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In two dimensions, it is the product of two such Gaussian functions, one in each dimension: [1] [2] [3] (,) = + where x is the distance from the origin in the horizontal axis, y is the distance from the origin in the vertical axis, and σ is the standard deviation of the Gaussian distribution. It is important to note that the origin on these ...
where the standard deviations are expressed in a number of samples and N is the total number of samples. The standard deviation of a filter can be interpreted as a measure of its size. The cut-off frequency of a Gaussian filter might be defined by the standard deviation in the frequency domain:
The graph of a Gaussian is a characteristic symmetric "bell curve" shape. The parameter a is the height of the curve's peak, b is the position of the center of the peak, and c (the standard deviation, sometimes called the Gaussian RMS width) controls the width of the "bell".
Note that the Laplacian of the Gaussian can be used as a filter to produce a Gaussian blur of the Laplacian of the image because = by standard properties of convolution. The relationship between the difference of Gaussians operator and the Laplacian of the Gaussian operator is explained further in Appendix A in Lindeberg (2015).
If the considered function is the density of a normal distribution of the form = [()] where σ is the standard deviation and x 0 is the expected value, then the relationship between FWHM and the standard deviation is [1] = .
The mean and the standard deviation of a set of data are descriptive statistics usually reported together. In a certain sense, the standard deviation is a "natural" measure of statistical dispersion if the center of the data is measured about the mean. This is because the standard deviation from the mean is smaller than from any other point.
Application of non-local means to an image corrupted by Gaussian noise Non-local means is an algorithm in image processing for image denoising . Unlike "local mean" filters, which take the mean value of a group of pixels surrounding a target pixel to smooth the image, non-local means filtering takes a mean of all pixels in the image, weighted ...
Another way is to define the cdf () as the probability that a sample lies inside the ellipsoid determined by its Mahalanobis distance from the Gaussian, a direct generalization of the standard deviation. [13] In order to compute the values of this function, closed analytic formula exist, [13] as follows.