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Histogram equalization

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Histogram equalization is a method in <a href="Image_processing" class="mw-redirect" title="Image processing">image processing</a> of <a href="Contrast_(vision)" title="Contrast (vision)">contrast</a> adjustment using the <a href="Image_histogram" title="Image histogram">image's histogram</a>.

Histogram equalization is a specific case of the more general class of histogram remapping methods. These methods seek to adjust the image to make it easier to analyze or improve visual quality (e.g., <a href="Retinex" class="mw-redirect" title="Retinex">retinex</a>).

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

This method usually increases the global contrast of many images, especially when the image is represented by a narrow range of intensity values. Through this adjustment, the <a href="Luminous_intensity" title="Luminous intensity">intensities</a> can be better distributed on the histogram utilizing the full range of intensities evenly. This allows for areas of lower local contrast to gain a higher contrast. Histogram equalization accomplishes this by effectively spreading out the highly populated intensity values, which tend to degrade image contrast.

The method is useful in images with backgrounds and foregrounds that are both bright or both dark. In particular, the method can lead to better views of <a href="Bone" title="Bone">bone</a> structure in <a href="X-ray" title="X-ray">x-ray</a> images and to better detail in <a href="Photograph" title="Photograph">photographs</a> that are either over or under-exposed. A key advantage of the method is that it is a fairly straightforward technique adaptive to the input image and an <a href="Invertible_operation" class="mw-redirect" title="Invertible operation">invertible operation</a>. So, in theory, if the histogram equalization <a href="Function_(mathematics)" title="Function (mathematics)">function</a> is known, then the original histogram can be recovered. The calculation is not <a href="Computation" title="Computation">computationally</a> intensive. A disadvantage of the method is that it is indiscriminate. It may increase the contrast of background <a href="Signal_noise" class="mw-redirect" title="Signal noise">noise</a>, while decreasing the usable <a href="Signal" title="Signal">signal</a>. In scientific imaging where spatial correlation is more important than intensity of signal (such as separating <a href="DNA" title="DNA">DNA</a> fragments of quantized length), the small <a href="Signal-to-noise_ratio" title="Signal-to-noise ratio">signal-to-noise ratio</a> usually hampers visual detections.

Histogram equalization often produces unrealistic effects in photographs; however it is very useful for scientific images like <a href="Thermography" title="Thermography">thermal</a>, <a href="Satellite" title="Satellite">satellite</a> or <a href="X-ray" title="X-ray">x-ray</a> images, often the same class of images to which one would apply <a href="False-color" class="mw-redirect" title="False-color">false-color</a>. Also histogram equalization can produce undesirable effects (like visible <a href="Image_gradient" title="Image gradient">image gradient</a>) when applied to images with low <a href="Color_depth" title="Color depth">color depth</a>. For example, if applied to 8-bit image displayed with <a href="List_of_8-bit_computer_hardware_palettes" class="mw-redirect" title="List of 8-bit computer hardware palettes">8-bit gray-scale palette</a> it will further reduce <a href="Color_depth" title="Color depth">color depth</a> (number of unique shades of gray) of the image. Histogram equalization will work the best when applied to images with much higher <a href="Color_depth" title="Color depth">color depth</a> than <a href="Palette_(computing)" title="Palette (computing)">palette</a> size, like <a href="Continuous_function" title="Continuous function">continuous</a> data or 16-bit gray-scale images.

There are two ways to think about and implement histogram equalization, either as image change or as <a href="Palette_(computing)" title="Palette (computing)">palette</a> change. The operation can be expressed as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(M(I))}">

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<annotation encoding="application/x-tex">{\displaystyle P(M(I))}</annotation>

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</math><img src="./7ac8397dfa9c1199de24eeaeac80384c872f21af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.978ex; height:2.843ex;" alt="{\displaystyle P(M(I))}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

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

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</math><img src="./535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"> is the original image, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> is histogram equalization mapping operation and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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

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</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> is a palette. If we define a new palette as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P'=P(M)}">

<msup>

</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle P'=P(M)}</annotation>

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</math><img src="./2cbb61d7dce3b9e061da2a545f963e8feaf8a1e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.602ex; height:3.009ex;" alt="{\displaystyle P'=P(M)}" loading="lazy"> and leave image <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

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

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</math><img src="./535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"> unchanged then histogram equalization is implemented as <a href="Palette_(computing)" title="Palette (computing)">palette</a> change or mapping change. On the other hand, if palette <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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

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</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> remains unchanged and image is modified to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I'=M(I)}">

<msup>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle I'=M(I)}</annotation>

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</math><img src="./886a083d628b41da73f174a6dfb3457c88e640e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.423ex; height:3.009ex;" alt="{\displaystyle I'=M(I)}" loading="lazy"> then the implementation is accomplished by image change. In most cases palette change is preferred as it preserves the original data.

Modifications of this method use multiple histograms, called subhistograms, to emphasize local contrast rather than overall global contrast. Examples of such methods include <a href="Adaptive_histogram_equalization" title="Adaptive histogram equalization">adaptive histogram equalization</a> and variations including, contrast limited adaptive histogram equalization, multipeak histogram equalization, and multipurpose beta-optimized bihistogram equalization (MBOBHE). The goal of these methods, especially MBOBHE, is to modifying the algorithm to improve the contrast without producing brightness mean-shift and detail loss artifacts.<a href="#cite_note-1">[1]</a>

A signal transform equivalent to histogram equalization also seems to happen in <a href="Biological_neural_networks" class="mw-redirect" title="Biological neural networks">biological neural networks</a> so as to maximize the output firing rate of the neuron as a function of the input statistics. This has been proved in particular in the <a href="Fly" title="Fly">fly</a> <a href="Retina" title="Retina">retina</a>.<a href="#cite_note-2">[2]</a>

<div class="mw-heading mw-heading3"><h3 id="Back_projection">Back projection</h3></div>

The back projection of a histogrammed image is the re-application of the modified histogram to the original image, functioning as a look-up table for pixel brightness values.

For each group of pixels taken from the same position from all input single-channel images, the function puts the histogram bin value to the destination image, where the coordinates of the bin are determined by the values of pixels in this input group. In terms of statistics, the value of each output image pixel characterizes the probability that the corresponding input pixel group belongs to the object whose histogram is used.<a href="#cite_note-3">[3]</a>

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

Consider a discrete <a href="Grayscale_image" class="mw-redirect" title="Grayscale image">grayscale image</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> and let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n_{i}}</annotation>

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</math><img src="./57f87f905ba5a4d8c691ccaecd65fc47bd007ba4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.194ex; height:2.009ex;" alt="{\displaystyle n_{i}}" loading="lazy"> be the number of occurrences of gray level <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

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

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</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">. The probability of a pixel value chosen uniformly randomly from image <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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</mrow>

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

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> being <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

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</mrow>

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

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</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">, is

<msub>

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<mfrac>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \ p_{X}(i)={\frac {n_{i}}{n}},\quad 0\leq i<L}</annotation>

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</math><img src="./925b0758b96cf56d54ece344133f79276d44ab8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.224ex; height:4.676ex;" alt="{\displaystyle \ p_{X}(i)={\frac {n_{i}}{n}},\quad 0\leq i<L}" loading="lazy"></dd></dl>

</mstyle>

</mrow>

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

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</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> being the total number of gray levels in the image, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n_{i}}</annotation>

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</math><img src="./57f87f905ba5a4d8c691ccaecd65fc47bd007ba4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.194ex; height:2.009ex;" alt="{\displaystyle n_{i}}" loading="lazy"> being the number of pixels in the image with value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

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

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</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> being the total number of pixels in the image. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{X}(i)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{X}(i)}</annotation>

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</math><img src="./136a26ab9730593180b8a636bf9e5fc0e57ec0fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:5.503ex; height:2.843ex;" alt="{\displaystyle p_{X}(i)}" loading="lazy"> is the image's histogram value for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

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

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</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">, with the histogram normalized to have a total area of 1.

Let us then define the <a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> of pixels in image <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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</mrow>

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

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">. For value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"> it is

<msub>

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</msub>

</mrow>

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</munderover>

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</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {cdf} _{X}(i)=\sum _{j=0}^{i}p_{X}(j)}</annotation>

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</math><img src="./dd0226bb34292b89c434e8dfd19c4b653b0a6e10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:19.691ex; height:7.509ex;" alt="{\displaystyle \operatorname {cdf} _{X}(i)=\sum _{j=0}^{i}p_{X}(j)}" loading="lazy">,</dd></dl>

which is also the image's accumulated normalized histogram.

We would like to create a transformation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T:[0,L-1]\rightarrow [0,L-1]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T:[0,L-1]\rightarrow [0,L-1]}</annotation>

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</math><img src="./5932b9eb3939aacf5b173bac6265a856662fd1bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.339ex; height:2.843ex;" alt="{\displaystyle T:[0,L-1]\rightarrow [0,L-1]}" loading="lazy"> to produce a new image <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">, with a flat histogram. Such an image would have a linearized cumulative distribution function (CDF) across the value range, i.e.

<msub>

</mrow>

</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {cdf} _{Y}(i)=(i+1)K}</annotation>

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</math><img src="./4c60363235e9ea3adccb6fd6383210eb98607c68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.914ex; height:2.843ex;" alt="{\displaystyle \operatorname {cdf} _{Y}(i)=(i+1)K}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq i<L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\leq i<L}</annotation>

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</math><img src="./7c8e93f13cf4adb1f905e5f7804125edce10d559.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.745ex; height:2.343ex;" alt="{\displaystyle 0\leq i<L}" loading="lazy"></dd></dl>

for some constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

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

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</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">. The properties of the CDF allow us to perform such a transform (see <a href="Cumulative_distribution_function#Inverse_distribution_function_(quantile_function)" title="Cumulative distribution function">Inverse distribution function</a>). It is defined as

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \ T(i)=\operatorname {cdf} _{X}(i)}</annotation>

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</math><img src="./627221fa58b696ca58244e8f513f695aafb35f32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.208ex; height:2.843ex;" alt="{\displaystyle \ T(i)=\operatorname {cdf} _{X}(i)}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ i}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./bc8d2dfe9ddfc5489e5ce676d223cc41a9489763.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.383ex; height:2.176ex;" alt="{\displaystyle \ i}" loading="lazy"> is in the range <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,L-1]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0,L-1]}</annotation>

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</math><img src="./519ec65a9436d3a0094d9c7df1fb388723f0098f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.076ex; height:2.843ex;" alt="{\displaystyle [0,L-1]}" loading="lazy">. Notice that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ T}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./630996e0f4ea15c1347439ec69ac78fc5ae2f755.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:2.176ex;" alt="{\displaystyle \ T}" loading="lazy"> maps the levels into the range <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation>

</semantics>

</math><img src="./738f7d23bb2d9642bab520020873cccbef49768d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}" loading="lazy">, since we used a normalized histogram of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">. In order to map the values back into their original range, the following simple transformation needs to be applied to each transformed image value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">:

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ k^{\prime }=k\cdot (\max(i)-\min(i))+\min(i)=k\cdot (L-1)}</annotation>

</semantics>

</math><img src="./526c4eb8beb53ae1cc7d864b6a7e5aca0e24fcce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.251ex; height:3.009ex;" alt="{\displaystyle \ k^{\prime }=k\cdot (\max(i)-\min(i))+\min(i)=k\cdot (L-1)}" loading="lazy"><a href="#cite_note-4">[4]</a></dd></dl>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> is a real value while<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ k^{\prime }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ k^{\prime }}</annotation>

</semantics>

</math><img src="./9a75fc60a92cd4cb21343e289a32f9b9978fc74b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.477ex; height:2.509ex;" alt="{\displaystyle \ k^{\prime }}" loading="lazy"> has to be an integer. An intuitive and popular method<a href="#cite_note-5">[5]</a> is applying the round operation:

<msup>

</mrow>

</msup>

<mi>round</mi>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ k^{\prime }=\operatorname {round} (k\cdot (L-1))}</annotation>

</semantics>

</math><img src="./77f95a3f25dba1ebe65c7c32de078d29ac114dc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.621ex; height:3.009ex;" alt="{\displaystyle \ k^{\prime }=\operatorname {round} (k\cdot (L-1))}" loading="lazy">.</dd></dl>

However, detailed analysis results in slightly different formulation. The mapped value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k^{\prime }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k^{\prime }}</annotation>

</semantics>

</math><img src="./3b43b207c13b82ee7555c77f8128366b2fa46bf8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.896ex; height:2.509ex;" alt="{\displaystyle k^{\prime }}" loading="lazy"> should be 0 for the range of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<k\leq 1/L}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0<k\leq 1/L}</annotation>

</semantics>

</math><img src="./8e5c9db0439517ae204698d47d64c6c7036c24d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.478ex; height:2.843ex;" alt="{\displaystyle 0<k\leq 1/L}" loading="lazy">. And <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k^{\prime }=1}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k^{\prime }=1}</annotation>

</semantics>

</math><img src="./1661bab61a3f4cab27363752156c9c22a8a74ed2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.157ex; height:2.509ex;" alt="{\displaystyle k^{\prime }=1}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/L<k\leq 2/L}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1/L<k\leq 2/L}</annotation>

</semantics>

</math><img src="./6297b2a4f7a0b3ac0fc63f4af5216ab795e10cbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.224ex; height:2.843ex;" alt="{\displaystyle 1/L<k\leq 2/L}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k^{\prime }=2}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k^{\prime }=2}</annotation>

</semantics>

</math><img src="./3d585efcfd9e1a74c81224e264edff2d2c325968.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.157ex; height:2.509ex;" alt="{\displaystyle k^{\prime }=2}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2/L<k\leq 3/L}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2/L<k\leq 3/L}</annotation>

</semantics>

</math><img src="./033423ce8f28a951b0ea6cdd312e9a4abd1e96ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.224ex; height:2.843ex;" alt="{\displaystyle 2/L<k\leq 3/L}" loading="lazy">, ...., and finally <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k^{\prime }=L-1}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k^{\prime }=L-1}</annotation>

</semantics>

</math><img src="./fe49ec6ed86f1cfaaf2a027c90036a7876591ffc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.58ex; height:2.676ex;" alt="{\displaystyle k^{\prime }=L-1}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (L-1)/L<k\leq 1}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (L-1)/L<k\leq 1}</annotation>

</semantics>

</math><img src="./ad58b465e490862a3878ec76841ac34f95e484a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.711ex; height:2.843ex;" alt="{\displaystyle (L-1)/L<k\leq 1}" loading="lazy">. Then the quantization formula from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k^{\prime }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k^{\prime }}</annotation>

</semantics>

</math><img src="./3b43b207c13b82ee7555c77f8128366b2fa46bf8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.896ex; height:2.509ex;" alt="{\displaystyle k^{\prime }}" loading="lazy"> should be

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k^{\prime }=\operatorname {ceil} (L\cdot k)-1}</annotation>

</semantics>

</math><img src="./12b4f7d46c3e29ba9687d77cce14ecc7489baa7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.638ex; height:3.009ex;" alt="{\displaystyle k^{\prime }=\operatorname {ceil} (L\cdot k)-1}" loading="lazy">.

(Note: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k^{\prime }=-1}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k^{\prime }=-1}</annotation>

</semantics>

</math><img src="./958ba37a4307cc6812d656b743d83053e4468ebd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.965ex; height:2.676ex;" alt="{\displaystyle k^{\prime }=-1}" loading="lazy"> when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k=0}</annotation>

</semantics>

</math><img src="./6307c8a99dad7d0bcb712352ae0a748bd99a038b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=0}" loading="lazy">, however, it does not happen just because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k=0}</annotation>

</semantics>

</math><img src="./6307c8a99dad7d0bcb712352ae0a748bd99a038b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=0}" loading="lazy"> means that there is no pixel corresponding to that value.)

<div class="mw-heading mw-heading2"><h2 id="On_color_images">On color images</h2></div>

The above-described histogram equalization works on a grayscale image. It can also be used on color images. One option is applying the method separately to the red, green and blue components of the <a href="RGB" class="mw-redirect" title="RGB">RGB</a> color values of the image, which likely produces dramatic changes in the image's <a href="Color_balance" title="Color balance">color balance</a> since the relative distributions of the color channels change as a result of applying the algorithm. However, if the image is first converted to another color space, <a href="Lab_color_space" class="mw-redirect" title="Lab color space">Lab</a>, or <a href="HSL_and_HSV" title="HSL and HSV">HSL/HSV</a> in particular, then the algorithm can be applied to the <a href="Luminance" title="Luminance">luminance</a> or value channel without resulting in changes to color properties of the image.<a href="#cite_note-6">[6]</a>

There are several histogram equalization methods in 3D space<a href="#cite_note-7">[7]</a> which result in "whitening" where the probability of bright pixels are higher than that of dark ones.<a href="#cite_note-8">[8]</a> Han et al. proposed to use a new CDF defined by the ISO-luminance plane, which results in uniform gray distribution.<a href="#cite_note-9">[9]</a>

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

For consistency with <a href="Statistical" class="mw-redirect" title="Statistical">statistical</a> usage, "CDF" (i.e. Cumulative distribution function) should be replaced by "cumulative histogram", especially since the article links to <a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> which is derived by dividing values in the cumulative histogram by the overall amount of pixels. The equalized CDF is defined in terms of <a href="Ranking" title="Ranking">rank</a> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle rank/pixelcount}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle rank/pixelcount}</annotation>

</semantics>

</math><img src="./863fc1448586da5a102454dcab1781ad784f9641.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.824ex; height:2.843ex;" alt="{\displaystyle rank/pixelcount}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Small_image">Small image</h3></div>

The 8-bit grayscale image shown has the following values:

<th style="background-color:#343434; color:#ffffff;">52

</th>

<th style="background-color:#373737; color:#ffffff;">55

</th>

<th style="background-color:#3d3d3d; color:#ffffff;">61

</th>

<th style="background-color:#3b3b3b; color:#ffffff;">59

</th>

<th style="background-color:#464646; color:#ffffff;">79

</th>

<th style="background-color:#3d3d3d; color:#ffffff;">61

</th>

<th style="background-color:#4c4c4c; color:#ffffff;">76

</th>

<th style="background-color:#3d3d3d; color:#ffffff;">61

</th></tr>

<tr>

<td style="background-color:#3e3e3e; color:#ffffff;">62

</td>

<td style="background-color:#3b3b3b; color:#ffffff;">59

</td>

<td style="background-color:#373737; color:#ffffff;">55

</td>

<td style="background-color:#686868; color:#ffffff;">104

</td>

<td style="background-color:#5e5e5e; color:#ffffff;">94

</td>

<td style="background-color:#555555; color:#ffffff;">85

</td>

<td style="background-color:#3b3b3b; color:#ffffff;">59

</td>

<td style="background-color:#474747; color:#ffffff;">71

</td></tr>

<tr>

<td style="background-color:#3f3f3f; color:#ffffff;">63

</td>

<td style="background-color:#414141; color:#ffffff;">65

</td>

<td style="background-color:#424242; color:#ffffff;">66

</td>

<td style="background-color:#717171; color:#ffffff;">113

</td>

<td style="background-color:#909090; color:#000000;">144

</td>

<td style="background-color:#686868; color:#ffffff;">104

</td>

<td style="background-color:#3f3f3f; color:#ffffff;">63

</td>

<td style="background-color:#484848; color:#ffffff;">72

</td></tr>

<tr>

<td style="background-color:#404040; color:#ffffff;">64

</td>

<td style="background-color:#464646; color:#ffffff;">70

</td>

<td style="background-color:#464646; color:#ffffff;">70

</td>

<td style="background-color:#7e7e7e; color:#ffffff;">126

</td>

<td style="background-color:#9a9a9a; color:#000000;">154

</td>

<td style="background-color:#6d6d6d; color:#ffffff;">109

</td>

<td style="background-color:#474747; color:#ffffff;">71

</td>

<td style="background-color:#454545; color:#ffffff;">69

</td></tr>

<tr>

<td style="background-color:#434343; color:#ffffff;">67

</td>

<td style="background-color:#494949; color:#ffffff;">73

</td>

<td style="background-color:#444444; color:#ffffff;">68

</td>

<td style="background-color:#6a6a6a; color:#ffffff;">106

</td>

<td style="background-color:#7a7a7a; color:#ffffff;">122

</td>

<td style="background-color:#585858; color:#ffffff;">88

</td>

<td style="background-color:#444444; color:#ffffff;">68

</td>

<td style="background-color:#444444; color:#ffffff;">68

</td></tr>

<tr>

<td style="background-color:#444444; color:#ffffff;">68

</td>

<td style="background-color:#4f4f4f; color:#ffffff;">79

</td>

<td style="background-color:#3c3c3c; color:#ffffff;">60

</td>

<td style="background-color:#464646; color:#ffffff;">70

</td>

<td style="background-color:#4d4d4d; color:#ffffff;">77

</td>

<td style="background-color:#424242; color:#ffffff;">66

</td>

<td style="background-color:#3a3a3a; color:#ffffff;">58

</td>

<td style="background-color:#4b4b4b; color:#ffffff;">75

</td></tr>

<tr>

<td style="background-color:#454545; color:#ffffff;">69

</td>

<td style="background-color:#555555; color:#ffffff;">85

</td>

<td style="background-color:#404040; color:#ffffff;">64

</td>

<td style="background-color:#3a3a3a; color:#ffffff;">58

</td>

<td style="background-color:#373737; color:#ffffff;">55

</td>

<td style="background-color:#3d3d3d; color:#ffffff;">61

</td>

<td style="background-color:#414141; color:#ffffff;">65

</td>

<td style="background-color:#535353; color:#ffffff;">83

</td></tr>

<tr>

<td style="background-color:#464646; color:#ffffff;">70

</td>

<td style="background-color:#575757; color:#ffffff;">87

</td>

<td style="background-color:#454545; color:#ffffff;">69

</td>

<td style="background-color:#444444; color:#ffffff;">68

</td>

<td style="background-color:#414141; color:#ffffff;">65

</td>

<td style="background-color:#494949; color:#ffffff;">73

</td>

<td style="background-color:#4e4e4e; color:#ffffff;">78

</td>

<td style="background-color:#5a5a5a; color:#ffffff;">90

</td></tr></tbody></table>

The histogram for this image is shown in the following table. Pixel values that have a zero count are excluded for the sake of brevity.

<th>Value</th>

<th>Count

</th>

<th>Value</th>

<th>Count

</th>

<th>Value</th>

<th>Count

</th>

<th>Value</th>

<th>Count

</th>

<th>Value</th>

<th>Count

</th></tr>

<tr>

<td>1

</td>

<td>2

</td>

<td>1

</td>

<td>2

</td>

<td>1

</td></tr>

<tr>

<td>3

</td>

<td>3

</td>

<td>2

</td>

<td>1

</td>

<td>1

</td></tr>

<tr>

<td>2

</td>

<td>2

</td>

<td>1

</td>

<td>1

</td>

<td>1

</td></tr>

<tr>

<td>3

</td>

<td>1

</td>

<td>1

</td>

<td>1

</td>

<td>1

</td></tr>

<tr>

<td>1

</td>

<td>5

</td>

<td>1

</td>

<td>1

</td>

<td>1

</td></tr>

<tr>

<td>4

</td>

<td>3

</td>

<td>1

</td>

<td>2

</td>

</td></tr>

<tr>

<td>1

</td>

<td>4

</td>

<td>2

</td>

<td>1

</td></tr>

<tr>

<td>2

</td>

<td>2

</td>

<td>1

</td>

<td>1

</td></tr></tbody></table></dd></dl>

The <a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> (CDF) is shown below. Again, pixel values that do not contribute to an increase in the function are excluded for brevity.

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy">, Pixel Intensity</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {CDF} (v)}</annotation>

</semantics>

</math><img src="./bb1457c6d88869818745541b4a45c4df4a726e27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.908ex; height:2.843ex;" alt="{\displaystyle \operatorname {CDF} (v)}" loading="lazy"></th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(v)}</annotation>

</semantics>

</math><img src="./5642c579b0a2d0271b1ecb418390f78902d5de45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.276ex; height:2.843ex;" alt="{\displaystyle h(v)}" loading="lazy">, equalized <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy">

</th></tr>

<tr>

<td>0

</td></tr>

<tr>

<td>12

</td></tr>

<tr>

<td>20

</td></tr>

<tr>

<td>32

</td></tr>

<tr>

<td>36

</td></tr>

<tr>

<td>53

</td></tr>

<tr>

<td>57

</td></tr>

<tr>

<td>65

</td></tr>

<tr>

<td>73

</td></tr>

<tr>

<td>85

</td></tr>

<tr>

<td>93

</td></tr>

<tr>

<td>97

</td></tr>

<tr>

<td>117

</td></tr>

<tr>

<td>130

</td></tr>

<tr>

<td>146

</td></tr>

<tr>

<td>154

</td></tr>

<tr>

<td>158

</td></tr>

<tr>

<td>166

</td></tr>

<tr>

<td>170

</td></tr>

<tr>

<td>174

</td></tr>

<tr>

<td>178

</td></tr>

<tr>

<td>182

</td></tr>

<tr>

<td>190

</td></tr>

<tr>

<td>194

</td></tr>

<tr>

<td>202

</td></tr>

<tr>

<td>206

</td></tr>

<tr>

<td>210

</td></tr>

<tr>

<td>215

</td></tr>

<tr>

<td>219

</td></tr>

<tr>

<td>227

</td></tr>

<tr>

<td>231

</td></tr>

<tr>

<td>235

</td></tr>

<tr>

<td>239

</td></tr>

<tr>

<td>243

</td></tr>

<tr>

<td>247

</td></tr>

<tr>

<td>251

</td></tr>

<tr>

<td>255

</td></tr></tbody></table> (Please note that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(v)=\operatorname {ceil} (\operatorname {CDF} (v))-1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(v)=\operatorname {ceil} (\operatorname {CDF} (v))-1}</annotation>

</semantics>

</math><img src="./a846c668c3adfc508c503d3c3bd0ef2686d30cc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.453ex; height:2.843ex;" alt="{\displaystyle h(v)=\operatorname {ceil} (\operatorname {CDF} (v))-1}" loading="lazy"> version is not illustrated yet.)</dd></dl>

This CDF shows that the minimum value in the subimage is 52 and the maximum value is 154. The CDF of 64 for value 154 coincides with the number of pixels in the image. The CDF must be normalized to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,255]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0,255]}</annotation>

</semantics>

</math><img src="./0b92f49fdc420e36b9d62c711c3c6ebe7d9fcebc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.977ex; height:2.843ex;" alt="{\displaystyle [0,255]}" loading="lazy">. The general histogram equalization formula is:

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(v)=\mathrm {round} \left({\frac {\operatorname {CDF} (v)-\operatorname {CDF} _{\min }}{(M\times N)-\operatorname {CDF} _{\min }}}\times (L-1)\right)}</annotation>

</semantics>

</math><img src="./4db888b4d695e56c720e6e211dc8ca9e0b99ceb3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:48.146ex; height:6.509ex;" alt="{\displaystyle h(v)=\mathrm {round} \left({\frac {\operatorname {CDF} (v)-\operatorname {CDF} _{\min }}{(M\times N)-\operatorname {CDF} _{\min }}}\times (L-1)\right)}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {CDF} _{\min }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {CDF} _{\min }}</annotation>

</semantics>

</math><img src="./8e6d2fec86fecbff6adc2d267b1a82e00a4742dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.944ex; height:2.509ex;" alt="{\displaystyle \operatorname {CDF} _{\min }}" loading="lazy"> is the minimum non-zero value of the cumulative distribution function (in this case 1), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\times N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./77b09b633429cadcb3f0ddeadc6dcbe7bd65fb13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.346ex; height:2.176ex;" alt="{\displaystyle M\times N}" loading="lazy"> gives the image's number of pixels (for the example above 64, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> is width and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> the height) and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> is the number of grey levels used (in most cases, like this one, 256).

Note that to scale values in the original data that are above 0 to the range 1 to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L-1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L-1}</annotation>

</semantics>

</math><img src="./7fe61101dd489eb8e1a974ab6c409190a76541bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.586ex; height:2.343ex;" alt="{\displaystyle L-1}" loading="lazy">, inclusive, the above equation would instead be:

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(v)=\mathrm {round} \left({\frac {\operatorname {CDF} (v)-\operatorname {CDF} _{\min }}{(M\times N)-\operatorname {CDF} _{\min }}}\times (L-2)\right)+1}</annotation>

</semantics>

</math><img src="./e5d344dcf29fbbf9a55c370a2bf80aa3f8db1ca1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:52.149ex; height:6.509ex;" alt="{\displaystyle h(v)=\mathrm {round} \left({\frac {\operatorname {CDF} (v)-\operatorname {CDF} _{\min }}{(M\times N)-\operatorname {CDF} _{\min }}}\times (L-2)\right)+1}" loading="lazy"></dd></dl>

where cdf(v) > 0. Scaling from 1 to 255 preserves the non-zero-ness of the minimum value.

The equalization formula for the example scaling data from 0 to 255, inclusive, is:

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(v)=\mathrm {round} \left({\frac {\operatorname {cdf} (v)-1}{63}}\times 255\right)}</annotation>

</semantics>

</math><img src="./6d6e19d15ca29a2cad47437cb75f3626ae8da6c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.275ex; height:6.343ex;" alt="{\displaystyle h(v)=\mathrm {round} \left({\frac {\operatorname {cdf} (v)-1}{63}}\times 255\right)}" loading="lazy"></dd></dl>

For example, the CDF of 78 is 46. (The value of 78 is used in the bottom row of the 7th column.) The normalized value becomes:

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(78)=\mathrm {round} \left({\frac {46-1}{63}}\times 255\right)=\mathrm {round} \left(0.714286\times 255\right)=182}</annotation>

</semantics>

</math><img src="./85c3e2b258f6d32cdbc8831753ca8d450b0e79ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:64.767ex; height:6.176ex;" alt="{\displaystyle h(78)=\mathrm {round} \left({\frac {46-1}{63}}\times 255\right)=\mathrm {round} \left(0.714286\times 255\right)=182}" loading="lazy"></dd></dl>

Once this is done then the values of the equalized image are directly taken from the normalized CDF to yield the equalized values:

<th style="background-color:#000000; color:#FFFFFF;">0

</th>

<th style="background-color:#0C0C0C; color:#FFFFFF;">12

</th>

<th style="background-color:#353535; color:#FFFFFF;">53

</th>

<th style="background-color:#202020; color:#FFFFFF;">32

</th>

<th style="background-color:#929292; color:#000000;">190

</th>

<th style="background-color:#353535; color:#FFFFFF;">53

</th>

<th style="background-color:#AEAEAE; color:#000000;">174

</th>

<th style="background-color:#353535; color:#FFFFFF;">53

</th></tr>

<tr>

<th style="background-color:#393939; color:#FFFFFF;">57

</th>

<th style="background-color:#202020; color:#FFFFFF;">32

</th>

<th style="background-color:#0C0C0C; color:#FFFFFF;">12

</th>

<th style="background-color:#E3E3E3; color:#000000;">227

</th>

<th style="background-color:#DBDBDB; color:#000000;">219

</th>

<th style="background-color:#CACACA; color:#000000;">202

</th>

<th style="background-color:#202020; color:#FFFFFF;">32

</th>

<th style="background-color:#9A9A9A; color:#000000;">154

</th></tr>

<tr>

<th style="background-color:#414141; color:#FFFFFF;">65

</th>

<th style="background-color:#555555; color:#FFFFFF;">85

</th>

<th style="background-color:#5D5D5D; color:#FFFFFF;">93

</th>

<th style="background-color:#EFEFEF; color:#000000;">239

</th>

<th style="background-color:#FBFBFB; color:#000000;">251

</th>

<th style="background-color:#E3E3E3; color:#000000;">227

</th>

<th style="background-color:#414141; color:#FFFFFF;">65

</th>

<th style="background-color:#9E9E9E; color:#000000;">158

</th></tr>

<tr>

<th style="background-color:#494949; color:#FFFFFF;">73

</th>

<th style="background-color:#929292; color:#000000;">146

</th>

<th style="background-color:#929292; color:#000000;">146

</th>

<th style="background-color:#F7F7F7; color:#000000;">247

</th>

<th style="background-color:#FFFFFF; color:#000000;">255

</th>

<th style="background-color:#EBEBEB; color:#000000;">235

</th>

<th style="background-color:#9A9A9A; color:#000000;">154

</th>

<th style="background-color:#828282; color:#000000;">130

</th></tr>

<tr>

<th style="background-color:#616161; color:#FFFFFF;">97

</th>

<th style="background-color:#A6A6A6; color:#000000;">166

</th>

<th style="background-color:#757575; color:#FFFFFF;">117

</th>

<th style="background-color:#E7E7E7; color:#000000;">231

</th>

<th style="background-color:#F3F3F3; color:#000000;">243

</th>

<th style="background-color:#D2D2D2; color:#000000;">210

</th>

<th style="background-color:#757575; color:#FFFFFF;">117

</th>

<th style="background-color:#757575; color:#FFFFFF;">117

</th></tr>

<tr>

<th style="background-color:#757575; color:#FFFFFF;">117

</th>

<th style="background-color:#BEBEBE; color:#000000;">190

</th>

<th style="background-color:#242424; color:#FFFFFF;">36

</th>

<th style="background-color:#BEBEBE; color:#000000;">146

</th>

<th style="background-color:#B2B2B2; color:#000000;">178

</th>

<th style="background-color:#5D5D5D; color:#FFFFFF;">93

</th>

<th style="background-color:#141414; color:#FFFFFF;">20

</th>

<th style="background-color:#AAAAAA; color:#000000;">170

</th></tr>

<tr>

<th style="background-color:#828282; color:#000000;">130

</th>

<th style="background-color:#CACACA; color:#000000;">202

</th>

<th style="background-color:#494949; color:#FFFFFF;">73

</th>

<th style="background-color:#141414; color:#FFFFFF;">20

</th>

<th style="background-color:#0C0C0C; color:#FFFFFF;">12

</th>

<th style="background-color:#353535; color:#FFFFFF;">53

</th>

<th style="background-color:#555555; color:#FFFFFF;">85

</th>

<th style="background-color:#C2C2C2; color:#000000;">194

</th></tr>

<tr>

<th style="background-color:#929292; color:#000000;">146

</th>

<th style="background-color:#CECECE; color:#000000;">206

</th>

<th style="background-color:#828282; color:#000000;">130

</th>

<th style="background-color:#757575; color:#FFFFFF;">117

</th>

<th style="background-color:#555555; color:#FFFFFF;">85

</th>

<th style="background-color:#A6A6A6; color:#000000;">166

</th>

<th style="background-color:#B6B6B6; color:#000000;">182

</th>

<th style="background-color:#D7D7D7; color:#000000;">215

</th></tr></tbody></table>

Notice that the minimum value 52 is now 0 and the maximum value 154 is now 255.

<td>

</td>

<td>

</td></tr>

<tr>

<td align="center">Original

</td>

<td align="center">Equalized

</td></tr></tbody></table></dd></dl>

<td>

</td>

<td>

</td></tr>

<tr>

<td align="center">Histogram of Original image

</td>

<td align="center">Histogram of Equalized image

</td></tr></tbody></table></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Full-sized_image">Full-sized image</h3></div>

<td>

</td>

<td>

</td></tr>

<td>

</td>

<td>

</td></tr></tbody></table>

<ul><li><a href="Histogram_matching" title="Histogram matching">Histogram matching</a></li>

<li><a href="Adaptive_histogram_equalization" title="Adaptive histogram equalization">Adaptive histogram equalization</a></li>

<li><a href="Normalization_(image_processing)" title="Normalization (image processing)">Normalization (image processing)</a></li>

<li><a href="Digital_image_processing" title="Digital image processing">Digital image processing</a></li>

<li><a href="Image_segmentation" title="Image segmentation">Image segmentation</a></li></ul>

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

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<li id="cite_note-8"><a href="#cite_ref-8">^</a> N. Bassiou and C. Kotropoulos, "<a rel="nofollow" class="external text" href="http://www.sciencedirect.com/science/article/pii/S1077314206002141">Color image histogram equalization by absolute discounting back-off</a>," Computer Vision and Image Understanding, vol. 107, no. 1-2, pp.108-122, Jul.-Aug. 2007

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<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite id="CITEREFHanYangLee2011" class="citation journal cs1">Han, Ji-Hee; Yang, Sejung; Lee, Byung-Uk (2011). "A Novel 3-D Color Histogram Equalization Method with Uniform 1-D Gray Scale Histogram". IEEE Transactions on Image Processing. 20 (2): 506–512. <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/2011ITIP...20..506H">2011ITIP...20..506H</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.2010.2068555">10.1109/TIP.2010.2068555</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/20801744">20801744</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:17972519">17972519</a>.</cite>

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<ul><li>Acharya and Ray, Image Processing: Principles and Applications, Wiley-Interscience 2005 <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-71998-6</bdi></li>

<li>Russ, The Image Processing Handbook: Fourth Edition, CRC 2002 <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8493-2532-3</bdi></li>

<li><a rel="nofollow" class="external text" href="http://www.generation5.org/content/2004/histogramEqualization.asp">"Histogram Equalization" at Generation5</a> (<a rel="nofollow" class="external text" href="https://archive.today/20160516012012/http://www.generation5.org/content/2004/histogramEqualization.asp">archive</a>)</li></ul>

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