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Issue No.02 - March/April (2008 vol.14)
pp: 313-328
We introduce a family of box splines for efficient, accurate and smooth reconstruction of volumetric data sampled on the Body Centered Cubic (BCC) lattice, which is the favorable volumetric sampling pattern due to its optimal spectral sphere packing property. First, we construct a box spline based on the four principal directions of the BCC lattice that allows for a linear $C^0$ reconstruction. Then, the design is extended for higher degrees of continuity. We derive the explicit piecewise polynomial representation of the $C^0$ and $C^2$ box splines that are useful for practical reconstruction applications. We further demonstrate that approximation in the shift-invariant space---generated by BCC-lattice shifts of these box splines---is \emph{twice} as efficient as using the tensor-product B-spline solutions on the Cartesian lattice (with comparable smoothness and approximation order, and with the same sampling density). Practical evidence is provided demonstrating that not only the BCC lattice is generally a more accurate sampling pattern, but also allows for extremely efficient reconstructions that outperform tensor-product Cartesian reconstructions.
Spline and piecewise polynomial approximation, Spline and piecewise polynomial interpolation, Splines, Signal processing, Reconstruction, Finite volume methods
Alireza Entezari, Dimitri Van De Ville, Torsten Möller, "Practical Box Splines for Reconstruction on the Body Centered Cubic Lattice", IEEE Transactions on Visualization & Computer Graphics, vol.14, no. 2, pp. 313-328, March/April 2008, doi:10.1109/TVCG.2007.70429
[1] R.N. Bracewell, The Fourier Transform and Its Applications, McGraw-Hill Series in Electrical Eng. Circuits and Systems, third ed. McGraw-Hill, 1986.
[2] G. Burns, Solid State Physics. Academic Press, 1985.
[3] I. Carlbom, “Optimal Filter Design for Volume Reconstruction and Visualization,” Proc. IEEE Conf. Visualization (VIS '93), pp. 54-61, Oct. 1993.
[4] H. Carr, T. Möller, and J. Snoeyink, “Artifacts Caused by Simplicial Subdivision,” IEEE Trans. Visualization and Computer Graphics, vol. 12, no. 2, pp. 231-242, Mar./Apr. 2006.
[5] L. Condat and D. Van De Ville, “Quasi-Interpolating Spline Models for Hexagonally-Sampled Data,” IEEE Trans. Image Processing, vol. 16, no. 5, pp. 1195-1206, May 2007.
[6] J.H Conway and N.J.A. Sloane, Sphere Packings, Lattices and Groups, third ed. Springer, 1999.
[7] B. Csébfalvi, “Prefiltered Gaussian Reconstruction for High-Quality Rendering of Volumetric Data Sampled on a Body-Centered Cubic Grid,” Proc. IEEE Visualization Conf. (VIS '05), pp.311-318, 2005.
[8] M. Dæhlen, “On the Evaluation of Box Splines,” Mathematical Methods in Computer Aided Geometric Design, pp. 167-179, 1989.
[9] C. de Boor, K. Höllig, and S. Riemenschneider, “Box Splines,” Applied Mathematical Sciences, vol. 98, Springer, 1993.
[10] C. de Boor, “On the Evaluation of Box Splines,” Numerical Algorithms, vol. 5, nos. 1-4, pp. 5-23, 1993.
[11] D.E. Dudgeon and R.M. Mersereau, Multidimensional Digital Signal Processing, first ed. Prentice Hall, 1984.
[12] S.C. Dutta Roy and B. Kumar, “Digital Differentiators,” Handbook of Statistics, vol. 10, Elsevier, pp. 159-205, 1993.
[13] A. Entezari, R. Dyer, and T. Möller, “Linear and Cubic Box Splines for the Body Centered Cubic Lattice,” Proc. IEEE Visualization Conf. (VIS '04), pp. 11-18, Oct. 2004.
[14] A. Entezari, T. Meng, S. Bergner, and T. Möller, “A Granular Three Dimensional Multiresolution Transform,” Proc. Eurographics/IEEE-VGTC Symp. Visualization (EuroVis '06), pp. 267-274, May 2006.
[15] T.C. Hales, “Cannonballs and Honeycombs,” Notices of the AMS, vol. 47, no. 4, pp. 440-449, Apr. 2000.
[16] R.G. Keys, “Cubic Convolution Interpolation for Digital Image Processing,” IEEE Trans. Acoustics, Speech, and Signal Processing, vol. 29, no. 6, pp. 1153-1160, Dec. 1981.
[17] L. Kobbelt, “Stable Evaluation of Box Splines,” Numerical Algorithms, vol. 14, no. 4, pp. 377-382, 1997.
[18] H.R. Künsch, E. Agrell, and F.A. Hamprecht, “Optimal Lattices for Sampling,” IEEE Trans. Information Theory, vol. 51, no. 2, pp. 634-647, Feb. 2005.
[19] T. Theußl, O. Mattausch, T. Möller, and E. Gröller, “Reconstruction Schemes for High Quality Raycasting of the Body-Centered Cubic Grid,” Technical Report TR-186-2-02-11, Inst. Computer Graphics and Algorithms, Vienna Univ. of Tech nology, Dec. 2002.
[20] T. Theußl, T. Möller, and E. Gröller, “Optimal Regular Volume Sampling,” Proc. IEEE Visualization Conf. (VIS '01), pp. 91-98, Oct. 2001.
[21] S.R. Marschner and R.J. Lobb, “An Evaluation of Reconstruction Filters for Volume Rendering,” Proc. IEEE Conf. Visualization (VIS'94), pp. 100-107, 1994.
[22] J.H. McClellan, “The Design of Two-Dimensional Digital Filters by Transformations,” Proc. Seventh Ann. Princeton Conf. Information Sciences and Systems, pp. 247-251, 1973.
[23] T. Meng, B. Smith, A. Entezari, A.E. Kirkpatrick, D. Weiskopf, L. Kalantari, and T. Möller, “On Visual Quality of Optimal 3D Sampling and Reconstruction,” Proc. 33rd Graphics Interface Conf. (GI '07), pp. 265-272, May 2007.
[24] R.M. Mersereau, “The Processing of Hexagonally Sampled Two-Dimensional Signals,” Proc. IEEE, vol. 67, no. 6, pp. 930-949, June 1979.
[25] D.P. Mitchell and A.N. Netravali, “Reconstruction Filters in Computer Graphics,” Computer Graphics (Proc. ACM SIGGRAPH '88), vol. 22, pp. 221-228, Aug. 1988.
[26] T. Möller, R. Machiraju, K. Mueller, and R. Yagel, “A Comparison of Normal Estimation Schemes,” Proc. IEEE Visualization Conf. (VIS '97), pp. 19-26, Oct. 1997.
[27] T. Möller, K. Mueller, Y. Kurzion, R. Machiraju, and R. Yagel, “Design of Accurate and Smooth Filters for Function and Derivative Reconstruction,” Proc. IEEE Symp. Volume Visualization (VVS '98), pp. 143-151, Oct. 1998.
[28] A.V. Oppenheim and R.W. Schafer, Discrete-Time Signal Processing. Prentice Hall, 1989.
[29] D.P. Petersen and D. Middleton, “Sampling and Reconstruction of Wave-Number-Limited Functions in $N\hbox{-}{\rm Dimensional}$ ${\rm Euclidean}\;{\rm Spaces}$ ,” Information and Control, vol. 5, no. 4, pp.279-323, Dec. 1962.
[30] G. Strang and G.J. Fix, “A Fourier Analysis of the Finite Element Variational Method,” Constructive Aspects of Functional Analysis, pp. 796-830, 1971.
[31] P. Thévenaz, T. Blu, and M. Unser, “Interpolation Revisited,” IEEE Trans. Medical Imaging, vol. 19, no. 7, pp. 739-758, July 2000.
[32] D. Van De Ville, T. Blu, M. Unser, W. Philips, I. Lemahieu, and R. Van de Walle, “Hex-Splines: A Novel Spline Family for Hexagonal Lattices,” IEEE Trans. Image Processing, vol. 13, no. 6, pp. 758-772, June 2004.
[33] Wikipedia, Minkowski Addition,, 2007 , June 2007.
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