TheInfoList

In
image processing Digital image processing is the use of a digital computer A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations automatically. Modern computers can perform generic se ...
, a kernel, convolution matrix, or mask is a small
matrix Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols, or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the material in between a eukaryot ...
used for blurring, sharpening, embossing,
edge detection Edge detection includes a variety of mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in S ... , and more. This is accomplished by doing a ''
convolution In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
'' between the kernel and an
image An image (from la, imago) is an artifact that depicts visual perception Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...
.

# Details

The general expression of a convolution is $g\left(x,y\right)= \omega *f\left(x,y\right)=\sum_^a,$ where $g\left(x,y\right)$ is the filtered image, $f\left(x,y\right)$ is the original image, $\omega$ is the filter kernel. Every element of the filter kernel is considered by $-a \leq dx \leq a$ and $-b \leq dy \leq b$. Depending on the element values, a kernel can cause a wide range of effects. The above are just a few examples of effects achievable by convolving kernels and images.

## Origin

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.

# Convolution

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 mathematical convolution. The matrix operation 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 The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
(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: 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: 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'' :*corresponding input image pixels are found relative to the kernel's origin. 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.http://www.songho.ca/dsp/convolution/convolution2d_example.html The general form for matrix convolution is $\begin x_ & x_ & \cdots & x_ \\ x_ & x_ & \cdots & x_ \\ \vdots & \vdots & \ddots & \vdots \\ x_ & x_ & \cdots & x_ \\ \end * \begin y_ & y_ & \cdots & y_ \\ y_ & y_ & \cdots & y_ \\ \vdots & \vdots & \ddots & \vdots \\ y_ & y_ & \cdots & y_ \\ \end = \sum^_ \sum^_ x_ y_$

## Edge Handling Kernel convolution usually requires values from pixels outside of the image boundaries. There are a variety of methods for handling image edges. ; Extend : 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. ; Wrap :The image is conceptually wrapped (or tiled) and values are taken from the opposite edge or corner. ; Mirror : 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. ; Crop : 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. ; Kernel Crop : Any pixel in the kernel that extends past the input image isn't used and the normalizing is adjusted to compensate.

## Normalization

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.

## Concrete implementation

Here a concrete convolution implementation done with the
GLSL OpenGL Shading Language (GLSL) is a high-level High-level and low-level, as technical terms, are used to classify, describe and point to specific Objective (goal), goals of a systematic operation; and are applied in a wide range of contexts, suc ...
shading language : // 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) // 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 region3x3(sampler2D sampler, vec2 uv) // 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) void mainImage( out vec4 fragColor, in vec2 fragCoord )

# References

* * * * {{cite book , last2= Stockman , first2= George C. , last1= Shapiro , first1= Linda G., author1-link=Linda Shapiro , date= February 2001 , title= Computer Vision , work= Prentice Hall , pages= 53–54 , isbn= 978-0130307965

* Multidimensional discrete convolution