A box blur is an image 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 and is a convolution

Convolution

In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...

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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.

Box blurs are frequently used to approximate a Gaussian blur

Gaussian blur

A Gaussian blur is the result of blurring an image by a Gaussian function. It is a widely used effect in graphics software, typically to reduce image noise and reduce detail...

. By the central limit theorem

Central limit theorem

In probability theory, the central limit theorem states conditions under which the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed. The central limit theorem has a number of variants. In its common...

, if applied 3 times on the same image, a box blur approximates the Gaussian kernel to within about 3%, yielding the same result as a quadratic convolution kernel.

In the frequency domain

Frequency domain

In electronics, control systems engineering, and statistics, frequency domain is a term used to describe the domain for analysis of mathematical functions or signals with respect to frequency, rather than time....

, a box blur has zeros and negative components. That is, a sine wave

Sine wave

The sine wave or sinusoid is a mathematical function that describes a smooth repetitive oscillation. It occurs often in pure mathematics, as well as physics, signal processing, electrical engineering and many other fields...

 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

Bokeh

In photography, bokeh is the blur, or the aesthetic quality of the blur, in out-of-focus areas of an image, or "the way the lens renders out-of-focus points of light."...

circles touch to form a bright spot where there would be a dark spot between two bright spots in the original image.