levallois, J.: Multigrid convergent principal curvature estimators in digital geometry. Comput. Vis. Image Underst. 129(4), 27–41 (2014). https://doi.org/10.1016/j.cviu.2014.04.013

    Article  Google Scholar 

  4. 4.

    Condat, L., Möller, T.: Quantitative error analysis for the reconstruction of derivatives. IEEE Trans. Signal Process. 59(6), 2965–2969 (2011). https://doi.org/10.1109/TSP.2011.2119316, http://ieeexplore.ieee.org/document/5720324/

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    de Boor, C.: On the evaluation of box splines. Numer. Algor. 5(1–4), 5–23 (1993). https://doi.org/10.1007/BF02109280

    MathSciNet  Article  Google Scholar 

  6. 6.

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

  7. 7.

    Elliott, H., Fischer, R.S., Myers, K.A., Desai, R.A., Gao, L., Chen, C.S., Adelstein, R.S., Waterman, C.M., Danuser, G.: Myosin II controls cellular branching morphogenesis and migration in three dimensions by minimizing cell-surface curvature. Nat. Cell Biol. 17(2), 137–147 (2015). https://doi.org/10.1038/ncb3092

    Article  Google Scholar 

  8. 8.

    Entezari, A.: 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). https://doi.org/10.1109/TVCG.2006.141

    Article  Google Scholar 

  9. 9.

    Fang, M., Lu, J., Peng, Q.: Volumetric data modeling and analysis based on seven-directional box spline. Sci. China Inform. Sci. 57(6), 1–14 (2014). https://doi.org/10.1007/s11432-013-4941-3

    Article  MATH  Google Scholar 

  10. 10.

    Gu, W., Fang, M.e., Ma, L.: High-quality topological structure extraction of volumetric data on Cˆ2-continuous framework. Comput. Aided Geom. Des. 35-36, 215–224 (2015). https://doi.org/10.1016/j.cagd.2015.03.004

    MathSciNet  Article  Google Scholar 

  11. 11.

    Kim, M.: Analysis of symmetry groups of box-splines for evaluation on GPUs. Graph. Model. 93, 14–24 (2017). https://doi.org/10.1016/j.gmod.2017.08.001

    MathSciNet  Article  Google Scholar 

  12. 12.

    Kim, M., Peters, J.: Fast and stable evaluation of box-splines via the BB-form. Numer. Algor. 50(4), 381–399 (2009). https://doi.org/10.1007/s11075-008-9231-6

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Kindlmann, G., Whitaker, R., Tasdizen, T., Moller, T.: Curvature-based transfer functions for direct volume rendering: methods and applications. In: IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, pp 513–520. IEEE (2003), https://doi.org/10.1109/VISUAL.2003.1250414

  14. 14.

    Kobbelt, L.: Stable evaluation of box-splines. Numer. Algor. 14(4), 377–382 (1997). https://doi.org/10.1023/A:1019133501773

    MathSciNet  Article  Google Scholar 

  15. 15.

    Lee, J., Nishikawa, R.M., Reiser, I., Boone, J.M., Lindfors, K.K.: Local curvature analysis for classifying breast tumors: preliminary analysis in dedicated breast CT. Med. Phys. 42(9), 5479–5489 (2015). https://doi.org/10.1118/1.4928479

    Article  Google Scholar 

  16. 16.

    Marschner, S.R., Lobb, R.J.: An evaluation of reconstruction filters for volume rendering. In: Proceedings of the Conference on Visualization ’94, VIS ’94, pp 100–107. IEEE Computer Society Press, Los Alamitos (1994), https://doi.org/10.1109/VISUAL.1994.346331

  17. 17.

    Musuvathy, S., Martin, T., Cohen, E.: Ridge extraction from isosurfaces of volumetric data using implicit B-splines. In: 2010 Shape Modeling International Conference, pp 163–174. IEEE (2010), https://doi.org/10.1109/SMI.2010.29

  18. 18.

    Okamoto, M., Kurokawa, K., Matsuura-Tokita, K., Saito, C., Hirata, R., Nakano, A.: High-curvature domains of the ER are important for the organization of ER exit sites in Saccharomyces cerevisiae. J. Cell Sci. 125(14), 3412–3420 (2012). https://doi.org/10.1242/jcs.100065

    Article  Google Scholar 

  19. 19.

    van Pelt, R., Vilanova, A., van de Wetering, H.: Illustrative volume visualization using GPU-based particle systems. IEEE Trans. Vis. Comput. Graph. 16(4), 571–582 (2010). https://doi.org/10.1109/TVCG.2010.32

    Article  Google Scholar 

  20. 20.

    van Pelt, R., Vilanova, A., van de Wetering, H.: Local geometry of isoscalar surfaces. IEEE Trans. Vis. Comput. Graph. 16(4), 571–582 (2010). https://doi.org/10.1103/PhysRevE.76.056316

    Article  Google Scholar 

  21. 21.

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

  22. 22.

    Pienaar, R., Fischl, B., Caviness, V., Makris, N., Grant, P.E.: A methodology for analyzing curvature in the developing brain from preterm to adult. Int. J. Imaging Syst. Technol. 18(1), 42–68 (2008). https://doi.org/10.1002/ima.20138

    Article  Google Scholar 

  23. 23.

    Pinter, C.C.: A Book of Abstract Algebra. Dover Publications (1990)

  24. 24.

    Senechal, M.: Which tetrahedra fill space? Math. Mag. 54(5), 227–243 (1981). https://doi.org/10.2307/2689983

    MathSciNet  Article  Google Scholar 

  25. 25.

    Sigg, C., Hadwiger, M.: Fast third-order texture filtering. In: Pharr, M., Fernando, R (eds.) GPU Gems 2, chap. 20, pp 313–330. Addison-Wesley (2005)

  26. 26.

    Stanford University: The stanford 3d scanning repository. http://graphics.stanford.edu/data/3Dscanrep (2014). [Online; accessed 13 Feb. 2018]

Download references

Acknowledgments

We appreciate Hyunjun Kim for generating the signed distance volumetric datasets used in this study.

Funding

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1B03932569).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Minho Kim.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kim, M. Fast and stable evaluation of splines and their derivatives generated by the seven-direction quartic box-spline. Numer Algor 86, 887–909 (2021). https://doi.org/10.1007/s11075-020-00916-7

Download citation

Keywords

  • Box-splines
  • Splines
  • Exact evaluation
  • Group theory
  • GPGPU