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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Multidimensional signal processing</span></span>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="Signal_processing" title="Signal processing">signal processing</a>, <b>multidimensional signal processing</b> 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 <a href="Image_processing" class="mw-redirect" title="Image processing">image processing</a> 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 <a href="Fast_Fourier_transform" title="Fast Fourier transform">fast Fourier transform</a> due to more degrees of freedom.<sup id="cite_ref-dudmer83_1-0" class="reference"><a href="#cite_note-dudmer83-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> In some cases, m-D signals and systems can be simplified into single dimension signal processing methods, if the considered systems are separable.
</p><p>Typically, multidimensional signal processing is directly associated with <a href="Digital_signal_processing" title="Digital signal processing">digital signal processing</a> because its complexity warrants the use of computer modelling and computation.<sup id="cite_ref-dudmer83_1-1" class="reference"><a href="#cite_note-dudmer83-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> A multidimensional signal is similar to a single dimensional signal as far as manipulations that can be performed, such as <a href="Sampling_(signal_processing)" title="Sampling (signal processing)">sampling</a>, <a href="Fourier_analysis" title="Fourier analysis">Fourier analysis</a>, and <a href="Filter_(signal_processing)" title="Filter (signal processing)">filtering</a>. The actual computations of these manipulations grow with the number of dimensions.
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<div class="mw-heading mw-heading2"><h2 id="Sampling">Sampling</h2></div>
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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Multidimensional_sampling" title="Multidimensional sampling">Multidimensional sampling</a></div>
<p>Multidimensional sampling requires different analysis than typical 1-D sampling. Single dimension sampling is executed 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 <a href="Lattice_(order)" title="Lattice (order)">lattice</a>, which is a "pattern" based on the sampling <a href="Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vectors</a> of the m-D data set.<sup id="cite_ref-mer83_2-0" class="reference"><a href="#cite_note-mer83-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> These vectors can be single dimensional or multidimensional depending on the data and the application.<sup id="cite_ref-mer83_2-1" class="reference"><a href="#cite_note-mer83-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p><p>Multidimensional sampling is similar to classical sampling as it must adhere to the <a href="Nyquist%E2%80%93Shannon_sampling_theorem" title="Nyquist–Shannon sampling theorem">Nyquist–Shannon sampling theorem</a>. It is affected by <a href="Aliasing" title="Aliasing">aliasing</a> and considerations must be made for eventual Multidimensional Signal Reconstruction.
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<div class="mw-heading mw-heading2"><h2 id="Fourier_Analysis">Fourier Analysis</h2></div>
<div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="Fourier_analysis" title="Fourier analysis">Fourier analysis</a>, <a href="Multidimensional_transform" title="Multidimensional transform">Multidimensional transform</a>, and <a href="Fast_Fourier_transform" title="Fast Fourier transform">Fast Fourier transform</a></div>
<p>A multidimensional signal can be represented in terms of sinusoidal components. This is typically done with a type of <a href="Fourier_transform" title="Fourier transform">Fourier transform</a>. The m-D <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> transforms a signal from a signal domain representation to a <a href="Frequency_domain" title="Frequency domain">frequency domain</a> 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:
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<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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^{-j2\pi k_{1}n_{1}}e^{-j2\pi k_{2}n_{2}}\cdots e^{-j2\pi k_{m}n_{m}}}">
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<annotation encoding="application/x-tex">{\displaystyle 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^{-j2\pi k_{1}n_{1}}e^{-j2\pi k_{2}n_{2}}\cdots e^{-j2\pi k_{m}n_{m}}}</annotation>
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</math></span><img src="./a1e120bc38aefa78396bad1fffc3c04938c3a3f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:90.795ex; height:7.009ex;" alt="{\displaystyle 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^{-j2\pi k_{1}n_{1}}e^{-j2\pi k_{2}n_{2}}\cdots e^{-j2\pi k_{m}n_{m}}}" loading="lazy"></span></dd></dl>
<p>where <i>X</i> stands for the multidimensional discrete Fourier transform, <i>x</i> stands for the sampled time/space domain signal, <i>m</i> stands for the number of dimensions in the system, <i>n</i> are sample indices and <i>k</i> are frequency samples.<sup id="cite_ref-dudmer83_2_3-0" class="reference"><a href="#cite_note-dudmer83_2-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>
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 <a href="Independence" title="Independence">independence</a> between <a href="Variable_(mathematics)" title="Variable (mathematics)">variables</a> of the multidimensional signal.<sup id="cite_ref-dudmer83_2_3-1" class="reference"><a href="#cite_note-dudmer83_2-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> In general, <a href="Fast_Fourier_transform" title="Fast Fourier transform">fast Fourier transforms</a> (FFTs), reduce the number of computations by a substantial factor. While there are a number of different implementations of this <a href="Algorithm" title="Algorithm">algorithm</a> for m-D signals, two often used variations are the vector-radix FFT and the row-column FFT.
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<div class="mw-heading mw-heading2"><h2 id="Filtering">Filtering</h2></div>
<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Filter_(signal_processing)" title="Filter (signal processing)">Filter (signal processing)</a></div>
<p>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 <a href="Digital_filters" class="mw-redirect" title="Digital filters">digital filters</a> 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.<sup id="cite_ref-dudmer83_2_3-2" class="reference"><a href="#cite_note-dudmer83_2-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Typically, a <a href="Prototype" title="Prototype">prototype</a> filter is designed in a single dimension and that filter is <a href="Extrapolate" class="mw-redirect" title="Extrapolate">extrapolated</a> to m-D using a <a href="Map_(mathematics)" title="Map (mathematics)">mapping function</a>.<sup id="cite_ref-dudmer83_2_3-3" class="reference"><a href="#cite_note-dudmer83_2-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> One of the original mapping functions from 1-D to 2-D was the McClellan Transform.<sup id="cite_ref-mer78_4-0" class="reference"><a href="#cite_note-mer78-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Both <a href="Finite_impulse_response" title="Finite impulse response">FIR</a> and <a href="Infinite_impulse_response" title="Infinite impulse response">IIR</a> filters can be transformed to m-D, depending on the application and the mapping function.
</p>
<div class="mw-heading mw-heading2"><h2 id="Applicable_Fields">Applicable Fields</h2></div>
<ul><li><a href="Audio_signal_processing" title="Audio signal processing">Audio signal processing</a></li>
<li><a href="Image_processing" class="mw-redirect" title="Image processing">Image processing</a></li>
<li><a href="Towed_array_sonar" title="Towed array sonar">Towed array sonar</a></li>
<li><a href="X-ray_computed_tomography" class="mw-redirect" title="X-ray computed tomography">X-ray computed tomography</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<li id="cite_note-dudmer83-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-dudmer83_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-dudmer83_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">D. Dudgeon and R. Mersereau, Multidimensional Digital Signal Processing, Prentice-Hall, First Edition, pp. 2, 1983.</span>
</li>
<li id="cite_note-mer83-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-mer83_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-mer83_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">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.</span>
</li>
<li id="cite_note-dudmer83_2-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-dudmer83_2_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-dudmer83_2_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-dudmer83_2_3-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-dudmer83_2_3-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text">D. Dudgeon and R. Mersereau, Multidimensional Digital Signal Processing, Prentice-Hall, First Edition, pp. 61,112, 1983.</span>
</li>
<li id="cite_note-mer78-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-mer78_4-0">^</a></b></span> <span class="reference-text">Mersereau, R.M.; Mecklenbrauker, W.; <a href="Thomas_F._Quatieri" title="Thomas F. Quatieri">Quatieri, T., Jr.</a>, "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.</span>
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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>
<ul><li><span class="noviewer" typeof="mw:File"></span> Media related to <a href="https://commons.wikimedia.org/wiki/Category:Multidimensional_signal_processing" class="extiw external" title="commons:Category:Multidimensional signal processing">Multidimensional signal processing</a> at Wikimedia Commons</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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