The Integral Image is used as a quick and effective way of calculating the sum of values (pixel values)
in a given image – or a rectangular subset of a grid (the given image).

In this article we will assume that concepts of integral image is known and then proceed
to see how it can be used to compute the mean and variance of a image patch.

Given a integral representation of an image,the sum of value of pixels in the rectangular
region R with vertices A,B,C,D is given by
\[
I = S(A) +S(D) -S(B) -S(C)
\]

Dividing this quantity by the number of pixels gives us the mean value
of pixels in the region.
\[
\mu = \frac{I}{N}
\]

Let us also consider the squared integral image.To obtain this all the pixel
values in the image are squared then integral image is computed.

consider the variance about a rectangulation regions
\[
v= \sum_i (x_i-\mu)^2
v= \sum_i x_i^2 - 2\sum_i x_i \mu + \mu^2
v= \sum_i x_i^2 - \mu^2
\]
The summation $x_i^2$ can obtained by the square integral image
and $\mu$ can be obtained by integral image computation.

This enables us to compute the variance of rectangular patch of image.

A similar method can be employed to compute the denominator variance term normalize cross correlation
formula.