Multidimensional Signal and Color Image Processing Using Lattices

9Random Field Models

9.1 Introduction

Many phenomena in imaging systems are random in nature and are most appropriately addressed using probabilistic techniques. The most notable example is random noise that may corrupt an image. However, many image structures and patterns, such as textures, can also be considered to be random phenomena. Essentially, anything that varies in a fashion that is not perfectly predictable can be modeled with probabilistic methods. Besides image structure and noise, this can include random variations due to scene lighting, object surface orientation, motion, etc. To correctly model these quantities, we must extend the techniques of probability and stochastic processes to random functions of two or more independent variables. We call such multidimensional random processes random fields. In this chapter we present a first introduction to some aspects of random field modeling of images. We assume a basic familiarity with topics in probability such as random variables, probability distributions, expectation, conditional distributions, etc. There are numerous textbooks covering these topics; some examples that are particularly relevant to signal and image processing are Priestley (1981), Papoulis and Unnikrishna Pillai (2002), Stark and Woods (2002) and Leon‐Garcia (2008).

Random field models have many applications in image processing. One of the most fundamental of these is image estimation: to estimate an unknown image (or selected image samples) given ...