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In [[digital signal processing]], '''Multidimensionalmultidimensional 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 bandlimited[[band-limited]] functions to higher-dimensional [[Euclidean spacesspace]]s.

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

As with the Nyquist–Shannon sampling theorem, this theorem also assumes an idealization of any real-world situation, as it only applies to functions that are sampled over an infinitude of points. Perfect reconstruction is mathematically possible for the idealized model but only an approximation for real-world functions and sampling techniques, albeit in practice often a very good one.

==Preliminaries==

{{No footnotes|section|date=October 2014}}

[[Image:Hexagonal_sampling_lattice.png|thumb|Fig. 1: A hexagonal sampling lattice <math>\Lambda</math> and its generating vectors ''v''₁ and ''v''₂|right|200px]]

[[Image:Reciprocal_latticeHexagonal sampling lattice.png|thumb|Fig. 21: TheA reciprocalhexagonal lattice <math>\Gamma</math> corresponding to thesampling lattice <math>\Lambda</math> of Fig. 1 and its generatingbasis vectors ''uv''₁ and ''uv''₂ (figure not to scale).|right|200px]]

[[Image:Hexagonal_sampling_latticeReciprocal lattice.png|thumb|Fig. 12: AThe reciprocal lattice <math>\Gamma</math> corresponding hexagonalto samplingthe lattice <math>\Lambda</math> of Fig. 1 and its generatingbasis vectors ''vu''₁ and ''vu''₂ (figure not to scale).|right|200px]]

The concept of a [[Bandlimiting|bandlimited]] function in one dimension can be generalized to the notion of a wavenumber-limited function in higher dimensions. Recall that the [[Fourier transform]] of an integrable function ''ƒ<math>f(.\cdot)''</math> on ''n''-dimensional Euclidean space is defined as:

:<math>\hat{f}(\xi) = \mathcal{F}(f)(\xi) = \int_{\Re^n} f(x) e^{-2\pi i \langle x,\xi \rangle} \, dx</math>

where ''x'' and ''ξ'' are ''n''-dimensional [[vector (mathematics)|vectors]], and <math>\langle x,\xi \rangle</math> is the [[inner product]] of the vectors. The function ''ƒ<math>f(.\cdot)''</math> is said to be wavenumber-limited to a set <math>\Omega</math> if the Fourier transform satisfies <math>\hat{f}(\xi) = 0</math> for <math>\xi \notin \Omega</math>.

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">< /ref> 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:Aliased_sampled_spectrum_in_2DUnaliased sampled spectrum in 2D.png|thumb|Fig. 43: 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. InThe thisblue example,circle represents the samplingsupport lattice<math>\Omega</math> isof notthe fineoriginal enoughwavenumber-limited field, and hence the discsgreen overlapcircles inrepresent the sampled spectrumrepetitions. ThusIn thethis spectrumexample withinthe <math>\Omega</math>spectral representedrepetitions bydo thenot blueoverlap circleand cannothence bethere recoveredis exactlyno duealiasing. toThe theoriginal overlapspectrum fromcan thebe repetitionsexactly (shownrecovered infrom green),the thussampled leading to aliasingspectrum.|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}}}}

where <math>|\Lambda| </math> represents the volume of the [[parallelepiped]] formed by the vectors {''v''₁, ..., ''v''_''n''}. This periodic function is often referred to as the sampled spectrum and can be interpreted as the analogue of the [[discrete-time Fourier transform]] (DTFT) in higher dimensions. If the original wavenumber-limited spectrum <math>\hat f(.\cdot)</math> is supported on the set <math>\Omega</math> then the function <math>\hat f_s(.\cdot)</math> is supported on periodic repetitions of <math>\Omega</math> shifted by points on the reciprocal lattice <math>\Gamma</math>. If the conditions of the Petersen-Middleton theorem are met, then the function <math>\hat f_s(\xi)</math> is equal to <math>\hat f(\xi)</math> for all <math>\xi \in \Omega</math>, and hence the original field can be exactly reconstructed from the samples. In this case therethe isreconstructed nofield aliasing inmatches the reconstruction.original Asfield anand examplecan supposebe thatexpressed <math>\Omega</math>in is a circular disc. Figure 3 illustrates the supportterms of <math>\hat f_s(.)</math> when the conditionssamples of the Petersen-Middleton theorem are met. We see that the spectral repetitions do not overlap. Figure 4 shows the scenario where the conditions are not met. In this case the spectral repetitions overlap leading to aliasing in the reconstruction.as

{{NumBlk|:|<math>f(x) = \sum_{y \in \Lambda} |\Lambda| f(y) \check \chi_\Omega(y - x)</math>,|{{EquationRef|Eq.2}}}}

where <math>\check \chi_\Omega(\cdot)</math> is the inverse Fourier transform of the [[Indicator function|characteristic function]] of the set <math>\Omega</math>. This interpolation formula is the higher-dimensional equivalent of the [[Whittaker–Shannon interpolation formula]].

As an example suppose that <math>\Omega</math> is a circular disc. Figure 3 illustrates the support of <math>\hat f_s(\cdot)</math> when the conditions of the Petersen-Middleton theorem are met. We see that the spectral repetitions do not overlap and hence the original spectrum can be exactly recovered.

==Implications==

Line 30 ⟶ 44:

===Aliasing===

{{main|Aliasing}}

[[Image:Unaliased_sampled_spectrum_in_2DAliased sampled spectrum in 2D.png|thumb|Fig. 34: 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. TheIn bluethis circle representsexample, the supportsampling <math>\Omega</math>lattice ofis thenot originalfine wavenumber-limited field,enough and hence the greendiscs circlesoverlap representin the repetitionssampled spectrum. InThus thisthe examplespectrum thewithin spectral<math>\Omega</math> repetitionsrepresented doby notthe overlapblue andcircle hencecannot therebe isrecovered noexactly aliasing.due Theto originalthe spectrumoverlap canfrom bethe exactlyrepetitions recovered(shown fromin thegreen), sampledthus spectrumleading to aliasing.|right|300px]]

[[Image:Aliased_sampled_spectrum_in_2D.png|thumb|Fig. 4: Support of the sampled spectrum <math>\hat f_s(.)</math> obtained by hexagonal sampling of a two-dimensional function wavenumber-limited to a circular disc. In this example, the sampling lattice is not fine enough and hence the discs overlap in the sampled spectrum. Thus the spectrum within <math>\Omega</math> represented by the blue circle cannot be recovered exactly due to the overlap from the repetitions (shown in green), thus leading to aliasing.|right|300px]]

[[File:Moire pattern of bricks small.jpg|thumb|205px|Fig. 5: Spatial aliasing in the form of a [[Moiré pattern]].]]

[[File:Moire pattern of bricks.jpg|thumb|205px|Fig. 6: Properly sampled image of brick wall.]]

The theorem gives conditions on sampling lattices for perfect reconstruction of the sampled. If the lattices are not fine enough to satisfy the Petersen-Middleton condition, then the field cannot be reconstructed exactly from the samples in general. In this case we say that the samples may be [[Aliasing|aliased]]. Again, consider the example in which <math>\Omega</math> is a circular disc. If the Petersen-Middleton conditions do not hold, the support of the sampled spectrum will be as shown in Figure 4. In this case the spectral repetitions overlap leading to aliasing in the reconstruction.

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 ''f(.)'' on the lattice <math>\Lambda</math> are sufficient to create a [[periodic summation]] of the function <math>\hat f(.)</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}}}}

where <math>|\Lambda| </math> represents the volume of the [[parallelepiped]] formed by the vectors {''v''₁, ..., ''v''_''n''}. This periodic function is often referred to as the sampled spectrum and can be interpreted as the analogue of the [[discrete-time Fourier transform]] (DTFT) in higher dimensions. If the original wavenumber-limited spectrum <math>\hat f(.)</math> is supported on the set <math>\Omega</math> then the function <math>\hat f_s(.)</math> is supported on periodic repetitions of <math>\Omega</math> shifted by points on the reciprocal lattice <math>\Gamma</math>. If the conditions of the Petersen-Middleton theorem are met, then the function <math>\hat f_s(\xi)</math> is equal to <math>\hat f(\xi)</math> for all <math>\xi \in \Omega</math>, and hence the original field can be exactly reconstructed from the samples. In this case there is no aliasing in the reconstruction. As an example suppose that <math>\Omega</math> is a circular disc. Figure 3 illustrates the support of <math>\hat f_s(.)</math> when the conditions of the Petersen-Middleton theorem are met. We see that the spectral repetitions do not overlap. Figure 4 shows the scenario where the conditions are not met. In this case the spectral repetitions overlap leading to aliasing in the reconstruction.

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.

===Optimal sampling lattices===

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_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 | 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==

{{Reflist}}

{{DSP}}

[[Category:Digital signal processing]]

[[Category:Multidimensional signal processing]]

[[Category:Theorems in Fourier analysis]]