White Noise.

Image intensities values always contain some degree of randomness, namely if we capture two images of the same scene one after the other then the images are not exactly the same. The variability can be due to random changes in the scene (small fluctuations in the illumination, for example) and/or to variations in the imaging device/eye such as the noise in the electronics. There is also noise in neural systems, namely cell membranes and spike trains do not respond in a deterministic way to a visual stimulus (even if the stimulus itself had no noise). Let's take the example that I(x) is a sampled " noise " function n(x), which we take to be a sequence of N independent and identically distributed random variables with mean 0 and variance σ 2 n , i.e., for any x, E{ n(x)} = 0 and E{ n(x) 2 } = σ 2 n. The symbol E stands for " expected value ". For those of you who have not taken a probability course, the expected value of a random variable is just its average over a large (infinite) number of trials. Note that the σ 2 n here refers to the variance in the intensity, not to a spatial spread/blur as we have previously used it. In fact, we won't need to assume the pixels are independent. It will be enough to assume the two pixels are statistically uncorrelated, 1 namely, for any d = 0, E{ n(x)n(x + d)} = 0. (1) More generally, the correlation of any two (mean 0) variables X, Y is the expected value of the product XY. As illustrated in the slides, in general, two (mean 0) variables X, Y are positively correlated if samples of (X, Y) tend to lie on a line of positive slope through the origin, negatively correlated if samples of (X, Y) tend to lie on a line of negative slope through the origin, and uncorrelated if they are not well fit by any line through the origin.