Umeå University's logo

Much like a backwards computed Sudoku puzzle, starting from the completed number grid and working ones way down to a partially completed grid without damaging the route back to the full unique solution, this thesis tackles the challenges behind setting up a number puzzle in the context of biomedical imaging. By leveraging sparse signal processing theory, we study the means of practical undersampling of positron emission tomography (PET) measurements, an imaging modality in nuclear medicine that visualises functional processes within the body using radioactive tracers. What are the rules for measurement removal? How many measurements can be removed without damaging the route back to the full solution? Moreover, how is the original solution retained once the data has been altered? This thesis aims to investigate and answer such questions in relation to PET data sampling, thereby creating a foundation for a PET Sampling Puzzle.

The objective is to develop intelligent data sampling strategies that allow for practical undersampling of PET measurements combined with sophisticated computational compensations to address the resulting data distortions. We focus on two main challenges in PET undersampling: low-count measurements due to reduced radioactive dose or reduced scan times and incomplete measurements from sparse PET detector configurations. The methodological framework is based on key aspects of sparse signal processing: sparse representations, sparsity patterns and sparse signal recovery, encompassing denoising and inpainting. Following the characteristics of PET measurements, all elements are considered with an underlying assumption of signal-dependent Poisson distributed noise.

The results demonstrate the potential of noise awareness, sparsity, and deep learning to enhance and restore measurements corrupted with signal-dependent Poisson distributed noise, such as those in PET imaging, thereby marking a notable step towards unravelling the PET Sampling Puzzle.