Multidimensional sampling: Difference between revisions

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In [[digital signal processing]], '''multidimensional sampling''' is the process of converting a function of a [[multidimensional variable]] into a discrete collection of values of the function measured on a discrete set of points. This article presents the basic result due to Petersen and Middleton<ref name="petmid62">D. P. Petersen and D. Middleton, "Sampling and Reconstruction of Wave-Number-Limited Functions in N-Dimensional Euclidean Spaces", Information and Control, vol. 5, pp. 279–323, 1962.</ref> on conditions for perfectly reconstructing a [[wavenumber]]-limited function from its measurements on a discrete [[Lattice (group)|lattice]] of points. This result, also known as the '''Petersen–Middleton theorem''', is a generalization of the [[Nyquist–Shannon sampling theorem]] for sampling one-dimensional [[band-limited]] functions to higher-dimensional [[Euclidean space]]s.

In essence, the Petersen–Middleton theorem shows that a wavenumber-limited function can be perfectly reconstructed from its values on an infinite lattice of points, provided the lattice is fine enough. The theorem provides conditions on the lattice under which perfect reconstruction is possible.

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==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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A simple illustration of aliasing can be obtained by studying low-resolution images. A gray-scale image can be interpreted as a function in two-dimensional space. An example of aliasing is shown in the images of brick patterns in Figure 5. The image shows the effects of aliasing when the sampling theorem's condition is not satisfied. If the lattice of pixels is not fine enough for the scene, aliasing occurs as evidenced by the appearance of the [[Moiré pattern]] in the image obtained. The image in Figure 6 is obtained when a smoothened version of the scene is sampled with the same lattice. In this case the conditions of the theorem are satisfied and no aliasing occurs.

S. P. Efimov from [[Bauman Moscow State Technical University]] in 1978 y. found an approach to ease the restrictions for spectrum ___domain.<ref>{{cite journal |last1=Efimov |first1=Sergei |title=Reconstruction of a field with finite-spectrum by samples of signals of filters |journal=Problemy Peredaci Informacii |date=1978 |volume=14 |issue=2 |pages=53-60 |url=https://mi.mathnet.ru/eng/ppi/p1534}}</ref> He considered N identical sampling lattices to be shifted arbitrarily to each other. Optimal sampling is valid for spectrum ___domain that shifted versions of is close-packed N times on reciprocal lattice. Therefore, ring can be overlapped by a set of hexagons instead of one. The [[James Webb Space Telescope]] array consists from 18 hexagons. Sampling on 18 shifted lattices is possible for 2-d Fourier transform of the array signal (i. e. for emitted signal).

===Optimal sampling lattices===

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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 | 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| 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| 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 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 | 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 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| 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==