For over 30 years, Bayesian image analysis has provided an important pathway to image reconstruction and enhancement, by balancing a priori expectations of image characteristics with a model for the noise process. The conventional Bayesian modeling approach defined in image space implements priors that describe inter-dependence between spatial locations (and can therefore be difficult to model and compute). However, similar models can be developed more conveniently in Fourier transformed space as a large set of independent processes. The originally complex high-dimensional estimation problem in image space, is thereby broken down into a series of (trivially parallelizable) independent one-dimensional problems in Fourier space. We will illustrate a range Bayesian image analysis in Fourier space (BIFS) models along with a Python package we have developed for their implementation. We will additionally look at how BIFS priors can be used to approximate a range of Markov random field models. Finally, we will give an overview of extending these Fourier space models into the wavelet domain using multifractal priors.
John Kornak is Professor of Biostatistics at the University of California, San Francisco. He earned his Undergraduate degree in Mathematics with Statistics at the University of Nottingham, UK in 1996. He subsequently obtained his PhD in Statistics, also at the University of Nottingham, studying the statistical image analysis of fMRI datasets. Dr. Kornak has held postdoc positions at the University of California, San Francisco (UCSF) working on the statistical image analysis of Magnetic Resonance Spectroscopic imaging data, and then at The Ohio State University working on spatial statistics modeling. Dr. Kornak then took up a faculty position at UCSF in the Department of Radiology and Biomedical Imaging, later moving to the Department of Epidemiology and Biostatistics. In addition to being a Professor of Biostatistics, he is also the Head of the Health Data Science Program and Director of the Clinical and Translational Sciences Institute Biostatistics Consulting Unit. Dr. Kornak's research at UCSF has primarily focused on the application of medical imaging to neurodegenerative diseases, and he is presently the PI of an NIH NIBIB R01 grant for developing Bayesian Image Analysis in Fourier Space, with applications in understanding perfusion MRI effects in Frontotemporal dementia and breast cancer detection with contrast-enhanced MRI.