Does logarithm of Gaussian image still gaussian distribution? I have an image 2D  that pixel intensity follows multi Gaussian distribution such as
$$p \left( I(x) \in \Omega_i \mid (I(x)\right)=\frac{1}{2\pi \sigma_i}\exp\left(-\frac {(I(x)-\mu_i)^2}{2\sigma_i^2} \right )$$
where $I(x):\Omega \to R$, $\sigma_i,\mu_i$ are std, and mean of region $\Omega_i$. (For example: Image in $\Omega$ domain can spearate into three region $\Omega_1,\Omega_2,\Omega_3$)
I will take the logarithm of image I such as 
$$I_{\log}=\log(I)$$
Then, could I said that the pixel intensity in logarithm space still follows multi Gaussian distribution such as?
$$p \left( I(x) \in \Omega_i \mid (I(x)\right)=\frac{1}{2\pi \sigma_i}\exp\left(-\frac {(I_{\log}-\mu_i)^2}{2\sigma_i^2} \right )$$
Update: This is image distribution


Since any linear transformation of a Gaussian variable is Gaussian, you get that "locally" (if your Gaussian distribution for $I$ is concentrated strongly enough around a small enough region) then $\log I$, or any differnentiable function $f(I)$ for that matter, will look Gaussian in most of the probability mass for $I$ and you can easily estimate the mean and variance for the new Gaussian distribution of $f(I)$. 
