Multidimensional sampling: Difference between revisions

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Similarly, the configuration of uniformly spaced sampling points in one-dimension can be generalized to a [[Lattice (group)|lattice]] in higher dimensions. A lattice is a collection of points <math>\Lambda \subset \Re^n</math> of the form

<math>

\Lambda = \left\{ \sum_{i=1}^n a_i v_i \; | \; a_i \in\Bbbmathbb{Z} \right\}

</math>

where {''v''₁, ..., ''v''_''n''} is a [[Basis (linear algebra)|basis]] for <math>\Re^n</math>. The [[reciprocal lattice]] <math>\Gamma</math> corresponding to <math>\Lambda</math> is defined by

:<math>

\Gamma = \left\{ \sum_{i=1}^n a_i u_i \; | \; a_i \in\Bbbmathbb{Z} \right\}

</math>

where the vectors <math>u_i</math> are chosen to satisfy <math>\langle u_i, v_j \rangle = \delta_{ij}</math>. That is, if the vectors <math>u_i</math> form columns of a matrix <math>A</math> and <math>v_i</math> the columns of a matrix <math>B</math>, then <math>A=B^{-T}</math>. An example of a sampling lattice in two dimensional space is a [[hexagonal lattice]] depicted in Figure 1. The corresponding reciprocal lattice is shown in Figure 2. The reciprocal lattice of a [[square lattice]] in two dimensions is another square lattice. In three dimensional space the reciprocal lattice of a

[[Close-packing of equal spheres#FCC_and_HCP_Lattices|face-centered cubic (FCC) lattice]] is a body centered cubic (BCC) lattice.

==The theorem==

Let <math>\Lambda</math> denote a lattice in <math>\Re^n</math> and <math>\Gamma</math> the corresponding reciprocal lattice. The theorem of Petersen and Middleton<ref name="petmid62" /> states that a function <math>f(\cdot)</math> that is wavenumber-limited to a set <math>\Omega \subset \Re^n</math> can be exactly reconstructed from its measurements on <math>\Lambda</math> provided that the set <math>\Omega</math> does not overlap with any of its shifted versions <math>\Omega + x </math> where the shift ''x'' is any nonzero element of the reciprocal lattice <math>\Gamma</math>. In other words, <math>f(\cdot)</math> can be exactly reconstructed from its measurements on <math>\Lambda</math> provided that <math>\Omega \cap \{x+y:y\in\Omega\} = \phiemptyset </math> for all <math>x \in \Gamma\setminus\{0\}</math>.

==Reconstruction==

[[Image:Unaliased sampled spectrum in 2D.png|thumb|Fig. 3: Support of the sampled spectrum <math>\hat f_s(\cdot)</math> obtained by hexagonal sampling of a two-dimensional function wavenumber-limited to a circular disc. The blue circle represents the support <math>\Omega</math> of the original wavenumber-limited field, and the green circles represent the repetitions. In this example the spectral repetitions do not overlap and hence there is no aliasing. The original spectrum can be exactly recovered from the sampled spectrum.|right|300px]]

The generalization of the [[Poisson summation formula]] to higher dimensions <ref name="stewei71">E. M. Stein and G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces", Princeton University Press, Princeton, 1971.</ref> can be used to show that the samples, <math>\{f(x): x \in \Lambda\} </math>, of the function <math>f(\cdot)</math> on the lattice <math>\Lambda</math> are sufficient to create a [[periodic summation]] of the function <math>\hat f(\cdot)</math>. The result is:

{{NumBlk|:|<math>\hat f_s(\xi)\ \stackrel{\mathrm{def}}{=} \sum_{y \in \Gamma} \hat f\left(\xi - y\right) = \sum_{x \in \Lambda} |\Lambda|f(x) \ e^{-i 2\pi \langle x, \xi \rangle},</math>|{{EquationRef|Eq.1}}}}

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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 doijournal | last1 = Kunsch | first1 = H. R. | last2 = Agrell | first2 = E. | last3 = Hamprecht | first3 = F. A. | doi = 10.1109/TIT.2004.840864 |noedit title = Optimal Lattices for Sampling | journal = IEEE Transactions on Information Theory | volume = 51 | issue = 2 | pages = 634 | year = 2005 | 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 doijournal | last1 = Entezari | first1 = A. | last2 = Van De Ville | first2 = D. | last3 = Moller | first3 = T. | doi = 10.1109/TVCG.2007.70429 |noedit 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| citeseerx = 10.1.1.330.3851 }}</ref> Hex-splines<ref>{{Cite doijournal | 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| bibcode = 2004ITIP...13..758V | 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 doijournal | last1 = Mirzargar | first1 = M. | last2 = Entezari | first2 = A. | doi = 10.1109/TSP.2010.2051808 |noedit title = Voronoi Splines | journal = IEEE Transactions on Signal Processing | volume = 58 | issue = 9 | pages = 4572 | year = 2010 | bibcode = 2010ITSP...58.4572M }}</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 doijournal | last1 = Ye | first1 = W. | last2 = Entezari | first2 = A. | doi = 10.1109/TIP.2011.2162421 |noedit 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| bibcode = 2012ITIP...21.2969Y }}</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" />

==Applications==

The Petersen–Middleton theorem is useful in designing efficient sensor placement strategies in applications involving measurement of spatial phenomena such as seismic surveys, environment monitoring and spatial audio-field measurements.<ref>{{Cite book |last=Bardan |first=V. |title=69th EAGE Conference and Exhibition incorporating SPE EUROPEC 2007 |date=2007-06-11 |chapter=The Petersen-Middleton Theorem and Sampling of Seismic Data |chapter-url=https://www.earthdoc.org/content/papers/10.3997/2214-4609.201401831 |language=en |publisher=European Association of Geoscientists & Engineers |pages=cp |doi=10.3997/2214-4609.201401831 |isbn=978-90-73781-54-2}}</ref>

==References==