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I am writing a Gaussian blur filter in graphics shader code. I want to make the blur parameterized by radius from the users perspective. The best method I can figure to do this is to pick a suitable stopping point for y, say .001, and solve for the variance to plug into the normal distribution function that will achieve that value of y.

Unfortunately I cannot for the life of me solve this equation for v...

$x = 10$ (blur radius)

$$.001 = \frac{1}{2 \pi v^2}e^{-\frac{x^2}{2v^2}}$$.

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I have tried to convert to latex (which you can do by enclosing in $ signs), so if things aren't what you expect, let me know. – Aryabhata Sep 4 '10 at 19:42
It looks like there is a typo here. If this is supposed to be the probability density function for an isotropic binormal distribution in 2d and $x$ is the radial coordinate, then it needs to be multiplied by $x$. (This won't change the solution methods already proposed, but it affects the details.) A few Newton-Raphson iterations should suffice to find the solution quickly. – whuber Sep 15 '10 at 3:26

To expand on the suggestion to use the Lambert function, I'll show how it arises in your equation of interest.

Starting with

$y=\frac1{2\pi v^2}\exp\left(-\frac{x^2}{2v^2}\right)$

we multiply both sides by $-\pi x^2$ to give

$-\pi x^2 y=-\frac{x^2}{2v^2}\exp\left(-\frac{x^2}{2v^2}\right)$

which can now be inverted to the Lambert function (recall that the Lambert function $W(z)$ is the inverse function of $z\exp(z)$, $W(z)\exp(W(z))=z$):

$-\frac{x^2}{2v^2}=W(-\pi x^2 y)$

which we can now solve for $v$

$v=\frac{x}{\sqrt{-2W(-\pi x^2 y)}}$

The choice of sign for the square root is motivated by the fact that variances are conventionally taken to be positive.

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+1. I suggest you add a brief explanation of the Lambert function. I will delete my answer. – Aryabhata Sep 5 '10 at 22:49
Moron: Done, hopefully the one line explanation of the Lambert function is crystal-clear. – J. M. Sep 5 '10 at 23:01
Heh, should've done that to begin with. Thank you Kaestur! – J. M. Sep 5 '10 at 23:30

I don't think the radius has to involve the variance. You could just use the radius to scale the x-axis.

So you could sample the distribution from -1 to 1, always. If the radius is 4, would sample 9 regions. For 10 you sample 21 times between -1 and 1. The variance can stay the same.

You might want to try this question here too

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