# Box blur

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A box blur (also known as a box linear filter) is a spatial domain linear filter in which each pixel in the resulting image has a value equal to the average value of its neighboring pixels in the input image. It is a form of low-pass ("blurring") filter. A 3 by 3 box blur can be written as 1/9 * determinant matrix:

${\displaystyle {\frac {1}{9}}{\begin{bmatrix}1&1&1\\1&1&1\\1&1&1\end{bmatrix}}}$

Due to its property of using equal weights it can be implemented using a much simpler accumulation algorithm which is significantly faster than using a sliding window algorithm.[1]

Box blurs are frequently used to approximate a Gaussian blur.[2] By the central limit theorem, repeated application of a box blur will approximate a Gaussian blur.

In the frequency domain, a box blur has zeros and negative components. That is: a sine wave with a period equal to the size of the box will be blurred away entirely and wavelengths shorter than the size of the box may be phase reversed, as seen when two bokeh circles touch to form a bright spot where there would be a dark spot between two bright spots in the original image.

## References

1. ^ Wojciech Jarosz. 2001. Fast Image Convolutions
2. ^ W3C SVG1.1 specification, 15.17 Filter primitive 'feGaussianBlur'