Stereo matching and disparity

I try implement algorithm from article "A stereo matching algorithm with an adaptive window : theory and experiment" by Takeo Kanade and Masatoshi Okutomi. (I am new user and I can't post link)

First problem is on page 7 (denoted as 2) - Taylor expansion with partial derivative. How compute this derivative? It will be delta neighbour pixel intensivity? But in Taylor series is only first derivative - it mean, derivative is constant over window?

Second problem: never exists formula to compute noise variance \sigma^2_n in formula (26) on page 11 (denoted as 6). If I assume this as zero in first step, or no noise at image (or even image1=image2) in denominator can bez zero, derivative/denominator will be infinity at (0,0) point, Each sum will be infinity, and 1/infinity will zero always for no noise.

Maybe it is error in Taylor series and derivative is not over f2 but f2-f1?