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Since optimal lattices, in general, are non-separable, designing [[interpolation]] and [[reconstruction filter|reconstruction filters]] requires non-tensor-product (i.e., non-separable) filter design mechanisms. [[Box spline|Box splines]] 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 doi| 10.1109/TVCG.2007.70429|noedit}}</ref>. Hex-splines<ref>{{Cite doi| 10.1109/TIP.2004.827231}}</ref> are the generalization of [[B-splines]] for 2-D hexagonal lattices. Similarly, in 3-D and higher dimensions, Voronoi splines<ref> {{Cite doi| 10.1109/TSP.2010.2051808|noedit}}</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|Brillouin zones]] (i.e., <math>\Omega</math> in above) of these lattices (which are [[zonohedron|zonotopes]])<ref name="mdsinc">{{Cite doi| 10.1109/TIP.2011.2162421 |noedit}}</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"></ref>.
==Applications==
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