In the introduction to this chapter we describe three ways of estimating the mean of a process: the arithmetic mean μ_A, the geometric mean μ_G, and the harmonic mean μ_H.

An image is generated that contains a certain number of pixels and whose intensity distribution is governed by one of five probability distributions. The histogram of the intensities is displayed in blue. One of the Pythagorean means will be indicated in red, one in green and one in purple. Choose the number of pixels in the image and the underlying probability distribution of the intensities.

1. At the smallest image size of four pixels, which distributions show the greatest spread among the Pythagorean means?


2. Which of the three means {μ_A, μ_G,  μ_H,} is shown in red? Which in green? Which in purple?


3. Why does the order of the three means in the window remain the same as you change distribution and/or image size?


4. Based upon your answer to the previous question, what might be a reasonable definition for the spread of the Pythagorean means?


5. As the number of pixels in the image increases, what characteristic of the distribution appears to control the spread among the Pythagorean means?


6. The random variable being investigated in this exercise is the intensity of pixels. What constraints does this impose on the random variable?


7. If we were to look at another stochastic signal such as speech, which of the Pythagorean means might be considered inappropriate and why?


8. If we were to look at a low-level pixel intensity where random variations were due to photon statistics, which of the Pythagorean means might be considered inappropriate and why? (Remember that photon statistics are characterized by a Poisson process.)