# Metric for Estimating Congruity between Quantum Images +
## 4 Experimental Validation of the QIFM Metric
### QIFM (\(F\)) versus [PERSON]'s Similarity Score (\(S\))  
As seen in Table 1, [PERSON] et al. 's probability-based image similarity score, which was defined in [15; 24] and highlighted in Section 2, produces results that are at best exiguous because the similarity scores for the 23 pairs (reported in Table 1) vary within a narrow band in the range from 0.8 to 1.0. This narrow range does not discernibly convey similarity and less so when two or more pairs are assessed. Actually, as defined in [15; 23], this image similarity score produces a minimum value of 0.5 for any random pairing of a reference and test image. In other words, using the definition in [15; 23] and Equation (8) it is guaranteed that even the most dissimilar pairing of a reference and test image will have a 50% similarity score, which appears very lopsided. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**S/No.** & **Pair** & **Fidelity, \(F\)** & **PSNR, \(P\)** & **Similarity Score, \(S\)** & _SIM_1  & _rNMSE_2  \\ \hline
1 & A–A & 1.00 & \(\infty\) & 1.00 & 1.00 & 1.00 \\  
–K & 0.45 & 8.61 & 0.84 & 0.20 & 0.44 \\ \hline \end{tabular}  
* \({}^{1}\) _SIM_ are recalibrated values of similarity (\(S\)) as obtained using Equation (16) and \({}^{2}\) rNMSE is obtained using Equation (17).  
\end{table} 
Table 1: Summary of FPS Analysis. In contrast, our proposed QIFM fidelity metric has a smoother (i.e., wider-range) interval that is well spaced out in a manner that realistically conveys congruity between each of the reference and test images that were paired. In order to widen the interval and expand the range of coverage for [PERSON] et al. 's image similarity score (\(S\)), here, we recalibrated the original formulation of the similarity score, which was defined in Equation (8) (and the discussions in [23]), so that a wider range that is normalised between zero to one (i.e., an interval [0,1]) is obtained. We achieve this recalibration using Equation (16):  
\[SIM=1-10\times P_{S_{0}}\left|1\right\rangle \tag{16}\]  
where \(P_{S_{0}}\left|1\right\rangle\), which is the probability of obtaining a state \(\left|1\right\rangle\) after measurement is made, retains its form as defined in Equation (7). The coefficient of ten (10) is added to ensure that a normalised value is obtained for the _SIM_ value between two or more images without significantly altering the requirements needed to realise the original results using Equation (8). Consequently, the values in the sixth column of Table 1 are the revised similarity values and, in order to distinguish it from the original notation for similarity (as defined in Equation (8)), we call this new measure the _SIM_ score between the pair \(r\) and \(T\). Using this recalibrated formulation of similarity, a wider range of values that more graphically convey the notion of congruity is obtained. This adjustment in the similarity scores (\(S\)) better mimics the explicitness expected in descriptions of congruity between different pairings of images in Figure 8.  
One example that clearly stands out in showing the importance of this revised formulation for [PERSON]'s probability-based image similarity score is the House-Bridge pairing in pair L-O. A simple visual inspection reveals that these two images that are glaringly discrepant. However, [PERSON]'s similarity score (\(S\)) attributes a high similarity score of 0.918 for this pair but on the recalibrated similarity (_SIM_) scale this has a score of only 0.59. Similar adjustments can be seen in the other reference and test images pairs, notably that in the Tree-House pair (i.e., pair P-L), which reduces from 0.843 using [PERSON]'s image similarity score to only 0.21 using the revised _SIM_ scale. All these adjustments suggest that the recalibrated _SIM_ index is a better similarity scale than [PERSON]'s original similarity score (\(S\)).