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Issue No.02 - Feb. (2013 vol.19)
pp: 319-330
Minho Kim , Sch. of Comput. Sci., Univ. of Seoul, Seoul, South Korea
ABSTRACT
This paper presents an alternative box-spline filter for the body-centered cubic (BCC) lattice, the seven-direction quartic box-spline M7 that has the same approximation order as the eight-direction quintic box-spline M8 but a lower polynomial degree, smaller support, and is computationally more efficient. When applied to reconstruction with quasi-interpolation prefilters, M7 shows less aliasing, which is verified quantitatively by integral filter metrics and frequency error kernels. To visualize and analyze distributional aliasing characteristics, each spectrum is evaluated on the planes and lines with various orientations.
INDEX TERMS
splines (mathematics), data visualisation, filtering theory, polynomial approximation, signal reconstruction, distributional aliasing characteristics, quartic box-spline reconstruction, BCC lattice, alternative box-spline filter, body-centered cubic lattice, seven-direction quartic box-spline, approximation order, eight-direction quintic box-spline, lower polynomial degree, quasiinterpolation prefilter reconstruction, integral filter metrics, frequency error kernels, Lattices, Spline, Polynomials, Approximation methods, Kernel, Rendering (computer graphics), FCC, quasi-interpolation, Volume reconstruction, BCC lattice, box-spline
CITATION
Minho Kim, "Quartic Box-Spline Reconstruction on the BCC Lattice", IEEE Transactions on Visualization & Computer Graphics, vol.19, no. 2, pp. 319-330, Feb. 2013, doi:10.1109/TVCG.2012.130
REFERENCES
 [1] T. Blu and M. Unser, "Quantitative Fourier Analysis of Approximation Techniques: Part I - Interpolators and Projectors," IEEE Trans. Signal Processing, vol. 47, no. 10, pp. 2783-2795, Oct. 1999. [2] A. Cavaretta, C. Micchelli, and W. Dahmen, Stationary Subdivision. Am. Math. Soc., 1991. [3] L. Condat and D. Van De Ville, "Three-Directional Box-Splines: Characterization and Efficient Evaluation," IEEE Signal Processing Letters, vol. 13, no. 7, pp. 417-420, July 2006. [4] J.H. Conway and N.J.A. Sloane, Sphere Packings, Lattices and Groups, third ed. Springer-Verlag, 1998. [5] B. Csébfalvi, "An Evaluation of Prefiltered B-Spline Reconstruction for Quasi-Interpolation on the Body-Centered Cubic Lattice," IEEE Trans. Visualization and Computer Graphics, vol. 16, no. 3, pp. 499-512, May/June 2010. [6] B. Csébfalvi and B. Domonkos, "3D Frequency-Domain Analysis of Non-Separable Reconstruction Schemes by Using Direct Volume Rendering," Proc. 26th Spring Conf. Computer Graphics (SCCG '10), pp. 51-60, 2010. [7] B. Csébfalvi and M. Hadwiger, "Prefiltered B-Spline Reconstruction for Hardware-Accelerated Rendering of Optimally Sampled Volumetric Data," Vision, Modeling, and Visualization, pp. 325-332, 2006. [8] C. de Boor, K. Höllig, and S. Riemenschneider, Box Splines. Springer-Verlag, 1993. [9] D.E. Dudgeon and R.M. Mersereau, Multidimensional Digital Signal Processing. Prentice-Hall, Inc., 1984. [10] A. Entezari, R. Dyer, and T. Möller, "Linear and Cubic Box Splines for the Body Centered Cubic Lattice," Proc. Conf. Visualization, pp. 11-18, 2004. [11] A. Entezari, M. Mirzargar, and L. Kalantari, "Quasi-Interpolation on the Body Centered Cubic Lattice," Computer Graphics Forum, vol. 28, no. 3, pp. 1015-1022, June 2009. [12] A. Entezari and T. Möller, "Extensions of the Zwart-Powell Box Spline for Volumetric Data Reconstruction on the Cartesian Lattice," IEEE Trans. Visualization and Computer Graphics, vol. 12, no. 5, pp. 1337-1344, Sept. 2006. [13] A. Entezari, D. Van De Ville, and T. Möller, "Practical Box Splines for Reconstruction on the Body Centered Cubic lattice," IEEE Trans. Visualization and Computer Graphics, vol. 14, no. 2, pp. 313-328, Mar. 2008. [14] B. Finkbeiner, A. Entezari, D. Van De Ville, and T. Möller, "Efficient Volume Rendering on the Body Centered Cubic Lattice Using Box Splines," Computers and Graphics, vol. 34, no. 4, pp. 409-423, Aug. 2010. [15] M. Kim, A. Entezari, and J. Peters, "Box Spline Reconstruction on the Face-Centered Cubic Lattice," IEEE Trans. Visualization and Computer Graphics, vol. 14, no. 6, pp. 1523-1530, Nov./Dec. 2008. [16] M. Kim and J. Peters, "Fast and Stable Evaluation of Box-Splines via the Bernstein-Bézier Form," Numerical Algorithms, vol. 50, no. 4, pp. 381-399, Apr. 2009. [17] M. Kim and J. Peters, "Symmetric Box-Splines on Root Lattices," J. Computational and Applied Math., vol. 235, no. 14, pp. 3972-3989, May 2011. [18] S.R. Marschner and R.J. Lobb, "An Evaluation of Reconstruction Filters for Volume Rendering," Proc. Conf. Visualization, pp. 100-107, Oct. 1994. [19] M. Mirzargar and A. Entezari, "Voronoi Splines," IEEE Trans. Signal Processing, vol. 58, no. 9, pp. 4572-4582, Sept. 2010. [20] Persistence of Vision Pty. Ltd., "Persistence of Vision Raytracer, 2004," http:/www.povray.org, 2012. [21] J. Peters, "${\rm{C}}^2$ Surfaces Built from Zero Sets of the 7-Direction Box Spline," Proc. IMA Conf. the Math. of Surfaces, pp. 463-474, 1994. [22] D.P. Petersen and D. Middleton, "Sampling and Reconstruction of Wave-Number-Limited Functions in N-Dimensional Euclidean Spaces," Information and Control, vol. 5, no. 4, pp. 279-323, 1962. [23] S. Roettger, "The Volume Library (Online), Jan. 2012," http://www9.informatik.uni-erlangen.de/External vollib, 2012. [24] D.M.Y. Sommerville, "Space-Filling Tetrahedra in Euclidean Space," Proc. Edinburgh Math. Soc., vol. 41, pp. 49-57, Feb. 1922.
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