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<title>Thresholding (image processing)</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Thresholding (image processing)</span></span>

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<p>In <a href="Digital_image_processing" title="Digital image processing">digital image processing</a>, <b>thresholding</b> is the simplest method of <a href="Image_segmentation" title="Image segmentation">segmenting images</a>. From a <a href="Grayscale" title="Grayscale">grayscale</a> image, thresholding can be used to create <a href="Binary_image" title="Binary image">binary images</a>.<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>

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

<p>The simplest thresholding methods replace each pixel in an image with a black pixel if the image intensity <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 I_{i,j}}">

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</math></span><img src="./348b9cc1f8f5c6d18fa888bf7ebcca546f47c6f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.958ex; height:2.843ex;" alt="{\displaystyle I_{i,j}}" loading="lazy"></span> is less than a fixed value called the threshold <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 T}">

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<div class="mw-heading mw-heading2"><h2 id="Automatic_thresholding">Automatic thresholding</h2></div>

<p>While in some cases, the threshold <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 T}">

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</math></span><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"></span> can be selected manually by the user, there are many cases where the user wants the threshold to be automatically set by an algorithm. In those cases, the threshold should be the "best" threshold in the sense that the partition of the pixels above and below the threshold should match as closely as possible the actual partition between the two classes of objects represented by those pixels (e.g., pixels below the threshold should correspond to the background and those above to some objects of interest in the image).

</p><p>Many types of automatic thresholding methods exist, the most famous and widely used being <a href="Otsu's_method" title="Otsu's method">Otsu's method</a>. Sezgin et al 2004 categorized thresholding methods into broad groups based on the information the algorithm manipulates.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Note however that such a categorization is necessarily fuzzy as some methods can fall in several categories (for example, Otsu's method can be both considered a histogram-shape and a clustering algorithm)

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<ul><li><b><a href="Histogram" title="Histogram">Histogram</a> shape</b>-based methods, where, for example, the peaks, valleys and curvatures of the smoothed histogram are analyzed.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Note that these methods, more than others, make certain assumptions about the image intensity probability distribution (i.e., the shape of the histogram),</li>

<li><b>Clustering</b>-based methods, where the gray-level samples are clustered in two parts as background and foreground,<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li>

<li><b><a href="Entropy_(information_theory)" title="Entropy (information theory)">Entropy</a></b>-based methods result in algorithms that use the entropy of the foreground and background regions, the cross-entropy between the original and binarized image, etc.,<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li>

<li><b>Object Attribute</b>-based methods search a measure of similarity between the gray-level and the binarized images, such as fuzzy shape similarity, edge coincidence, etc.,</li>

<li><b>Spatial</b> methods use higher-order probability distribution and/or correlation between pixels.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Global_vs_local_thresholding">Global vs local thresholding</h3></div>

<p>In most methods, the same threshold is applied to all pixels of an image. However, in some cases, it can be advantageous to apply a different threshold to different parts of the image, based on the local value of the pixels. This category of methods is called local or adaptive thresholding. They are particularly adapted to cases where images have inhomogeneous lighting, such as in the sudoku image on the right. In those cases, a neighborhood is defined and a threshold is computed for each pixel and its neighborhood. Many global thresholding methods can be adapted to work in a local way, but there are also methods developed specifically for local thresholding, such as the Niblack<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> or the Bernsen algorithms.

</p><p>Software such as <a href="ImageJ" title="ImageJ">ImageJ</a> propose a wide range of automatic threshold methods, both global and local.

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<div class="mw-heading mw-heading3"><h3 id="Benefits_of_Local_Thresholding_Over_Global_Thresholding[8]">Benefits of Local Thresholding Over Global Thresholding<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></h3></div>

<ul><li>Adaptability to Local Image Characteristics: Local thresholding can adapt to variations in illumination, contrast, and texture within different parts of the image. This adaptability helps in handling images with non-uniform lighting conditions or complex textures.</li>

<li>Preservation of Local Details: By applying tailored thresholds to different regions, local thresholding can preserve fine details and edges that might be lost in global thresholding, especially in areas with varying intensities or gradients.</li>

<li>Reduced Sensitivity to Noise: Local thresholding can be less sensitive to noise compared to global thresholding, as the thresholding decision is based on local statistics rather than the entire image.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Examples_of_Algorithms_for_Local_Thresholding">Examples of Algorithms for Local Thresholding</h3></div>

<ul><li>Niblack's Method:<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Niblack's algorithm computes a local threshold for each pixel based on the mean and standard deviation of the pixel's neighborhood. It adjusts the threshold based on the local characteristics of the image, making it suitable for handling variations in illumination.</li>

<li>Bernsen's Method:<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Bernsen's algorithm calculates the threshold for each pixel by considering the local contrast within a neighborhood. It uses a fixed window size and is robust to noise and variations in background intensity.</li>

<li>Sauvola's Method:<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Sauvola's algorithm extends Niblack's method by incorporating a dynamic factor that adapts the threshold based on the local contrast and mean intensity. This adaptive factor improves the binarization results, particularly in regions with varying contrasts.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Extensions_of_binary_thresholding">Extensions of binary thresholding</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Multi-band_images">Multi-band images</h3></div>

<p>Color images can also be thresholded. One approach is to designate a separate threshold for each of the <a href="RGB_color_model" title="RGB color model">RGB</a> components of the image and then combine them with an <a href="Binary_and" class="mw-redirect" title="Binary and">AND</a> operation. This reflects the way the camera works and how the data is stored in the computer, but it does not correspond to the way that people recognize color. Therefore, the <a href="HSL_and_HSV" title="HSL and HSV">HSL and HSV</a> color models are more often used; note that since <a href="Hue" title="Hue">hue</a> is a circular quantity it requires <a href="Circular_thresholding" title="Circular thresholding">circular thresholding</a>. It is also possible to use the <a href="CMYK_color_model" title="CMYK color model">CMYK</a> color model.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading3"><h3 id="Multiple_thresholds">Multiple thresholds</h3></div>

<p>Instead of a single threshold resulting in a binary image, it is also possible to introduce multiple increasing thresholds <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 T_{n}}">

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</math></span><img src="./4d9241493be76739f2400f258f32c24f9689161f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.576ex; height:2.509ex;" alt="{\displaystyle T_{n}}" loading="lazy"></span>. In that case, implementing <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 N}">

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</math></span><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> classes, where pixels with intensity <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 I_{ij}}">

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</math></span><img src="./b8e861c28675994decc6373478cade6f99e4f4df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.5ex; height:2.843ex;" alt="{\displaystyle I_{ij}}" loading="lazy"></span> such that <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 T_{n}<I_{ij}<T_{n+1}}">

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</math></span><img src="./4e812d5e82fd251210cdc883ed9f827801c101c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.95ex; height:2.843ex;" alt="{\displaystyle T_{n}<I_{ij}<T_{n+1}}" loading="lazy"></span> will be assigned to class <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 n}">

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</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>. Most of the binary automatic thresholding methods have a natural extension for multi-thresholding.

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

<p>Thresholding will work best under certain conditions :

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<ul><li>low level of noise</li>

<li>higher intra-class variance than inter-class variance, i.e., pixels from a same group have closer intensities to each other than to pixels of another group,</li>

<li>homogeneous lighting, etc.</li></ul>

<p>In difficult cases, thresholding will likely be imperfect and yield a binary image with <a href="False_positives_and_false_negatives" title="False positives and false negatives">false positives and false negatives</a>.

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

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</style><cite id="CITEREFShapiroStockman2001" class="citation book cs1">Shapiro, Linda G.; Stockman, George C. (2001). <i>Computer Vision</i>. Prentice Hall. p. 83. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-13-030796-5</bdi>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFSankur2004" class="citation journal cs1">Sankur, Bülent (2004). "Survey over image thresholding techniques and quantitative performance evaluation". <i>Journal of Electronic Imaging</i>. <b>13</b> (1): 146. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2004JEI....13..146S">2004JEI....13..146S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1117%2F1.1631315">10.1117/1.1631315</a>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFZackRogersLatt1977" class="citation journal cs1">Zack, G W; Rogers, W E; Latt, S A (July 1977). <a rel="nofollow" class="external text" href="https://doi.org/10.1177%2F25.7.70454">"Automatic measurement of sister chromatid exchange frequency"</a>. <i>Journal of Histochemistry & Cytochemistry</i>. <b>25</b> (7): <span class="nowrap">741–</span>753. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1177%2F25.7.70454">10.1177/25.7.70454</a></span>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/70454">70454</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:15339151">15339151</a>.</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite class="citation journal cs1">"Picture Thresholding Using an Iterative Selection Method". <i>IEEE Transactions on Systems, Man, and Cybernetics</i>. <b>8</b> (8): <span class="nowrap">630–</span>632. 1978. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTSMC.1978.4310039">10.1109/TSMC.1978.4310039</a>.</cite></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFBarghoutSheynin2013" class="citation journal cs1">Barghout, L.; Sheynin, J. (2013-07-25). <a rel="nofollow" class="external text" href="https://doi.org/10.1167%2F13.9.709">"Real-world scene perception and perceptual organization: Lessons from Computer Vision"</a>. <i>Journal of Vision</i>. <b>13</b> (9): 709. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1167%2F13.9.709">10.1167/13.9.709</a></span>.</cite></span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFKapurSahooWong1985" class="citation journal cs1">Kapur, J. N.; Sahoo, P. K.; Wong, A. K. C. (1985-03-01). "A new method for gray-level picture thresholding using the entropy of the histogram". <i>Computer Vision, Graphics, and Image Processing</i>. <b>29</b> (3): <span class="nowrap">273–</span>285. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0734-189X%2885%2990125-2">10.1016/0734-189X(85)90125-2</a>.</cite></span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite class="citation book cs1"><i>An introduction to digital image processing</i>. Prentice-Hall International. 1986. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-13-480600-X</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1244113797">1244113797</a>.</cite></span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Zhou, Huiyu., Wu, Jiahua., Zhang, Jianguo. Digital Image Processing: Part II. United States: Ventus Publishing, 2010.</span>

</li>

<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><cite id="CITEREFNiblack1986" class="citation book cs1">Niblack, Wayne (1986). <i>An introduction to digital image processing</i>. Prentice-Hall International. pp. <span class="nowrap">115–</span>116. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-13-480600-X</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1244113797">1244113797</a>.</cite></span>

</li>

<li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">Chaki, Nabendu., Shaikh, Soharab Hossain., Saeed, Khalid. Exploring Image Binarization Techniques. Germany: Springer India, 2014.</span>

</li>

<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><cite id="CITEREFSauvolaPietikäinen2000" class="citation journal cs1">Sauvola, J.; Pietikäinen, M. (February 2000). "Adaptive document image binarization". <i>Pattern Recognition</i>. <b>33</b> (2): <span class="nowrap">225–</span>236. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2000PatRe..33..225S">2000PatRe..33..225S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0031-3203%2899%2900055-2">10.1016/S0031-3203(99)00055-2</a>.</cite></span>

</li>

<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><cite id="CITEREFPhamMorrisonSchwockAviel-Ronen2007" class="citation journal cs1">Pham, Nhu-An; Morrison, Andrew; Schwock, Joerg; Aviel-Ronen, Sarit; Iakovlev, Vladimir; Tsao, Ming-Sound; Ho, James; Hedley, David W. (2007-02-27). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1810239">"Quantitative image analysis of immunohistochemical stains using a CMYK color model"</a>. <i>Diagnostic Pathology</i>. <b>2</b> (1): 8. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1186%2F1746-1596-2-8">10.1186/1746-1596-2-8</a></span>. <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1810239">1810239</a></span>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/17326824">17326824</a>.</cite></span>

</li>

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<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li>Gonzalez, Rafael C. & Woods, Richard E. (2002). Thresholding. In Digital Image Processing, pp. 595–611. Pearson Education. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>81-7808-629-8</bdi></li>

<li><cite id="CITEREFEichmann2009" class="citation journal cs1">Eichmann, Marco (2009). "Framework for efficient optimal multilevel image thresholding". <i>Journal of Electronic Imaging</i>. <b>18</b> (1): 013004–013004–10. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009JEI....18a3004L">2009JEI....18a3004L</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1117%2F1.3073891">10.1117/1.3073891</a>.</cite></li>

<li><cite id="CITEREFRosin2014" class="citation journal cs1">Rosin, Paul L. (March 2014). <a rel="nofollow" class="external text" href="https://orca.cardiff.ac.uk/id/eprint/61181/">"Efficient Circular Thresholding"</a>. <i>IEEE Transactions on Image Processing</i>. <b>23</b> (3): <span class="nowrap">992–</span>1001. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2014ITIP...23..992Y">2014ITIP...23..992Y</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTIP.2013.2297014">10.1109/TIP.2013.2297014</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/24464614">24464614</a>.</cite></li>

<li>Scott E. Umbaugh (2018). Digital Image Processing and Analysis, pp 93–96. CRC Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4987-6602-9</bdi></li></ul></div><div><div class="zim-footer">

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