Statistical Methods in Imaging

The theme of this chapter is statistical methods in imaging, with a marked emphasis on the Bayesian perspective. The application of statistical notions and techniques in imaging requires that images and the available data are redefined in terms of random variables, the genesis and interpretation of randomness playing a major role in deciding whether the approach will be along frequentist or Bayesian guidelines. The discussion on image formation from indirect information, which may come from non-imaging modalities, is coupled with an overview of how statistics can be used to overcome the hurdles posed by the inherent ill-posedness of the problem. The statistical counterpart to classical inverse problems and regularization approaches to contain the potentially disastrous effects of ill-posedness is the extraction and implementation of complementary information in imaging algorithms. The difficulty in expressing quantitative and uncertain notions about the imaging problem at hand in qualitative terms, which is a major challenge in a deterministic context, can be more easily overcome once the problem is expressed in probabilistic terms. An outline of how to translate some typical qualitative traits into a format which can be utilized by statistical imaging algorithms is presented. In line with the Bayesian paradigm favored in this chapter, basic principles for the construction of priors and likelihoods are presented, together with a discussion of numerous computational statistics algorithms, including maximum likelihood estimators, maximum a posteriori and conditional mean estimators, expectation maximization, Markov chain Monte Carlo, and hierarchical Bayesian models. Rather than aiming to be a comprehensive survey, the present chapter hopes to convey a wide and opinionated overview of statistical methods in imaging.