Multidimensional signal processing: Difference between revisions

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In [[signal processing]], '''multidimensional signal processing''' covers all signal processing done using [[multidimensional sampling]]signals and systems. While multidimensional signal processing is a subset of signal processing, it is unique in the sense that it deals specifically with data that can only be adequately detailed using more than one dimension. SpecificallyIn m-D digital signal processing, useful data is sampled in more than one dimension, such that multiple sensors are used to construct the data set. Examples of this are [[image processing]] and multi-sensor radar detection. Both of these examples use multiple sensors to sample signals in adjacent space and form images based on the manipulation of these multiple signals.

Multidimensional signals are part of [[multidimensional systems]], and as such are generally more complex than classical, single dimension signal processing. Processing in multi-dimension (m-D) requires more complex algorithms, compared to the 1-D case, to handle calculations such as the [[Fast Fourier Transform]] due to more degrees of freedom.<ref name="dudmer83">D. Dudgeon and R. Mersereau, Multidimensional Digital Signal Processing, Prentice-Hall, First Edition, pp. 2, 1983.</ref> In some cases, m-D signals and systems can be simplified into single dimension signal processing methods, utilizingif assumptionsthe suchconsidered assystems are symmetryseparable.

Typically, multidimensional signal processing is directly associated with [[digital signal processing]] because its complexity warrants the use of computer modelling and computation.<ref name="dudmer83"/> A multidimensional signal is similar to a single dimensional signal as far as manipulations that can be performed, such as [[Sampling (signal processing)|sampling]], [[Fourier analysis]], and [[Filter (signal processing)|filtering]]. The actual computations of these manipulations grow with the number of dimensions.

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== Sampling ==

{{main|Multidimensional sampling}}

Multidimensional sampling requires different analysis than typical 1-D sampling. Single dimension sampling is executing by selecting points along a continuous line and storing the values of this data stream. In the case of multidimensional sampling, the data is selected utilizing a [[lattice]], which is a "pattern" based on the sampling [[vector (mathematics and physics)|vectorvectors]] of the m-D data set.<ref name="mer83">Mersereau, R.; Speake, T., "The processing of periodically sampled multidimensional signals," Acoustics, IEEE Transactions on Speech and Signal Processing, IEEE Transactions on , vol.31, no.1, pp.188,-194, Feb 1983.</ref> These vectors can be single dimensional or multidimensional depending on the data and the application.<ref name="mer83" />

Multidimensional sampling is similar to classical sampling as it must adhere to the [[Nyquist–Shannon sampling theorem]]. It is affected by [[aliasing]] and considerations must be made for eventual reconstruction.

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== Fourier Analysis ==

{{main| Fourier Analysis| Multidimensional Transform| Fast Fourier Transform}}

A multidimensional signal can be represented in terms of sinusoidal components. This is typically done with a type of [[Fourier transform]]. The m-D [[Fourier transform]] transforms a signal from a [[timesignal ___domain]] representation to a [[frequency ___domain]] representation of the signal. In the case of digital processing, a [[discrete-time Fourier transform]]Transform (DFT) is utilized to transform a sampled timesignal ___domain representation into a frequency ___domain representation:

:<math> X(k_1,k_2,\dots,k_m) = \sum_{n_1=-\infty}^\infty \sum_{n_2=-\infty}^\infty \cdots \sum_{n_m=-\infty}^\infty x(n_1,n_2,\dots,n_m) e^{-j 2 \pi k_1 n_1} e^{-j 2 \pi k_2 n_2} \cdots e^{-j 2 \pi k_m n_m}</math>

where ''X'' stands for the multidimensional discrete Fourier transform, ''x'' stands for the sampled time/space ___domain signal, ''m'' stands for the number of dimensions in the system, ''n'' are time samplessample indices and ''k'' are frequency samples.<ref name="dudmer83_2">D. Dudgeon and R. Mersereau, Multidimensional Digital Signal Processing, Prentice-Hall, First Edition, pp. 61,112, 1983.</ref>

Computational complexity is usually the main concern when implementing any Fourier transform. For multidimensional signals, the complexity can be reduced by a number of different methods. The computation may be simplified if there is [[independence]] between [[variables]] of the multidimensional signal.<ref name="dudmer83_2"/> In general, [[Fast Fourier Transforms]] (FFTs), utilize efficiencies of the system to reduce the number of computations by a substantial factor. While there are a number of different implementations of this [[algorithm]] for m-D signals, two often used variations are the vector-radix FFT and the row-column FFT.

== Filtering ==

{{main|Filter (signal processing)}}

[[File:2-D filter frequency response and 1-D filter prototype frequency response.gif|thumb|500px1000px|rightcenter|A 2-D filter (left) defined by its 1-D prototype function (right) and a McClellan transformation.]]

Filtering is an important part of any signal processing application. Similar to typical single dimension signal processing applications, there are varying degrees of complexity within filter design for a given system. M-D systems utilize [[digital filters]] in many different applications. The actual implementation of these m-D filters can pose a design problem depending on whether the multidimensional polynomial is factorable.<ref name="dudmer83_2"/> Typically, a [[prototype]] filter is designed in a single dimension and that filter is [[extrapolate]]d to m-D using a [[map (mathematics)|mapping function]].<ref name="dudmer83_2"/> One of the original mapping functions from 1-D to 2-D was the McClellan Transform.<ref name="mer78">Mersereau, R.M.; Mecklenbrauker, W.; Quatieri, T., Jr., "McClellan transformations for two-dimensional digital filtering-Part I: Design," IEEE Transactions on Circuits and Systems, IEEE Transactions on , vol.23, no.7, pp.405,-414, Jul 1976.</ref> Both [[FIR]] and [[IIR]] filters can be transformed to m-D, depending on the application and the mapping function.

== Applicable Fields ==