How to Estimated the Noise Standard Deviation (STD - $ \sigma $) as a Function of Brightness from an Image?
I didn't read the article you referred to but I can try giving you some idea.

Run along the image for each pixel consider its 9 x 9 neighborhood.
For each pixel's neighbor hood calculate the STD and the Mean.  

Use the Mean as the brightness value.
Plot the STD as a function of the mean value.

Some ideas for improvement:


Set the neighborhood size in an adaptive manner to only include data from the same brightness level.  
For patches with the same value, aggregate the STD and average it.