Locally Adaptive Bayesian Modelling for Medical Imaging

Medical imaging plays a critical role in diagnosing and monitoring various medical conditions. However, the inherent limitations of imaging techniques often lead to blurred and noisy images, making accurate diagnosis challenging. Image processing, as an example of an inverse problem, has been described linearly between observed and true images. However, the large and ill-posed transformation matrices lead to impractical solutions. Hence, the theme of the thesis is to develop new image processing methods capable of delivering more robust results, increased image quality and reliable medical diagnosis.

To achieve the research goals, two different approaches under the Bayesian framework are proposed: one with locally adaptive hyper-prior parameters and one with a mixture prior distribution. Both approaches consider spatially varying smoothness, thereby enhancing the simulation flexibility and estimation accuracy of processed images in medical imaging applications.

For the first hierarchical Bayesian modelling, the estimation accuracy is improved by introducing locally adaptive hyperprior variances rather than the global hyperprior variance. The locally adaptive hyperprior variances account for the variation between individual pixels and their neighborhoods separately, confirming that the pixel density within the medical image is not constant.

The other improved Bayesian modelling with mixture prior distribution consists of two potential prior distributions: a novel Laplace and Gaussian mixture prior distribution which incorporates different smoothing strategies with the automatic model-based estimation of mixture component weightings, creating a locally adaptive model. The idea is to utilize the distinguished properties of probability distributions for processing different pixel densities. The introduction of spatial factor for prior distribution selection realizes the classification function since it classifies the pixels' environment into two clusters, smoothing and high-contrast areas respectively.

Furthermore, to refine localization accuracy and improve diagnostic confidence, compared to the use of a single image data resource, a combination of two (or more) medical images from different medical imaging techniques can be used but with a higher risk of misalignment. Therefore, as further exploration, the Bayes factor is employed for spatial parameters selections when the external spatial information provided.

Overall, the Bayesian approaches presented in this thesis utilize multi-level hierarchical modelling and Markov chain Monte Carlo (MCMC) estimation methods to sample from the posterior distribution and thereby perform estimation. In addition to image estimates, MCMC methods also provide hyperparameter estimates, enabling uncertainty quantification that considers potential sources of variability. The use of locally adaptive Bayesian modelling approaches is not confined to medical imaging applications; rather, it offers a broader framework for analyzing other inverse problems.