Gaussian blur is a widely used effect in graphics software such as Adobe Photoshop, The GIMP, Inkscape, and Paint.NET. It is typically used to reduce image noise and reduce detail levels. The visual effect of this blurring technique is a smooth blur resembling that of viewing the image through a translucent screen, distinctly different from the bokeh effect produced by an out-of-focus lens or the shadow of an object under usual illumination. Gaussian smoothing is also used as a pre-processing stage in computer vision algorithms in order to enhance image structures at different scales—see scale-space representation and scale-space implementation. Adobe Photoshop, or simply Photoshop, is a graphics editor developed and published by Adobe Systems. ... The GNU Image Manipulation Program, or GIMP, is a raster graphics editor application with some support for vector graphics. ... Inkscape is a vector graphics editor application. ... Paint. ... Image noise is unwanted and manifested in the pixels of an image. ... The astounding bokeh of a Helios-40 lens A photograph of jonquil flowers with background bokeh Compare a photograph of jonquil flowers with low background bokeh Bokeh (from the Japanese boke ぼけ, blur) is a photographic term describing the subjective aesthetic qualities of out-of-focus areas in an image produced... Computer vision is the science and technology of machines that see. ... Scale space theory is a framework for multi-scale signal representation developed by the computer vision and image processing communities. ... The linear scale space representation of an N-dimensional continuous signal is obtained by convolving with an N-dimensional Gaussian kernel When applying this operation to a discrete signal different approaches can be taken. ...

Mathematically speaking, applying a Gaussian blur to an image is the same as convolving the image with a Gaussian or normal distribution. (In contrast, convolving by a circle (i.e., a circular box blur) would more-accurately reproduce the bokeh effect.) Since the Fourier transform of a Gaussian is another Gaussian, applying a Gaussian blur has the effect of low pass filtering the image. The normal distribution, also called Gaussian distribution by scientists (named after Carl Friedrich Gauss due to his rigorous application of the distribution to astronomical data (Havil, 2003)) is a probability distribution of great importance in many fields. ... 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. ... The astounding bokeh of a Helios-40 lens A photograph of jonquil flowers with background bokeh Compare a photograph of jonquil flowers with low background bokeh Bokeh (from the Japanese boke ぼけ, blur) is a photographic term describing the subjective aesthetic qualities of out-of-focus areas in an image produced... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... A low-pass filter passes low frequencies fairly well, but attenuates high frequencies. ...

Mechanics

The Gaussian blur is a type of image-blurring filter that uses a normal distribution (also called "Gaussian distribution", thus the name "Gaussian blur") for calculating the transformation to apply to each pixel in the image. The equation of Gaussian distribution in N dimensions is The normal distribution, also called Gaussian distribution by scientists (named after Carl Friedrich Gauss due to his rigorous application of the distribution to astronomical data (Havil, 2003)) is a probability distribution of great importance in many fields. ... In mathematics, a transformation in elementary terms is any of a variety of different functions from geometry, such as rotations, reflections and translations. ...

or specifically in two dimensions

where r is the blur radius (r² = u² + v²), and σ is the standard deviation of the Gaussian distribution. When applied in two dimensions, this formula produces a surface whose contours are concentric circles with a Gaussian distribution from the center point. Pixels where this distribution is non-zero are used to build a convolution matrix, which is applied to the original image. Each pixel's value is set to a weighted average of that pixel's neighborhood. The original pixel's value receives the heaviest weight (having the highest Gaussian value), and neighboring pixels receive smaller weights as their distance to the original pixel increases. This results in a blur that preserves boundaries and edges better than other, more uniform blurring filters; see also scale-space implementation. In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. ... The Comet Nucleus Tour (CONTOUR) was a Discovery-class space mission. ... Concentric objects share the same center or origin. ... In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ... In statistics, given a set of data, X = { x1, x2, ..., xn} and corresponding weights, W = { w1, w2, ..., wn} the weighted mean is calculated as Note that if all the weights are equal, the weighted mean is the same as the arithmetic mean. ... A neighbourhood or neighborhood (see spelling differences) is a geographically localised community located within a larger city or suburb. ... The linear scale space representation of an N-dimensional continuous signal is obtained by convolving with an N-dimensional Gaussian kernel When applying this operation to a discrete signal different approaches can be taken. ...

In theory, the distribution at every point on the image will be non-zero, meaning that the entire image would need to be included in the calculations for each pixel. In practice, when computing a discrete approximation of the Gaussian function, pixels outside of approximately 3σ are small enough to be considered effectively zero. Thus, pixels outside of that range can be ignored. Typically, an imaging program need only calculate a matrix with dimensions to ensure all relevant pixels are accounted for.

In addition to being circularly symmetric, the Gaussian Blur can be applied to a two-dimensional image as two independent one-dimensional calculations, and so is termed Linearly Separable. That is, the effect of applying the two-dimensional matrix can also be achieved by applying a series of single-dimensional Gaussian matrices in the horizontal direction, then repeating the process in the vertical direction. In computational terms, this is a useful property, since the calculation can be performed in time, as opposed to for a non-separable kernel, where M,N are the dimensions of the image being filtered and m, n are the dimensions of the filter kernel.

Applying multiple, successive gaussian blurs to an image has the same effect as applying a single, larger gaussian blur, whose radius is the square root of the sum of the squares of the blur radii that were actually applied. For example, applying successive gaussian blurs with radii of 6 and 8 gives the same results as applying a single gaussian blur of radius 10, since . Because of this relationship, processing time cannot be saved by simulating a gaussian blur with successive, smaller blurs — the time required will be at least as great as performing the single large blur.

Gaussian blurring is commonly used when reducing the size of an image. When downsampling an image, it is common to apply a low-pass filter to the image prior to resampling. This is to ensure that spurious high-frequency information does not appear in the downsampled image (aliasing). Gaussian blurs have nice properties, such as having no sharp edges, and thus do not introduce ringing into the filtered image. In statistics, signal processing, computer graphics and related disciplines, aliasing is an effect that causes different continuous signals to become indistinguishable (or aliases of one another) when sampled. ...

Sample Gaussian matrix

This is a sample matrix, produced by calculating the Gaussian distribution for σ = 0.84089642. Note that the center element (at [4, 4]) has the largest value, decreasing symmetrically as distance from the center increases.

Note that 0.22508352 (the central one) is 1177 times larger than 0.00019117 which is just outside 3σ.

Example

The following example shows the effect of a Gaussian blur; Image 2 was created by applying a Gaussian blur filter (σ = 2) to Image 1.

Image 1: The original Image File history File links Both this H3224. ...

Image 2: The blurred image with σ = 2 Image File history File links H3224020. ...

See also

• Infinite impulse response (IIR)
• Scale-space implementation
• Gaussian function
• Gaussian filter

IIR (infinite impulse response) is a property of signal processing systems. ... The linear scale space representation of an N-dimensional continuous signal is obtained by convolving with an N-dimensional Gaussian kernel When applying this operation to a discrete signal different approaches can be taken. ... Gaussian curves parametrised by expected value and variance (see normal distribution) A Gaussian function (named after Carl Friedrich Gauss) is a function of the form: for some real constants a > 0, b, and c. ... In electronics and signal processing, A Gaussian filter is designed to give no overshoot to a step function input while maximising the rise and fall time. ...

References

• http://www.cee.hw.ac.uk/hipr/html/gsmooth.html