Image Statistics

ENDS489 Course Notes - Fall 2000
Section 1-5

Image Statistics

(Based on material from Digital Imaging: Theory and Applications, H. E. Burdick, McGraw-Hill, 1997)
(corrected 9/11/00)

Arithmetic Mean, Standard Deviation, and Variance

Useful statistical features of an image are its arithmetic mean, standard deviation, and variance. These are well known mathematical constructs that, when applied to a digital image, can reveal important information.

The arithmetic mean is the image's average value.
The standard deviation is a measure of the frequency distribution, or range of pixel values, of an image. If an image is supposed to be uniform throughout, the standard deviation should be small. A small standard deviation indicates that the pixel intensities do not stray very far from the mean; a large value indicates a greater range.
The variance is the square root of the standard deviation.

These values are calculated as shown below: mean = sum (p_x,y)/(x*y)

stdev =((sum(p_x,y*p_x,y)/(x*y)) - (mean * mean)) ^1/2

var = (stdev)*(stdev) In these equations, sum (P_x,y) represents the summation of all pixel values in the image, and sum (P_x,y*P_x,y) is the sum of the squares of all pixel values.

For some applications, such as machine vision, these statistical values can be very accurate indicators of image quality and may be used to make automated decisions.

Image Histogram

An important digital image tool is the histogram. A histogram is a statistical representation of the data within an image that shows how many pixels there are with each of the possible values. An image and its histogram are shown below. The histogram is a bar graph where each entry on the horizontal axis is one of the possible values that a pixel can have. In an 8-bit image, those values range from 0 to 255. Each vertical bar in the graph indicates the number of pixels of that value. The sum of all vertical bars is equal to the total number of pixels in the image. Usually, the absolute value of each vertical bar, or number of pixels at a specific value, is not important. What is important is the number of pixels at a specific value relative to the number of pixels at other values. Often, the histogram is never actually graphed, but is processed by other programs to make decisions as to how to manipulate the image.


	Image's histogram in Photoshop

A histogram represents a statistical analysis of an image. It indicates the distribution of the image data values. From this statistical data:

we can determine the best contrast for an image
we can determine the overall brightness or darkness of an image
we can determine how to apply a function, such as thresholding, to an image
we can judge the effectiveness of an algorithm, such as contrast enhancement, applied to an image.