# Metric for Estimating Congruity between Quantum Images +
## 4 Experimental Validation of the QIFM Metric
### PSNR versus Quantum Image Quality Metrics (QIFM (F), Similarity Score (S) and SIM Values)
Now, we present a statistical analysis for the degree of association between the _rNMSE_ (\(r\)) on one side and the trio of the QIFM (\(F\)), [PERSON] et al. 's probability-based score (\(S\)) and its recalibrated version (_SIM_) values on the other. For this task, we shall employ the widely used statistical correlation tests--i.e., the Spearman, Pearson and Kendal rank correlation coefficients--or simply the SPK correlation test. Details of these tests can be found in standard statistics texts and other available literature such as [28]. Table 2 presents results of our SPK correlation test to ascertain the degree of association between the rNMSE digital image quality measure and the three quantum image quality metrics (i.e., \(F\), \(S\) and _SIM_). In interpreting the results in Table 2, we see that the proposed QIFM metric (\(F\)) has the best correlation with the _rNMSE_ than any of the other quantum image metrics (i.e., [PERSON] et al. 's probability-based score (\(S\)) and the _SIM_ values) for each of the correlation indices that we employed \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Correlation Coefficient**} \\ \cline{2-4}
**Image Metric** & **Spearman** & **Pearson** & **Kendal** \\ \hline _rNMSE_ vs. QIFM & 0.5939 & 0.7268 & 0.4172 \\ _rNMSE_ vs. _SIM_ & 0.4866 & 0.7027 & 0.3747 \\ _rNMSE_ vs. Similarity & 0.4819 & 0.7106 & 0.3701 \\ QIFM vs. Similarity & 0.5596 & 0.7368 & 0.4426 \\ QIFM vs. _SIM_ & 0.3794 & 0.6467 & 0.2705 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: SPK correlation test between digital and quantum image metrics (for image pairs in Table 1). in the SPK test. In other words, using the classical versions of the quantum test and cover images in Figure 8 (as paired in Table 1) as benchmark, the assessment of congruity based on the QIFM fidelity metric (\(F\)) offers the least correlation in terms of deviations (aka congruity) between the test and cover images in each pair than is obtainable via both [PERSON]'s probability score (\(S\)) and its recalibrated version (\(SIM\)). Table 2 allows us to make the deduction that in spite of the quantum mechanical requirements for its execution, the QIFM metric yields the best correlation with expectations from classical (digital) image quality measures. In concluding our assessment using the PSNR, we revisit the lack of explicit lexicality in descriptions of congruity (or likeness) between images based on the PSNR as measure of image quality. To demonstrate this, we considered descriptions of congruity between the watermarked versions of the Lena and Blonde Lady images in Figure 9D,E and their pristine versions in Figure 9B,C (which were originally presented in Figure 8A,C). Using the WaQI quantum image watermarking protocol [13; 21; 27], we paired each of the two images with the PSAU logo in Figure 9A as watermark signal in order to obtain the watermarked versions of Lena and Blonde Lady images (i.e., in Figure 9D,E). The PSNR values and fidelity scores conveying the notion of congruity (or likeness) between the original and watermarked versions of the two images are also presented in Figure 9.  
As seen from the results, at 97% fidelity score the QIFM fidelity metric offers better explicitness in describing congruity (or likeness) between the original and watermarked versions of the images. From this description, it is trivial to deduce that the watermarked and original versions of the two images are somewhat the same, or that they have high fidelity. On the contrary, the PSNR values of 31.59 and 32.33 dB (for the Lena and Blonde Lady images respectively) present ambiguity in terms of descriptions of congruity between the original and watermarked images. Therefore, the quantum image metrics offer better lexical explicitness in their descriptions of congruity between two or more images than the PSNR in its standard form