Statistical image reconstruction

This chapter considers the use of statistical modelling approaches and stochastic estimation algorithms for the reconstruction of images from data. A key distinguishing feature of statistical image reconstruction, compared to classical deterministic approaches, is the explicit use of statistical descriptions and probability models. This provides an intuitiveand flexible framework within which the whole modelling and reconstruction process can be described. In essence, the relationship between data and parameters is described by a likelihood and relationships between parameters described by prior distributions. These are combined using Bayes theorem to produce a posterior distribution which is used as the basis for reconstruction. The resulting estimation task is now too complicated to admit explicit solution, and so numerical algorithms, designed using the Markov chain Monte Carlo method, are often used. As well as leading to image reconstruction, these techniques also allow a fuller investigation, in terms of estimation reliability and uncertainty, than other approaches. In the next section, general motivating and historical background is given. In section 22.3 the general approach to statistical image reconstruction will be introduced including basic notation and definitions. Then, in section 22.4, different types of data models will be discussed and corresponding definitions of likelihood functions given. Prior modelling will be the theme of sections 22.5, 22.6 and 22.7. In the first two, pixel-based models are considered, whereas in the third, some comments on feature-based approaches are made. Numerical estimation methods, and the Markov chain Monte Carlo approach in particular, are described in section 22.8 with examples in 22.9. Finally, discussion of future directions and some concluding remarks are given in section 22.10, followed by references.