Convolution matrix

The following important information is missing from this article or section:

Scientific sources for the theory are completely absent. Please complete

Help Wikipedia by researching and pasting it .

Faltungsmatrizen (including core , filter core , filter operator , filter mask or convolution kernel called english convolution kernel ) will be in digital image processing for the filter used. It is mostly about square matrices of odd dimensions in different sizes. Many image processing operations can be represented as a linear system using discrete convolution , a linear operation. For discrete two-dimensional functions (digital images) the following calculation formula results for the discrete convolution:

${\displaystyle I^{*}(x,y)=\sum _{i=1}^{n}\sum _{j=1}^{n}I(x-i+a,\;y-j+a)k(i,j)}$

${\displaystyle I^{*}(x,y)}$here is the result pixel, is the image to which the filter is applied, is the coordinate of the center point in the square convolution matrix, and is an element of the convolution matrix. In order to be able to clearly define the center point, uneven dimensions of the folding matrices are necessary. ${\displaystyle I}$ ${\displaystyle a}$ ${\displaystyle k(i,j)}$

For 3 × 3 convolution matrices, and . For 5 × 5 convolution matrices, and . ${\displaystyle n=3}$ ${\displaystyle a=2}$ ${\displaystyle n=5}$ ${\displaystyle a=3}$

Examples

• Smoothing filter, mean value filter ( soft focus )

${\displaystyle {\frac {1}{9}}\cdot {\begin{pmatrix}1&1&1\\1&1&1\\1&1&1\end{pmatrix}}}$

• Sharpening filter

${\displaystyle {\begin{pmatrix}0&-1&0\\-1&5&-1\\0&-1&0\end{pmatrix}}}$

• Edge filter, Laplace

${\displaystyle {\begin{pmatrix}0&1&0\\1&-4&1\\0&1&0\end{pmatrix}}}$

• Relief filter

${\displaystyle {\begin{pmatrix}-2&-1&0\\-1&1&1\\0&1&2\end{pmatrix}}}$

Convolution theorem

With the help of the convolution theorem , the effort to compute a discrete convolution can be reduced from the complexity class to . ${\displaystyle {\mathcal {O}}(n^{2})}$ ${\displaystyle {\mathcal {O}}(n\cdot \log n)}$

literature

• Gary Bradski, Adrian Kaehler: Learning OpenCV: Computer Vision with the OpenCV Library . O'Reilly Media, ISBN 978-0596516130 .

See also

• Prewitt operator
• Roberts operator
• Sobel operator
• Laplace filter