image processing - Understanding the Bilateral Filter - Neighbors and Sigma

I have been reading about the Bilateral Filter in - Fast Bilateral Filtering for the Display of High Dynamic- Range Images and have felt crippled and confused while understanding its working and the parameters.

In the equation for Bilateral Filter given as

• How to determine the number of neighboring pixels?


• Are all adjacent pixels the neighboring pixels? (top, right, bottom, left, diagonal)


• What does it really mean when people say the following (underlined):

Please help me understand this. I feel lost as I try to understand and gain an intuition of the Bilateral Filter.

I plotted a graph of the Gaussian function (y-axis represents the value of Gaussian function;) with three-sigma values 0.2, 0.5 and 0.9 to help me understand but got nothing.

(blue = 0.2, green = 0.9)