# I have some questions about background subtraction with Gaussian Mixture Model (GMM)

I'm learning about Gaussian Mixture Model (GMM) for background subtraction.. And have some questions about it..

1. In Chris Stauffer paper "Adaptive background mixture models for real-time tracking" Gaussian probability density function is $$\mathcal{N}(X_t,\mu,\Sigma)=\dfrac{1}{(2\pi)^\frac{n}{2} \left|{\Sigma}\right|^\frac{1}{2}}\exp\left(-\dfrac{1}{2}(X_t-\mu_t)^T\Sigma^{-1}(X_t-\mu_t)\right)$$
where $n$ in multi dimension, is $n$ in 3-D which is pixel position in x,y, and intensity of pixel?
2. In the same paper, they said: $$\Sigma_{k,t}=\sigma_k^2.I$$ Is $I$ is Indentity matrix?
3. They also said red,green, and blue pixel values are independent and have the same variances, if it independent why R,G, and B have same variances? shouldn't it have different variances because the value of each channel is different..
4. If I set the algorithm with number of distribution is 3, so it's only 1 distribution that chosen as background model and the rest is foreground model? Because they said the first B distribution are chosen as the background model, where $$B=\arg \min_b(\sum\limits_{k=1}^b w_k>T)$$

1. Yes, the dimension of $X$ should be 3, one for each of the red, green, and blue channels. You can see this from their Figure 2, where they show plots of the distributions of red and green channels only (to motivate why they want to use a Gaussian Mixture Model in the first place --- the distribute of pixel values is multi-modal).
2. Yes, $I$ is the identity matrix.
4. Whether only 1 distribution is chosen when $K=3$ will depend on the threshold $T$, surely? Yes, it is possible that only one distribution will be chosen.