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One of the objects of interest in designing a sampling scheme for wavenumber-limited fields is to identify the configuration of points that leads to the minimum sampling density, i.e., the density of sampling points per unit spatial volume in <math>\Re^n</math>. Typically the cost for taking and storing the measurements is proportional to the sampling density employed. Often in practice, the natural approach to sample two-dimensional fields is to sample it at points on a [[Lattice (group)|rectangular lattice]]. However, this is not always the ideal choice in terms of the sampling density. The theorem of Petersen and Middleton can be used to identify the optimal lattice for sampling fields that are wavenumber-limited to a given set <math>\Omega \subset \Re^d</math>. For example, it can be shown that the lattice in <math>\Re^2</math> with minimum spatial density of points that admits perfect reconstructions of fields wavenumber-limited to a circular disc in <math>\Re^2</math> is the hexagonal lattice.<ref name="mer79">D. R. Mersereau, “The processing of hexagonally sampled two-dimensional signals,” Proceedings of the IEEE, vol. 67, no. 6, pp. 930 – 949, June 1979.</ref> As a consequence, hexagonal lattices are preferred for sampling [[Isotropy|isotropic fields]] in <math>\Re^2</math>.
Optimal sampling lattices have been studied in higher dimensions.<ref>{{Cite journal | last1 = Kunsch | first1 = H. R. | last2 = Agrell | first2 = E. | last3 = Hamprecht | first3 = F. A. | doi = 10.1109/TIT.2004.840864 | title = Optimal Lattices for Sampling | journal = IEEE Transactions on Information Theory | volume = 51 | issue = 2 | pages = 634 | year = 2005 | pmid = | pmc = | url = https://research.chalmers.se/en/publication/11977 }}</ref> Generally, optimal [[sphere packing]] lattices are ideal for sampling smooth stochastic processes while optimal sphere covering lattices<ref>J. H. Conway, N. J. A. Sloane. Sphere packings, lattices and groups. Springer, 1999.</ref> are ideal for sampling rough stochastic processes.
Since optimal lattices, in general, are non-separable, designing [[interpolation]] and [[reconstruction filter]]s requires non-tensor-product (i.e., non-separable) filter design mechanisms. [[Box spline]]s provide a flexible framework for designing such non-separable reconstruction [[Finite impulse response|FIR]] filters that can be geometrically tailored for each lattice.<ref>A. Entezari. Optimal sampling lattices and trivariate box splines. [Vancouver, BC.]: Simon Fraser University, 2007. <http://summit.sfu.ca/item/8178>.</ref><ref name="fourDir">{{Cite journal | last1 = Entezari | first1 = A. | last2 = Van De Ville | first2 = D. | last3 = Moller | first3 = T. | doi = 10.1109/TVCG.2007.70429 | title = Practical Box Splines for Reconstruction on the Body Centered Cubic Lattice | journal = IEEE Transactions on Visualization and Computer Graphics | volume = 14 | issue = 2 | pages = 313–328 | year = 2008 | pmid = 18192712| pmc = | citeseerx = 10.1.1.330.3851 }}</ref> Hex-splines<ref>{{Cite journal | last1 = Van De Ville | first1 = D. | last2 = Blu | first2 = T. | last3 = Unser | first3 = M. | last4 = Philips | first4 = W. | last5 = Lemahieu | first5 = I. | last6 = Van De Walle | first6 = R. | doi = 10.1109/TIP.2004.827231 | title = Hex-Splines: A Novel Spline Family for Hexagonal Lattices | journal = IEEE Transactions on Image Processing | volume = 13 | issue = 6 | pages = 758–772 | year = 2004 | pmid = 15648867| pmc = | url = http://infoscience.epfl.ch/record/63112 }}</ref> are the generalization of [[B-splines]] for 2-D hexagonal lattices. Similarly, in 3-D and higher dimensions, Voronoi splines<ref>{{Cite journal | last1 = Mirzargar | first1 = M. | last2 = Entezari | first2 = A. | doi = 10.1109/TSP.2010.2051808 | title = Voronoi Splines | journal = IEEE Transactions on Signal Processing | volume = 58 | issue = 9 | pages = 4572 | year = 2010 | pmid = | pmc = }}</ref> provide a generalization of [[B-splines]] that can be used to design non-separable FIR filters which are geometrically tailored for any lattice, including optimal lattices.
Explicit construction of ideal low-pass filters (i.e., [[sinc]] functions) generalized to optimal lattices is possible by studying the geometric properties of [[Brillouin zone]]s (i.e., <math>\Omega</math> in above) of these lattices (which are [[zonohedron|zonotopes]]).<ref name="mdsinc">{{Cite journal | last1 = Ye | first1 = W. | last2 = Entezari | first2 = A. | doi = 10.1109/TIP.2011.2162421 | title = A Geometric Construction of Multivariate Sinc Functions | journal = IEEE Transactions on Image Processing | volume = 21 | issue = 6 | pages = 2969–2979 | year = 2012 | pmid = 21775264| pmc = }}</ref> This approach provides a closed-form explicit representation of <math>\check \chi_\Omega(\cdot)</math> for general lattices, including optimal sampling lattices. This construction provides a generalization of the [[Lanczos filter]] in 1-D to the multidimensional setting for optimal lattices.<ref name="mdsinc" />
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