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Traditionally we define a gaussian function at a point x (assuming mean to be 0) as follows

$$g_{\sigma}(x) = \frac{1}{\sqrt{2\pi \sigma^{2}}} \exp\left(\frac{x^{2}}{2\sigma^{2}}\right)$$

In some sources however, the exact form is given as follows :

$$g_{\sigma}(x) = \frac{1}{2\pi \sigma^{2}}\exp\left(\frac{x^{2}}{2\sigma^{2}}\right)$$

Why these two forms are used and where which one should be employed ?

For example, when I am working with images, a little difference produces great numerical variations which have an effect in terms of learning.

martini
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Ujjwal Aryan
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  • It may be a wild guess but I do know that the Gaussian kernel differs in the way you wrote it in the second formula for a $2-D$ system, where perhaps $x^{2}=|x|^{2}$ – Autolatry Jun 22 '15 at 11:47
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    There are missing negation signs inside the $\exp$ function. Apparently, SE is stupid and won't let me do an edit which changes just 2 characters. – Meni Rosenfeld Jun 22 '15 at 12:34

1 Answers1

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A gaussian function is any function of the form $$ a\,e^{-b(x-c)^2}. $$ A gaussian probability distribution with mean $0$ and variance $\sigma^2$ corresponds to the first of your definitions.

Julián Aguirre
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