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<title>Structuring element</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Structuring element</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="Mathematical_morphology" title="Mathematical morphology">mathematical morphology</a>, a <b>structuring element</b> is a shape, used to probe or interact with a given image, with the purpose of drawing conclusions on how this shape fits or misses the shapes in the image. It is typically used in morphological operations, such as <a href="Dilation_(morphology)" title="Dilation (morphology)">dilation</a>, <a href="Erosion_(morphology)" title="Erosion (morphology)">erosion</a>, <a href="Opening_(morphology)" title="Opening (morphology)">opening</a>, and <a href="Closing_(morphology)" title="Closing (morphology)">closing</a>, as well as the <a href="Hit-or-miss_transform" title="Hit-or-miss transform">hit-or-miss transform</a>.

</p><p>According to <a href="Georges_Matheron" title="Georges Matheron">Georges Matheron</a>, knowledge about an object (e.g., an image) depends on the manner in which we probe (observe) it.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> In particular, the choice of a certain structuring element for a particular morphological operation influences the information one can obtain. There are two main characteristics that are directly related to structuring elements:

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<ul><li><b>Shape</b>. For example, the structuring element can be a "ball" or a line; convex or a ring, etc. By choosing a particular structuring element, one sets a way of differentiating some objects (or parts of objects) from others, according to their shape or spatial orientation.</li>

<li><b>Size</b>. For example, one structuring element can be a <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 3\times 3}">

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<annotation encoding="application/x-tex">{\displaystyle 3\times 3}</annotation>

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</math></span><img src="./ddc0d4d6106875f8006be1d898512ca5843bad8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 3\times 3}" loading="lazy"></span> square or a <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 21\times 21}">

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<annotation encoding="application/x-tex">{\displaystyle 21\times 21}</annotation>

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</math></span><img src="./f7d049ae966505193a9cf88fa4d52a50dd746193.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.49ex; height:2.176ex;" alt="{\displaystyle 21\times 21}" loading="lazy"></span> square. Setting the size of the structuring element is similar to setting the observation scale, and setting the criterion to differentiate image objects or features according to size.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Mathematical_particulars_and_examples">Mathematical particulars and examples</h2></div>

<p>Structuring elements are particular cases of binary images, usually being small and simple. In mathematical morphology, <a href="Binary_image" title="Binary image">binary images</a> are <a href="Subset" title="Subset">subsets</a> of a <a href="Euclidean_space" title="Euclidean space">Euclidean space</a> <i>R</i><sup><i>d</i></sup> or the integer grid <i>Z</i><sup><i>d</i></sup>, for some dimension <i>d</i>. Here are some examples of widely used structuring elements (denoted by <i>B</i>):

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<ul><li>Let <i>E</i>=<i>R</i><sup>2</sup>; <i>B</i> is an open disk of radius <i>r</i>, centered at the origin.</li>

<li>Let <i>E</i>=<i>Z</i><sup>2</sup>; <i>B</i> is a 3x3 square, that is, <i>B</i>={(-1,-1),(-1,0),(-1,1),(0,-1),(0,0),(0,1),(1,-1),(1,0),(1,1)}.</li>

<li>Let <i>E</i>=<i>Z</i><sup>2</sup>; <i>B</i> is the "cross" given by: <i>B</i>={(-1,0),(0,-1),(0,0),(0,1),(1,0)}.</li></ul>

<p>In the discrete case, a structuring element can also be represented as a set of <a href="Pixel" title="Pixel">pixels</a> on a <a href="Grid_(spatial_index)" title="Grid (spatial index)">grid</a>, assuming the values 1 (if the pixel belongs to the structuring element) or 0 (otherwise).

</p><p>When used by a hit-or-miss transform, usually the structuring element is a composite of two disjoint sets (two simple structuring elements), one associated to the foreground, and one associated to the background of the image to be probed. In this case, an alternative representation of the composite structuring element is as a set of <a href="Pixel" title="Pixel">pixels</a> which are either set (1, associated to the foreground), not set (0, associated to the background) or "<a href="Don't_care" class="mw-redirect" title="Don't care">don't care</a>".

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<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">See (<a href="#dougherty92">Dougherty 1992</a>), chapter 1, page 1.</span>

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><cite id="dougherty92">Edward R. Dougherty, <i>An Introduction to Morphological Image Processing</i>, <style data-mw-deduplicate="TemplateStyles:r1238218222">

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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8194-0845-X</bdi> (1992)</cite></li>

<li>Jean Serra, <i>Image Analysis and Mathematical Morphology, Volume 1</i>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-12-637241-1</bdi> (1982)</li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2018-09-25" href="https://en.wikipedia.org/wiki/?title=Structuring_element&oldid=861164435">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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