Computing image integral

Question

How do one find the mean, std dev, and gradient from image integral? Given an image such as follows:

Answer 1

If C+A-B-C is the sum of all gray levels in the zone, then the mean is not

mean = C+A-B-D/4

but

mean = C+A-B-D/K

where K is the number of graylevels in the zone.

Also,

dev = sqrt( C'+A'-B'-D'/4 - (mean*mean) )

is not the stdev, because

dev = sqrt( (1/N)*sum_N ( x_i - u )^2 )

the equation here is equivalent to

dev = sqrt( (1/N)*sum_N ( (x_i)^2 ) - u^2 )

those equations are not equivalent.