Bayesian computation of Markov random fields and its application in medical imaging

Abstract

PET imaging has been an active area of research over the recent years and has been used to facilitate disease diagnosis, for instance, in cancer/tumor detection. Given the fact that PET images are extremely noisy, researchers have encountered difficulties in the analyses of PET images. Mixture models have been widely utilized in PET images analyses due to their flexibility and capabilities of modelling heterogeneous data. When spatial dependence has to be considered in the modelling, challenges arise in the subsequent parameter estimations. In particular, the large sizes of the images often lead to computational intractability. This thesis has been largely focused on the inferential problems resulting from the intractable normalizing constants in the spatial mixture models involving the Potts/Ising models.

In Chapter 2, a Bayesian spatial mixture model was employed to estimate kinetic parameters in compartmental model of the myocardium. Our results suggested that Bayesian inference can provide more robust estimations than the conventional methods. In addition, Bayesian inference naturally provided uncertainty estimations for the parameters. The uncertainty estimations are particularly important due to the extremely noisy nature of the data. The spatial dependence between voxels was incorporated by employing the Potts model as the prior in the spatial mixture model where Thermodynamic integration (\shortciteN{green2002hidden}) was utilized to solve the inferential problems related to the spatial correlation.

Motivated by the need to develop computationally efficient and accurate inferential methods for the spatial mixture models, in Chapter 3, a novel method was proposed to overcome intractable normalizing constant problem in the Potts model. The proposed method took advantage of conditional independence of the Markov random field (MRF) and the original lattice of Potts model was recursively split into sublattices. Two sublattices were generated at each split. The first sublattice consisted of pixels which were mutually independent given the second sublattice, and vice versa. Therefore, it became tractable to calculate the conditional density function of the first sublattice given the second one according to the property of the MRF. The second sublattice was then approximated by a new Potts model. The second sublattice was split again and two new smaller sublattices were generated. The decomposition procedure was repeated until some preset criterion was satisfied. The original lattice of Potts model was eventually decomposed into many sublattices of different sizes. The original density function can be calculated by multiplying all the conditional density functions of the sublattices. The procedure avoids the calculation of the normalizing constant entirely. It has been shown that the new method is able to deal with Potts models of large dimensions which cause problems in many existing methods. The ability of dealing with large lattices becomes more and more useful as the size of available data nowadays has increased exponentially. The algorithms which can handle large dataset are needed more urgently.

In Chapter 4, an alternative method was proposed to overcome the normalizing constant problem. In the suggested method, the intractable density function was decomposed into a series of conditional density functions. Subject to some assumptions, each conditional density can be approximated by a Monte Carlo approximation of conditional distribution of the corresponding summary statistics. The method has been demonstrated to be faster than most of the competitors in the empirical studies. In addition, this method is extendable to irregular lattices.

Finally, when a mixture model is used in conjunction with Markov random field, label switching arises since posterior distributions are invariant with respect to the permutation of MCMC samples. Various methods were developed to solve the label switching problem. However, it was difficult to find an algorithm which is suitable for the spatial mixture models involving large sized Potts models. In Chapter 5, a new method was suggested to solve the label switching problem for the spatial mixture models. We concluded with some discussions in Chapter 6.