# Image Quality Assessment for Gibbs Ringing Reduction
## 5 The Effect of Varying the Threshold, \(\epsilon\), on \(\ell_{e}^{0}\) and Huber Loss  
In this section, we focus on the significance of the parameter \(\epsilon\). For two of the metrics discussed in Section 3, namely \(\ell_{e}^{0}\) and Huber loss, \(\epsilon\) is the boundary between small errors--which can be either completely ignored (\(\ell_{e}^{0}\)) or diminished by squaring (Huber loss)--and larger errors which contribute to the measurement of error. As ringing is an oscillating artefact with zero mean, we have speculated that any error metric that weights larger errors more highly than lower ones may also be able to distinguish ringing from other errors such as blur. Our investigation in Section 3.1 demonstrates that this is indeed the case for those two metrics. In our previous work, when we proposed \(\ell_{e}^{0}\) as a metric in Gibbs suppression, we chose to use the median error of the Fourier reconstruction as a somewhat arbitrary but easily obtained value for \(\epsilon\), which seemed to work satisfactorily. We now wish to set that parameter based on something more evidence-based. Ideally, that evidence base would have some relationship to human perception. We note the concept of \"just noticeable difference\" (JND) [28]. The JND is \"the minimum amount by which a stimulus intensity must be changed relative to a background intensity in order to produce a noticeable variation in sensory experience\" [29]. Such perceptual thresholds depend on ambient light, the display screen and the idiosyncrasies of the vision of the observer, frustrating the desire for a single definitive answer. With that caveat, we now wish to address the question: how can we choose \(\epsilon\)? Our standard problem to evaluate a metric, as depicted in, e.g., Figure 3, is truncate a test signal or image in the Fourier domain, use Gaussian filters of different variance, \(\sigma\), to suppress the ringing in the Fourier reconstructed image and plot the metric as a function of  
Figure 7: The behaviours of PSR applied to the whole image (blue) and to the RoI (red). Applied to the RoI, the metric shows a local maximum and a local minimum. \(\sigma\). The \(\sigma\) for which the peak (or minimum) value of the metric is observed is then taken to characterize a 'best' filter according to that metric. We have observed that both \(\epsilon\)-dependent metrics exhibit a peak for many possible values of \(\epsilon\). However, the location of this peak varies with \(\epsilon\). We can obtain a continuum of test images by treating the fraction of the Fourier coefficients that are set to zero as a variable. This 'truncation percentage' was fixed for Figure 3 at 80% (i.e., only the central 20% of coefficients are retained). In Figure 8, we plot the 'best' \(\sigma\) according to our two \(\epsilon\)-dependent metrics as a function of truncation percentage for a variety of values of \(\epsilon\). We also plot the 'best' \(\sigma\) according to SSIM as a benchmark against which to compare the other metrics. This, it must be acknowledged, is a somewhat arbitrary benchmark, but (a) it has been shown to be a useful metric for this problem and (b) it has been shown to correlate with human perception. In the absence of a large user evaluation study, we are forced to rely on a proxy metric of this kind. We do not claim that the results are therefore a definitive determination of the best \(\epsilon\) but rather they are indicative of the general trends associated with varying that parameter. We also note that our test images are normalized to have maximum values of 1. The results in Figure 8 are for the Shepp-Logan phantom only in order to simplify the presentation of results, but we have also performed these simulations for the other two test images described in Section 2 with similar results. First let us consider \(\ell_{\epsilon}^{0}\), in the upper plot of Figure 8. We observe near-linear plots for small \(\epsilon\), small here meaning \(\leq 10^{-2}\).