# Image Quality Assessment for Gibbs Ringing Reduction
## 6 Conclusions  
Gibbs ringing is an imaging problem that could be suppressed through both traditional and learning approaches. These methods, including the combined methods, all intend to provide images with less ringing. In the literature, the merits of Gibbs suppression algorithms have been assessed with a mixture of qualitative and a variety of quantitative metrics. In a recent paper, the authors surveyed the quantitative metrics used in previous reports of novel Gibbs suppression algorithms and found them wanting [17]. However, it is only with good, consistent and convenient quantitative metrics can we provide critical comparisons between traditional and learning methods. We proposed the metric \(\ell_{e}^{0}\) for this purpose in our previous paper [17]. In this paper, we have extended our analysis to consider SSIM, MS-SSIM, Huber loss, MAE and a corrected definition of entropy. We have shown that those metrics join \(\ell_{e}^{0}\) in passing our test: given a variety of Gaussian filters, they can identify a 'best' one which trades off ringing  
Figure 9: **(top)** Mean squared error as a function of \(\log_{10}\epsilon\) between SSIM and \(\ell_{e}^{0}\) (blue), and between SSIM and Huber loss (red). Best values of \(\epsilon\) are identified by the minima. (**bottom**) Best \(\sigma\) for the Gaussian filter as a function of truncation percentage, as recommended by the three metrics (using \(\epsilon\) as identified above). suppression against blur. This is a necessary requirement of a good quantitative metric, though we do not claim it is a sufficient one. We had good reasons to investigate a region of interest limited to parts of the image close to edges as a means of enhancing or rehabilitating the 17 metrics we have investigated. We proposed an algorithm for finding such a region, but have found the approach had limited effect on the metrics. Finally, two of the metrics depended on an error threshold. We have investigated the effects of varying that threshold. Our results show that Huber loss is optimized by minimizing the threshold; in the limit, it becomes the \(\ell^{1}\) norm. For this minimal threshold, it provides good agreement with the SSIM. We have also seen that \(\ell_{e}^{0}\) has good agreement with SSIM for a suitable choice of \(\epsilon\). For smaller values of \(\epsilon\), it tolerates less error and so recommends more aggressive filters. It does so quite consistently: a small increase in Fourier domain truncation results in a small increase in the recommended filtering. For larger values of \(\epsilon\), this consistency is lost and the metric may not be as reliable as we might like. Thus, larger values of the threshold are recommended against. The work in this paper will help provide a foundation of evidence-led best practice in the comparison of Gibbs suppression algorithms, both traditional and learning approaches included. Conceptualization, [PERSON] and [PERSON]; methodology, [PERSON]; software, [PERSON]; validation, [PERSON]; formal analysis, [PERSON] and [PERSON]; investigation, [PERSON]; resources, [PERSON]; data curation, [PERSON]; writing--original draft preparation, [PERSON] and [PERSON]; writing--review and editing, [PERSON] and [PERSON]; visualization, [PERSON] and [PERSON]; supervision, [PERSON]; project administration, [PERSON]; funding acquisition, [PERSON] and [PERSON]. All authors have read and agreed to the published version of the manuscript. This research was funded by GliMR EU COST Action grant number CA18206. [PERSON] was funded by Irish Research Council Government of Ireland Postgraduate Scholarship. Not applicable. The authors would like to thank University College Dublin for support. The authors would like to thank our colleague [PERSON] for constructive conversations in relation to this work. The authors declare no conflict of interest. The following abbreviations are used in this manuscript: