Multidimensional signal processing: Difference between revisions

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In [[signal processing]], '''multidimensional signal processing''' covers all signal processing done using multidimensional 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. In m-D digital signal processing, useful data is sampled in more than one dimension. Examples of this are [[image processing]] and multi-sensor radar detection. Both of these examples use multiple sensors to sample signals and form images based on the manipulation of these multiple signals.

Processing in multi-dimension (m-D) requires more complex algorithms, compared to the 1-D case, to handle calculations such as the [[Fastfast Fourier Transformtransform]] 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, if the considered systems are separable.

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 executingexecuted 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 (order)|lattice]], which is a "pattern" based on the sampling [[vector (mathematics and physics)|vectors]] 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, 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[[Multidimensional Signal Reconstruction]].

== Fourier Analysis ==

{{main| Fourier Analysisanalysis| Multidimensional Transformtransform| Fast Fourier Transformtransform}}

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 signal ___domain representation to a [[frequency ___domain]] representation of the signal. In the case of digital processing, a discrete Fourier Transform (DFT) is utilized to transform a sampled signal ___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 sample 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 [[variable (mathematics)|variables]] of the multidimensional signal.<ref name="dudmer83_2"/> In general, [[Fastfast Fourier Transformstransform]]s (FFTs), 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|1000px|center|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.; [[Thomas F. Quatieri|Quatieri, T., Jr.]], "McClellan transformations for two-dimensional digital filtering-Part I: Design," IEEE Transactions on Circuits and Systems, vol.23, no.7, pp.405-414, Jul 1976.</ref> Both [[Finite impulse response|FIR]] and [[Infinite impulse response|IIR]] filters can be transformed to m-D, depending on the application and the mapping function.

== Applicable Fields ==

* [[Category:MultidimensionalAudio signal processing]]

* [[Image processing]]

* [[Towed array sonar]]

* [[X-ray computed tomography]]

== References ==

{{Reflist}}

==External links==

[[Category:Multidimensional signal processing]]

*{{Commonscat-inline}}

[[Category:Multidimensional signal processing| ]]

[[Category:Signal processing]]