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Tunnel-free voxelisation of rational Bézier surfaces

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Abstract

To synthesize natural and artificial objects into a hybrid graphics scene represented by a set of voxels, voxelisation of geometric models is necessary. Rational parametric surfaces have been widely used in the representation of free-form surfaces. Voxelisation of these surfaces is therefore of great importance in the development of a voxel-based modeling system. A key issue is to develop a tunnel-free voxelisation algorithm for these continuous surfaces. In this paper, we propose such an algorithm for a rational Bézier surface. We derive the bound of the parametric steps to ensure that the voxelised rational Bézier surface is, by our algorithm, 6-tunnel-free, and we give the mathematical proof of this property. For efficient computation, we employ the forward difference technique in homogeneous form in the implementation of the algorithm. For more general applications, we show that voxelisation of a NURBS surface can be realised by first converting it into a piecewise rational Bézier surface and then voxelising each of the rational Bézier surfaces. We indicate the advantages carrying out this procedure.

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References

  1. Andres E, Nehlig P, Francon J (1997) Tunnel-free supercover 3D polygon and polyhedra. Comput Graphics Forum 16(3):C3–C14

    Article  Google Scholar 

  2. Chang SL, Shantz M, Rocchetti R (1989) Rendering cubic curves and surfaces with integer adaptive forward differencing. Comput Graphics 23(3):157–166

    Article  Google Scholar 

  3. Cohen-or D, Kaufman A (1991) Scan conversion algorithms for linear and quadric objects. In: A. Kaufman (ed) Volume visualization. IEEE Computer Society Press, Los Alamitos, pp 280–301

  4. Cohen-or D, Kaufman A, Wang YX (1994) Generating a smooth voxel-based model from an irregular polygon mesh. Visual Comput 10(6):295–305

    Article  Google Scholar 

  5. Cohen-or D, Kaufman A, Kong TY (1996) On the soundness of surface voxelisations. In: T.Y. Kong, A. Rosenfeld (eds) Topological algorithms for digital image processing. Elsevier Science, Amsterdam, New York, pp 181–204

  6. Cohen-or D, Kaufman A (1997) 3D line voxelisation and connectivity control. IEEE Comput Graphics Appl Nov/Dec 17(6):80–87

    Google Scholar 

  7. Cohen-or D, Kaufman A (1995) Fundamentals of surface voxelisation. Graphical Models Image Process 57(6):453–461

    Article  Google Scholar 

  8. Farin G (ed) (1991) NURBS for curve and surface design. SIAM, Philadelphia

  9. Floater MS (1992) Derivatives of rational Bézier curves. CAGD 9:161–174

    MathSciNet  Google Scholar 

  10. Hersch RD (1986) Descriptive contour fill of partly degenerated shapes. IEEE Comput Graphics Appl 6/7:61–70

    Google Scholar 

  11. Huang J, Yagel R, Filippov V (1998) Accurate methods for the voxelisation of planar objects. IEEE VOLVIS ’98, October 19–20, Imperial IV Research Triangle Park, NC

  12. Jones MW (1996) The production of volume data from triangular meshes using voxelisation. Comput Graphics Forum 15(5):311–318

    Article  Google Scholar 

  13. Kaufman A, Cohen-or D, Yagel R (1993) Volume graphics. IEEE Comput 26(7):51–64

    Article  Google Scholar 

  14. Kaufman A (1987) An algorithm for 3D scan-conversion of polygons. EUROGRAPHICS ’87 pp 197–208

  15. Kaufman A (1987) Efficient algorithms for 3D scan-conversion of parametric curves, surfaces, and volumes. Comput Graphics 21(3):171–179

    Article  MathSciNet  Google Scholar 

  16. Kumar S, Manocha D, Lastra A (1996) Interactive display of large-scaled NURBS models. IEEE Trans Visual Comput Graphics 2(4):323–336

    Article  Google Scholar 

  17. Piegle L, Tiller W (1997) The NURBS book. Springer-Verlag, Berlin Heidelberg

  18. Lin F, Pan YH (1992) A stack-based approach for shading of region. Comput Graphics 16(1):79–84

    Article  Google Scholar 

  19. Lin F, Seah HS (1998) An effective 3D seed fill algorithm. Comput Graphics 22(5):641–644

    Article  Google Scholar 

  20. Pavlidis T (1985) Scan conversion of regions bounded by parabolic splines. IEEE Comput Graphics Appl 5/6:47–53

    Google Scholar 

  21. Rockwood A (1987) A generalized scanning technique for display of parametrically defined surfaces. IEEE Comput Graphics Appl 8:15–26

    Article  Google Scholar 

  22. Saito T, Wang GJ, Sederberg TW (1995) Hodographs and normals of rational curves and surfaces. CAGD 12:417–430

    Google Scholar 

  23. Sederbeg TW, Zundel AK (1989) Scan line display of algebraic surfaces. Comput Graphics 23(3):147–156

    Article  Google Scholar 

  24. Stolte N, Caubet R (1997) Comparison between different rasterisation methods for implicit surfaces. In:R.A. Earnshaw, J.A. Vince, H. Jones (eds) Visualization and modeling. Academic Press, San Diego, pp 191–201

  25. Taubin G (1994) Rasterizing algebraic curves and surfaces. IEEE Comput Graphics Appl 14(2):14–23

    Article  Google Scholar 

  26. van Wyk CJ (1984) Clipping to the boundary of a circular-arc polygon. Comput Vision, Graphics Image Process 25:383–392

  27. Wang GJ, Sederberg TW, Saito T (1997) Partial derivative of rational Bézier surfaces. CAGD 14:377–381

    Google Scholar 

  28. Wu ZK, Seah HS, Lin F (2000) NURBS volume for modelling complex objects. In: Volume graphics. Springer-Verlag, London, pp 159–167

  29. Yao CF, Rokne JG (1997) Applying rounding-up integral linear interpolation to the scan-conversion of filled polygon. Comput Graphics Forum 16(2):101–106

    Article  Google Scholar 

  30. Zhou JW, Lin F, Seah HS (2001) A volume modeling component of CAD. International Workshop on Volume Graphics, NY, pp 63–70

  31. Avila R, He T, Hong L, Kaufman A, Pfister H, Silva C, Sobierajski L, Wang S (1994) VolVis: a diversified volume visualization system. In: IEEE Visualization Proceedings. IEEE Computer Society Press, Washington DC, USA, pp 31–38

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Zhongke, W., Feng, L. & Soon, S. Tunnel-free voxelisation of rational Bézier surfaces. Vis Comput 19, 505–512 (2003). https://doi.org/10.1007/s00371-003-0215-1

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