Sometimes algorithms talk about the mean or variance of a region of an image (sometimes called a neighbourhood, or window). But what does this refer to? The mean is the average of a series of numbers, and the variance measures the average degree to which each number is different from the mean. So the mean of a 3×3 pixel neighbourhood is average of the 9 pixels within it, and the variance is the degree to which every pixel in the neighbourhood varies from the mean. To obtain a better perspective, it’s best to actually explore some images. Consider the grayscale image in Fig.1.

Four 200×200 regions have been extracted from the image, they are shown in Fig.2.

Statistics are given at the end of the post. Region B represents a part of the background sky. Region A is the same region processed using a median filter to smooth out discontinuities. In comparing region A and B, they both have similar means: 214.3 and 212.37 respectively. Yet their appearance is different – one is uniform, and the other seems to contain some variance, something we might attribute to noise in some circumstances. The variance of A is 2.65, versus 70.57 for B. Variance is a poor descriptive statistic, because it is hard to visualize, so many times it is converted to standard deviation (SD), which is just the square root of variance. For A and B, this is 1.63 and 8.4 respectively.

As shown in region C and D, more complex scenes introduce an increased variance, and SD. The variances can be associated with the distribution of the histograms shown below.

When comparing two regions, a lower variance (or in reality a lower SD) usually implies a more uniform region of pixels. Generally, mean and variance are not good estimators for image content because two totally different images can have same mean, and variance.

Image	Mean	Variance	SD
A	214.3	2.65	1.63
B	212.37	70.57	8.4
C	87.22	4463.11	66.81
D	56.33	2115.24	46.0