In the context of scale space image pyramids using Gaussian filters I noticed that it's common to downsample the image after blurring with $\sigma = 2*\sigma_{init}$ . My question is: What is the relation between variance and downsampling? What is the proof that I'm not losing information when the image is downsampled after blurring with this factor?
I think this can easily be shown using Nyquist and the cut-off frequency of the Gaussian filter, but I'm not sure how.
You're correct, it has to do with the Cut Off frequency of the Gaussian Blur Filter in its Frequency Domain.
In order to see it, just apply a DFT (Using MATLAB it can be achieved by fft / fft2) and look on the absolute value.
Look for the -3dB point and you'll see.
There is also an intuitive explanation on the original article which say that blurring using Gaussian Kernel with $ \sigma $ means removing any detail which its size is smaller than $ \sqrt{ \sigma } $.