On the Epistemology of Gibbs Ringing Reduction Algorithm Performance

Gibbs ringing is an artifact that appears around sharp transitions in a signal reconstructed from Fourier data. The reconstruction does not converge correctly but instead oscillates. For example, it appears at tissue boundaries in magnetic resonance imaging where it has the ability to mislead clinicians or image processing algorithms. There are many ringing reduction algorithms proposed in the literature. How do we make a systematic and quantitative comparison of ringing suppression algorithms? This includes our testing procedure, test signals, metrics, baselines, ground truths, datasets, and other factors. In this thesis, we analyse the commonly used baselines, metrics, test signals, etc. in respect of specific ringing suppression algorithms. For example, metrics are investigated in terms of parameter selection for filtered Fourier reconstruction, which allows us to interrogate the quality of evidence for algorithms that are compared with sometimes poorly chosen filters. Test signals are investigated in terms of the quality of evidence for Gegenbauer reconstruction. The discussion on metrics led us to consider questions of signal sparsity, which then led us to cast reconstruction of a discontinuous signal from Fourier data as a compressive sensing problem, a novel approach. Finally, metrics also led us to a discussion of cost functions in machine learning approaches to ringing suppression, along with the nature of the data used to train those algorithms. We learned that many metrics used in comparisons of ringing suppression algorithms are not fit for purpose. Many papers make comparisons of some other algorithm with substandard filters. Many papers provide evidence based on unsuitable test signals. Machine learning algorithms are often trained (1) using unsatisfactory cost functions (2) on one homogeneous dataset (3) with ringing introduced in an under-specified or limited fashion. Ultimately, this thesis is a story of poor epistemology in an image processing problem. Our results will inform the quantitative comparison of ringing suppression algorithms. They will also be important in aspects of machine learning for this problem.