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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.

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    The indication about $X$ is dubious since one needs a whole process $(X(x))_x$ and not just one random variable $X$. In fact it seems necessary to define the global distribution of this process $(X(x))_x$.2012-05-12
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    Didier, I have updated the problem formulation using a stochastic process instead of a random variable.2012-05-12
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    As I said in my first comment, one needs the *joint distribution* of the process $(X_x)_x$. This piece of information is still missing.2012-05-12
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    The effort you put into modifying your post is appreciated, but please, read what I wrote: to answer the question, one needs to know the distribution of $(X_x,X_y)$ for every $x\ne y$ (and much more!).2012-05-12
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    A subquestion that may be easier to answer: are the $X_x$ for different $x$ independent?2012-05-12
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    Rahul, yes the $X_x$ are all independent for different x and all $X_x$ are identically distributed. That's why I don't understand why I need to define the joint distribution of the process $(X_x)_x$. I use X indexed by x solely for the fact that $X_0$ shall be constant within the integral. But for the variance calculation also $X_0$ will be random.2012-05-13
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    **Because you never said so before**. // Unfortunately, i.i.d. random variables indexed by the reals cannot be integrated, see post below, hence the object which interests you is not defined.2012-05-13

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