# Metric for Estimating Congruity between Quantum Images +
## 4 Experimental Validation of the QIFM Metric
### FPS Analysis of Congruity between Quantum Images  
In the introductory sections of this study we promised readers an analysis that compares congruity between two or more FRQI quantum images in terms of three metrics--our proposed QIFM fidelity metric (\(F\)), the well-established classical PSNR (\(P\)) image quality measure, and [PERSON]'s probability-based image similarity score (\(S\)) [15]. The remainder of this section is devoted to this \(F\), \(P\) and \(S\) or simply FPS analysis.  
In presenting the FPS analysis, we adopt simulation-based techniques similar to those in [11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23] in order to simulate a dataset comprising of 16 images, each of size 512 \(\times\) 512 (presented in Figure 8 below), that are widely used in image processing literature. Similar simulation-based approaches were used to simulate the circuitry to execute the \(BCO\) and \(BO\) sub-circuits that coalesce to execute the \(FCO\) operation. Using this layout, the efficacy of our proposed QIFM metric as an acceptable measure to assess fidelity between two or more FRQI quantum images can be validated as presented and discussed in the sequel.  
Using the labels assigned to the images in our database (Figure 8), a notation \(r\)-\(T\) is used to indicate a \(j\)th pair consisting of a reference image, \(r\), and its test image, \(T\). It is assumed that a strip (defined in Section 2) consisting of the 16 images in our database has been captured, prepared and initialised as required in FRQI QIP [17; 19]. For brevity, however, the images in our database are presented neither as a vertically nor as a horizontally-oriented strip [9]. Instead, it is presented as a stack of images labelled from A to P (in Figure 8).  
Figure 8: Dataset of images paired for the FPS analysis. (**A**) Lena; (**B**) Inverted Lena; (**C**) Blonde Lady; (**D**) Peppers; (**E**) Scarfed Lady; (**F**) Baboon; (**G**) Brunette Lady; (**H**) Cameraman; (**I**) Man; (**J**) Couple; (**K**) Aeroplane; (**L**) House; (**M**) Pentagon; (**N**) Fingerprint; (**O**) Bridge; (**P**) Trees.  
Next, the 16 images that make up the strip are randomly paired into twenty-three (23) pairs each consisting of a reference and test image. The \"likeness\" between these pairs was evaluated in terms of the fidelity, \(F_{r,T}\); the PSNR, \(P_{r,T}\); and the Similarity, \(S_{r,T}\), between the pairing of each reference image (\(r\)) and a test image (\(T\)). These results are summarised in Table 1 while the detailed FPS analysis is presented in the sequel.  
The peak-signal-to-noise-ratio--PSNR (\(P\)) is widely adopted as an image reconstruction and quality measure in both digital image processing and more recently in QIP. It is most easily defined via the mean squared error (\(MSE\)), which for two \(m\times n\) monochrome images the original or input image, \(I\) and its watermarked version \(K\) in Equation (15):  
\[P=20\text{log}_{10}\left[\frac{MAX_{i}}{\sqrt{MSE}}\right] \tag{15}\]  
Here, \(MAX_{i}\) is the maximum possible pixel value of the image [13; 17; 25; 28].