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

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Kernel (image processing)

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</style><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="Kernel_(disambiguation)" class="mw-redirect mw-disambig" title="Kernel (disambiguation)">Kernel (disambiguation)</a>.</div>

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In <a href="Image_processing" class="mw-redirect" title="Image processing">image processing</a>, a kernel, convolution matrix, or mask is a small <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> used for blurring, sharpening, <a href="Image_embossing" title="Image embossing">embossing</a>, <a href="Edge_detection" title="Edge detection">edge detection</a>, and more. This is accomplished by doing a <a href="Convolution" title="Convolution">convolution</a> between the kernel and an <a href="Bitmap_image" class="mw-redirect" title="Bitmap image">image</a>. Or more simply, when each pixel in the output image is a function of the nearby pixels (including itself) in the input image, the kernel is that function.

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

The general expression of a convolution is

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<annotation encoding="application/x-tex">{\displaystyle g_{x,y}=\omega *f_{x,y}=\sum _{i=-a}^{a}{\sum _{j=-b}^{b}{\omega _{i,j}f_{x-i,y-j}}},}</annotation>

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where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x,y)}">

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</math><img src="./bf358a54b0375e22ae5f3ab2c3e1a22c0c87e11c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.444ex; height:2.843ex;" alt="{\displaystyle g(x,y)}" loading="lazy"> is the filtered image, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x,y)}">

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

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

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Depending on the element values, a kernel can cause a wide range of effects:

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<th>Kernel ω

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<th>Image result g(x,y)

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<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\ \ 0&\ \ 0&\ \ 0\\\ \ 0&\ \ 1&\ \ 0\\\ \ 0&\ \ 0&\ \ 0\end{bmatrix}}}</annotation>

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<td rowspan="2"><a href="Ridge_detection" title="Ridge detection">Ridge</a> or <a href="Edge_detection" title="Edge detection">edge detection</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}0&-1&0\\-1&\ \ 4&-1\\0&-1&0\end{bmatrix}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}-1&-1&-1\\-1&\ \ 8&-1\\-1&-1&-1\end{bmatrix}}}</annotation>

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<td>Sharpen

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<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\ \ 0&-1&\ \ 0\\-1&\ \ 5&-1\\\ \ 0&-1&\ \ 0\end{bmatrix}}}</annotation>

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<td><a href="Box_blur" title="Box blur">Box blur</a> (<a href="Normalization_(image_processing)" title="Normalization (image processing)">normalized</a>)

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{9}}{\begin{bmatrix}\ \ 1&\ \ 1&\ \ 1\\\ \ 1&\ \ 1&\ \ 1\\\ \ 1&\ \ 1&\ \ 1\end{bmatrix}}}</annotation>

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<td><a href="Gaussian_blur" title="Gaussian blur">Gaussian blur</a> 3 × 3 (approximation)

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{16}}{\begin{bmatrix}\ \ 1&\ \ 2&\ \ 1\\\ \ 2&\ \ 4&\ \ 2\\\ \ 1&\ \ 2&\ \ 1\end{bmatrix}}}</annotation>

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<td><a href="Gaussian_blur" title="Gaussian blur">Gaussian blur</a> 5 × 5 (approximation)

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{256}}{\begin{bmatrix}1&4&6&4&1\\4&16&24&16&4\\6&24&36&24&6\\4&16&24&16&4\\1&4&6&4&1\end{bmatrix}}}</annotation>

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<td><a href="Unsharp_masking" title="Unsharp masking">Unsharp masking</a> 5 × 5 Based on Gaussian blur with amount as 1 and threshold as 0 (with no <a href="Image_mask" class="mw-redirect" title="Image mask">image mask</a>)

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<annotation encoding="application/x-tex">{\displaystyle {\frac {-1}{256}}{\begin{bmatrix}1&4&\ \ 6&4&1\\4&16&\ \ 24&16&4\\6&24&-476&24&6\\4&16&\ \ 24&16&4\\1&4&\ \ 6&4&1\end{bmatrix}}}</annotation>

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The above are just a few examples of effects achievable by convolving kernels and images.

<div class="mw-heading mw-heading3"><h3 id="Origin">Origin</h3></div>

The origin is the position of the kernel which is above (conceptually) the current output pixel. This could be outside of the actual kernel, though usually it corresponds to one of the kernel elements. For a symmetric kernel, the origin is usually the center element.

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

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Symmetric_convolution" title="Symmetric convolution">Symmetric convolution</a></div>

Convolution is the process of adding each element of the image to its local neighbors, weighted by the kernel. This is related to a form of <a href="Convolution" title="Convolution">mathematical convolution</a>. <a href="Hadamard_product_(matrices)" title="Hadamard product (matrices)">The matrix operation</a> being performed—convolution—is not traditional matrix multiplication, despite being similarly denoted by *.

For example, if we have two three-by-three matrices, the first a kernel, and the second an image piece, convolution is the process of flipping both the rows and columns of the kernel and multiplying locally similar entries and summing. The element at coordinates [2, 2] (that is, the central element) of the resulting image would be a weighted combination of all the entries of the image matrix, with weights given by the kernel:

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<annotation encoding="application/x-tex">{\displaystyle \left({\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}}*{\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix}}\right)[2,2]=}</annotation>

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The other entries would be similarly weighted, where we position the center of the kernel on each of the boundary points of the image, and compute a weighted sum.

The values of a given pixel in the output image are calculated by multiplying each kernel value by the corresponding input image pixel values. This can be described algorithmically with the following pseudo-code:

<pre>for each image row in input image:

for each pixel in image row:

set accumulator to zero

for each kernel row in kernel:

for each element in kernel row:

if element position corresponding* to pixel position then

multiply element value corresponding* to pixel value

add result to accumulator

endif

set output image pixel to accumulator

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<dl><dd>*corresponding input image pixels are found relative to the kernel's origin.</dd></dl>

If the kernel is symmetric then place the center (origin) of the kernel on the current pixel. The kernel will overlap the neighboring pixels around the origin. Each kernel element should be multiplied with the pixel value it overlaps with and all of the obtained values should be summed. This resultant sum will be the new value for the current pixel currently overlapped with the center of the kernel.

If the kernel is not symmetric, it has to be flipped both around its horizontal and vertical axis before calculating the convolution as above.<a href="#cite_note-1">[1]</a>

The general form for matrix convolution is

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<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}x_{11}&x_{12}&\cdots &x_{1n}\\x_{21}&x_{22}&\cdots &x_{2n}\\\vdots &\vdots &\ddots &\vdots \\x_{m1}&x_{m2}&\cdots &x_{mn}\\\end{bmatrix}}*{\begin{bmatrix}y_{11}&y_{12}&\cdots &y_{1n}\\y_{21}&y_{22}&\cdots &y_{2n}\\\vdots &\vdots &\ddots &\vdots \\y_{m1}&y_{m2}&\cdots &y_{mn}\\\end{bmatrix}}=\sum _{i=0}^{m-1}\sum _{j=0}^{n-1}x_{(m-i)(n-j)}y_{(1+i)(1+j)}}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading3"><h3 id="Edge_handling">Edge handling</h3></div>

Kernel convolution usually requires values from pixels outside of the image boundaries. There are a variety of methods for handling image edges.

<dl><dt>Extend</dt>

<dd>The nearest border pixels are conceptually extended as far as necessary to provide values for the convolution. Corner pixels are extended in 90° wedges. Other edge pixels are extended in lines.</dd>

<dd>The image is conceptually wrapped (or tiled) and values are taken from the opposite edge or corner.</dd>

<dt>Mirror</dt>

<dd>The image is conceptually mirrored at the edges. For example, attempting to read a pixel 3 units outside an edge reads one 3 units inside the edge instead.</dd>

<dt>Crop / Avoid overlap</dt>

<dd>Any pixel in the output image which would require values from beyond the edge is skipped. This method can result in the output image being slightly smaller, with the edges having been cropped. Move kernel so that values from outside of image is never required. Machine learning mainly uses this approach. Example: Kernel size 10x10, image size 32x32, result image is 23x23.</dd>

<dt>Kernel Crop</dt>

<dd>Any pixel in the kernel that extends past the input image isn't used and the normalizing is adjusted to compensate.</dd>

<dt>Constant</dt>

<dd>Use constant value for pixels outside of image. Usually black or sometimes gray is used. Generally this depends on application.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Normalization">Normalization</h3></div>

Normalization is defined as the division of each element in the kernel by the sum of all kernel elements, so that the sum of the elements of a normalized kernel is unity. This will ensure the average pixel in the modified image is as bright as the average pixel in the original image.

<div class="mw-heading mw-heading3"><h3 id="Optimization">Optimization</h3></div>

Fast convolution algorithms include:

<ul><li>separable convolution</li></ul>

<div class="mw-heading mw-heading4"><h4 id="Separable_convolution">Separable convolution</h4></div>

2D convolution with an M × N kernel requires M × N multiplications for each sample (pixel). If the kernel is separable, then the computation can be reduced to M + N multiplications. Using separable convolutions can significantly decrease the computation by doing 1D convolution twice instead of one 2D convolution.<a href="#cite_note-2">[2]</a>

<div class="mw-heading mw-heading3"><h3 id="Implementation">Implementation</h3></div>

Here a concrete convolution implementation done with the <a href="OpenGL_Shading_Language" title="OpenGL Shading Language">GLSL</a> shading language :<div class="mw-highlight mw-highlight-lang-glsl mw-content-ltr" dir="ltr"><pre>// author : csblo

// Work made just by consulting :

// https://en.wikipedia.org/wiki/Kernel_(image_processing)

// Define kernels

#define identity mat3(0, 0, 0, 0, 1, 0, 0, 0, 0)

#define edge0 mat3(1, 0, -1, 0, 0, 0, -1, 0, 1)

#define edge1 mat3(0, 1, 0, 1, -4, 1, 0, 1, 0)

#define edge2 mat3(-1, -1, -1, -1, 8, -1, -1, -1, -1)

#define sharpen mat3(0, -1, 0, -1, 5, -1, 0, -1, 0)

#define box_blur mat3(1, 1, 1, 1, 1, 1, 1, 1, 1) * 0.1111

#define gaussian_blur mat3(1, 2, 1, 2, 4, 2, 1, 2, 1) * 0.0625

#define emboss mat3(-2, -1, 0, -1, 1, 1, 0, 1, 2)

// Find coordinate of matrix element from index

vec2 kpos(int index)

{

return vec2[9] (

vec2(-1, -1), vec2(0, -1), vec2(1, -1),

vec2(-1, 0), vec2(0, 0), vec2(1, 0),

vec2(-1, 1), vec2(0, 1), vec2(1, 1)

)[index] / iResolution.xy;

}

// Extract region of dimension 3x3 from sampler centered in uv

// sampler : texture sampler

// uv : current coordinates on sampler

// return : an array of mat3, each index corresponding with a color channel

mat3[3] region3x3(sampler2D sampler, vec2 uv)

{

// Create each pixels for region

vec4[9] region;

for (int i = 0; i < 9; i++)

region[i] = texture(sampler, uv + kpos(i));

// Create 3x3 region with 3 color channels (red, green, blue)

mat3[3] mRegion;

for (int i = 0; i < 3; i++)

mRegion[i] = mat3(

region[0][i], region[1][i], region[2][i],

region[3][i], region[4][i], region[5][i],

region[6][i], region[7][i], region[8][i]

);

return mRegion;

}

// Convolve a texture with kernel

// kernel : kernel used for convolution

// sampler : texture sampler

// uv : current coordinates on sampler

vec3 convolution(mat3 kernel, sampler2D sampler, vec2 uv)

{

vec3 fragment;

// Extract a 3x3 region centered in uv

mat3[3] region = region3x3(sampler, uv);

// for each color channel of region

for (int i = 0; i < 3; i++)

{

// get region channel

mat3 rc = region[i];

// component wise multiplication of kernel by region channel

mat3 c = matrixCompMult(kernel, rc);

// add each component of matrix

float r = c[0][0] + c[1][0] + c[2][0]

+ c[0][1] + c[1][1] + c[2][1]

+ c[0][2] + c[1][2] + c[2][2];

// for fragment at channel i, set result

fragment[i] = r;

}

return fragment;

}

void mainImage( out vec4 fragColor, in vec2 fragCoord )

{

// Normalized pixel coordinates (from 0 to 1)

vec2 uv = fragCoord/iResolution.xy;

// Convolve kernel with texture

vec3 col = convolution(emboss, iChannel0, uv);

// Output to screen

fragColor = vec4(col, 1.0);

}

</pre></div>

<ul><li><a href="Convolution" title="Convolution">Convolution in mathematics </a></li>

<li><a href="Multidimensional_discrete_convolution" title="Multidimensional discrete convolution">Multidimensional discrete convolution</a></li></ul>

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

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/* start https://en.wikipedia.org/ */

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/* end https://en.wikipedia.org/ */

</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.songho.ca/dsp/convolution/convolution2d_example.html">"Example of 2D Convolution"</a>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.songho.ca/dsp/convolution/convolution.html">"Convolution"</a>. www.songho.ca. Retrieved 2022-11-19.</cite>

</li>

</ol></div></div>

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

<ul><li><cite id="CITEREFLudwig" class="citation book cs1">Ludwig, Jamie (n.d.). <a rel="nofollow" class="external text" href="http://web.pdx.edu/~jduh/courses/Archive/geog481w07/Students/Ludwig_ImageConvolution.pdf">Image Convolution</a> (PDF). Portland State University.</cite></li>

<li><cite id="CITEREFLecarmeDelvare2013" class="citation book cs1">Lecarme, Olivier; Delvare, Karine (January 2013). The Book of GIMP: A Complete Guide to Nearly Everything. No Starch Press. p. 429. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1593273835</bdi>.</cite></li>

<li><cite id="CITEREFGumsterShimonski2012" class="citation book cs1">Gumster, Jason van; Shimonski, Robert (March 2012). GIMP Bible. John Wiley & Sons. pp. 438–442. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0470523971</bdi>.</cite></li>

<li><cite id="CITEREFShapiroStockman2001" class="citation book cs1"><a href="Linda_Shapiro" title="Linda Shapiro">Shapiro, Linda G.</a>; Stockman, George C. (February 2001). Computer Vision. Prentice Hall. pp. 53–54. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0130307965</bdi>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=38lj0VQci7E">Implementing 2d convolution on FPGA</a></li>

<li><a rel="nofollow" class="external text" href="https://developer.apple.com/library/content/documentation/Performance/Conceptual/vImage/ConvolutionOperations/ConvolutionOperations.html">vImage Programming Guide: Performing Convolution Operations</a></li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20121005005358/http://williamson-labs.com/convolution-2d.htm">Image Processing using 2D-Convolution</a></li>

<li><a rel="nofollow" class="external text" href="https://docs.gimp.org/2.8/en/plug-in-convmatrix.html">GNU Image Manipulation Program - User Manual - 8.2. Convolution Matrix</a></li>

<li><a rel="nofollow" class="external text" href="https://www.shadertoy.com/view/3sGXWh">GLSL Demonstration of 3x3 Convolution Kernels</a></li>

<li><a rel="nofollow" class="external text" href="https://github.com/Geof23/kernel_filter_2d">Complete C++ open source project</a></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-05-20" href="https://en.wikipedia.org/wiki/?title=Kernel_(image_processing)&oldid=1291268789">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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