Multidimensional sampling

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 on conditions for perfectly reconstructing a wavenumber-limited function from its measurements on a discrete 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 functions to higher-dimensional Euclidean spaces.

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.

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

Fig. 1: A hexagonal sampling lattice $\Lambda$ and its generating vectors v₁ and v₂

Fig. 2: The reciprocal lattice $\Gamma$ corresponding to the lattice $\Lambda$ of Fig. 1 and its generating vectors u₁ and u₂ (figure not to scale).

The concept of a 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(.) on n-dimensional Euclidean space is defined as:

${\hat {f}}(\xi )={\mathcal {F}}(f)(\xi )=\int _{\Re ^{n}}f(x)e^{-2\pi i\langle x,\xi \rangle }\,dx$

where x and ξ are n-dimensional vectors, and $\langle x,\xi \rangle$ is the inner product of the vectors. The function ƒ(.) is said to be wavenumber-limited to a set $\Omega$ if the Fourier transform satisfies ${\hat {f}}(\xi )=0$ for $\xi \notin \Omega$.

Similarly, the configuration of uniformly spaced sampling points in one-dimension can be generalized to a lattice in higher dimensions. A lattice is a collection of points $\Lambda \subset \Re ^{n}$ of the form $\Lambda =\left\{\sum _{i=1}^{n}a_{i}v_{i}\;|\;a_{i}\in {\mathbb {Z}}\right\}$ where {v₁, ..., v_n} is a basis for $\Re ^{n}$. The reciprocal lattice $\Gamma$ corresponding to $\Lambda$ is defined by

$\Gamma =\left\{\sum _{i=1}^{n}a_{i}u_{i}\;|\;a_{i}\in {\mathbb {Z}}\right\}$

where the vectors $u_{i}$ are chosen to satisfy $\langle u_{i},v_{j}\rangle =2\pi \delta _{ij}$. An example of a sampling lattice is depicted in Figure 1. The corresponding reciprocal lattice is shown in Figure 2.

The theorem

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

Implications

Aliasing

Main article: Aliasing

Spatial aliasing in the form of a Moiré pattern.

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 aliased.

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 on the right. The top 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 lower image 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 bandlimited 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 $\Re ^{n}$. 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 $\Omega \subset \Re ^{d}$. For example, it can be shown that the lattice in $\Re ^{2}$ with minimum spatial density of points that admits perfect reconstructions of fields wavenumber-limited to a circular disc in $\Re ^{2}$ is the hexagonal lattice. As a consequence, hexagonal lattices are preferred for sampling isotropic fields in $\Re ^{2}$.

Applications

The Petersen–Middleton theorem is useful in designing efficient sensor placement strategies in applications involving measurement of spatial phenomena such as seismic surveys and spatial audio-field measurements.