Summary
In this paper we introduce reconstruction kernels for the 3D optimal sampling lattice and demonstrate a practical realisation of a few. First, we review fundamentals of mul- tidimensional sampling theory. We derive the optimal regular sampling lattice in 3D, namely the Body Centered Cubic (BCC) lattice, based on a spectral sphere packing argument. With the introduction of this sampling lattice, we review some of its geometric properties and its dual lattice. We introduce the ideal reconstruction kernel in the space of bandlimited func- tions on this lattice. Furthermore, we introduce a family of box splines for reconstruction on this sampling lattice. We conclude the paper with some images and results of sampling on the BCC lattice and contrast it with equivalent samplings on the traditionally used Cartesian lattice. Our experimental results confirm the theory that BCC sampling yields a more accurate discrete representation of a signal comparing to the commonly used Cartesian sampling.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
P. Brémaud. Mathematical Principles of Signal Processing. Springer, Berlin, 2002.
G. Burns. Solid State Physics. Academic, New York, 1985.
J.H Conway and N.J.A. Sloane. Sphere Packings, Lattices and Groups, 3rd edition. Springer, Berlin, 1999.
C. de Boor, K Höllig, and S Riemenschneider. Box Splines. Springer, Berlin, 1993.
D. E. Dudgeon and R. M. Mersereau. Multidimensional Digital Signal Processing, 1st edition. Prentice-Hall, Englewood-Cliffs, NJ, 1984.
T.C. Hales. Cannonballs and honeycombs. Notices of the AMS, 47(4):440–449, April 2000.
C. Lanczos. Discourse on Fourier Series. New York, Hafner, 1966.
S.R. Marschner and R.J. Lobb. An evaluation of reconstruction filters for volume rendering. In R. Daniel Bergeron and Arie E. Kaufman, editors, Proceedings of the IEEE Conference on Visualization 1994, pages 100–107, Los Alamitos, CA, USA, October 1994. IEEE Computer Society Press.
T. Möller, R. Machiraju, K. Mueller, and R. Yagel. A comparison of normal estimation schemes. In Proceedings of the IEEE Conference on Visualization 1997, pages 19–26, October 1997.
G. Strang and G.J. Fix. A Fourier analysis of the finite element variational method. In Construct. Aspects of Funct. Anal., pages 796–830, 1971.
T. Theußl, T. Möller, and E. Gröller. Optimal regular volume sampling. In Proceedings of the IEEE Conference on Visualization 2001, pages 91–98, Oct. 2001.
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 Transactions on Image Processing, 13(6):758–772, June 2004.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Entezari, A., Dyer, R., Möller, T. (2009). From Sphere Packing to the Theory of Optimal Lattice Sampling. In: Möller, T., Hamann, B., Russell, R.D. (eds) Mathematical Foundations of Scientific Visualization, Computer Graphics, and Massive Data Exploration. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/b106657_12
Download citation
DOI: https://doi.org/10.1007/b106657_12
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-25076-0
Online ISBN: 978-3-540-49926-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)