I am glad to have found this great site. There is a problem I am trying to solve for a while. I want to analyze the noise attenuation behavior of the bilateral filter. So given the unnormalized Gaussian function $\phi(z)=\exp(-\frac{1}{2}z^2)$ I define weights $w(x)=\phi\left(\frac{x}{\sigma_s}\right)\phi\left(\frac{X_0-X_x}{\sigma_r}\right)$ which are basically the product of two Gaussian functions. $X_x$ is a stationary random process which is independent and identically normal distributed for all $x$, i.e., $\{X_x\sim \mathcal{N}(y,\sigma_n^2) \,:\, \forall x \in \mathbb{R}\}.$ Now I am interested to find an analytic expression of the variance $\sigma_b^2$ defined as $\sigma_b^2=Var\left\{\int\limits_{-\infty}^\infty \frac{w(x)}{\int_{-\infty}^\infty w(\tilde{x})d\tilde{x}}X_x dx\right\}$ as a function of $(\sigma_n,\sigma_s,\sigma_r)$.
For the simpler linear filtering case $w(x)=\phi(\frac{x}{\sigma_s})$ I have got the solution, which is:
$\sigma_b^2=\int\limits_{-\infty}^{\infty} \left(\frac{\exp\left(-\frac{x^2}{2\sigma_s^2}\right)}{\sqrt{2\pi}\sigma_s}\right)^2\sigma_n^2 dx =\frac{\sigma_n^2}{2\pi\sigma_s^2}|\sigma_s|\sqrt{\pi}=\frac{\sigma_n^2}{2\sqrt{\pi}\sigma_s }.$
But I don't know how to use the bilateral weights defined above.
Any advise is highly appreciated.