# Image Quality Assessment for Gibbs Ringing Reduction
###### Abstract  
Gibbs ringing is an artefact that is inevitable in any imaging modality where the measurement is Fourier band-limited. It impacts the quality of the image by creating a ringing appearance around discontinuities. Many novel ways of suppressing the artefact have been proposed, including machine learning methods, but the quantitative comparisons of the results have frequently been lacking in rigour. In this paper, we examine image quality assessment metrics on three test images with different complexity. We determine six metrics which show promise for simultaneously assessing severity of Gibbs ringing and of other error such as blurring. We examined applying metrics to a region of interest around discontinuities in the image and use the metrics on the resulting region of interest. We demonstrate that the region of interest approach does not improve the performance of the metrics. Finally, we examine the effect of the error threshold parameter in two metrics. Our results will aid development of best practice in comparison of algorithms for the suppression of Gibbs ringing. image quality assessment; Gibbs ringing; loss functions +
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phantom images with introduced ringing and 200 MRI images [14]. [PERSON] et al. proposed a novel method based on deep CNNs and transfer learning. They trained their model first on natural images with introduced ringing and then retrained this network on medical images [15]. The classical approaches and learning approaches can be combined, such as in [16], which combined a deep learning model with the sub-voxel shifting method. Reconstruction results--regardless of whether a classical approach, a learning approach or a combination of both was used--are often evaluated subjectively, which is time-consuming and requires professional knowledge. With objective image quality assessment metrics, it is possible to automate the process of determining the quality of reconstruction results, giving us essential tools to determine which approach is the most suitable. Many metrics have been used for indicating the success or failure of the reconstruction. However, it is not clear what metrics would be able to give the most useful feedback. Machine learning algorithms are as good as their cost functions and the metrics can be used as cost functions for machine learning approaches. Hence, with the rise of machine learning techniques, the refinement of automatic assessment of image quality becomes ever more important. Metrics can be full reference, reduced reference or no reference. In this paper, we mostly focus on full reference metrics specifically for evaluating Gibbs ringing. In [17], we evaluated twelve metrics--such as mean squared error and entropy--that were used in previous Gibbs ringing related studies and rejected them all as unfit for purpose. We plotted these metrics for a signal with ringing that was processed with a Gaussian filter. All of the metrics were monotonic with the \(\sigma\) parameter of the filter, meaning that they could measure blur, but not identify an inflection point where blur was balanced against ringing reduction. We proposed a new full reference metric for evaluating Gibbs ringing suppression results called \(\ell_{e}^{0}\). That paper did not compare the inflection point of the \(\ell_{e}^{0}\) curve with human perception of the image quality. The Structural Similarity Index (SSIM) is known to be correlated with human perception. We now make the comparison between \(\ell_{e}^{0}\), SSIM, a corrected definition of entropy and other loss functions commonly used in regression problems. The Multi-scale SSIM index (MS-SSIM) is usually performed at multiple scales through a multi-step downsampling process. It is a more advanced form of SSIM and also investigated in this paper. The entropy analysed in [17] was based on its used in [18; 19]. The definition of entropy in those papers is incorrect, being applied to the pixel values rather than the image histogram.