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Statistical and Learning Techniques in Computer Vision Lecture 1 : Markov Random Fields

Up to now we have considered distributions of a single random variable x. Recall that we wish to be in the position to treat each image as being an outcome of a stochastic process. Even in case the size of the image is small, say 256×256, we would need to model a distribution with 65536 random variables. Now consider Figure 1: Correlation of neighbouring pixels. Using the statistic g(δ) (1) we computed a value of g(δ = 1) = 25.65 while E[I(x) 2 ] = 255.8. These values indicate that the distributions of neighbouring pixels are strongly correlated. the statistic that was computed as part of exercise 3 on homework 1. In a first step you determined a set S(δ) = {(x, y) | ||x − y|| = δ}. Subsequently you computed the expectation of the difference in intensity values as a function of the distance between pixels. Let us rewrite this expression as (1) which holds under the assumption that E[I(x) 2 ] = E[I(y) 2 ]. Notice, as a result you computed as estimate of the correlation of the intensity values of pixel locations for which ||x − y|| = δ. Based on the results we conclude that this correlation cannot be ignored in case δ is small.