## 2 Methods
### Effect of Incorrect Estimation
\begin{table}
\begin{tabular}{c c} \hline \hline
**Name** & **Equation** \\ \hline AC–DC Ratio [71] & \(\phi^{\text{E}}_{\text{R}}=\left\langle\frac{\sum_{i,m\in(0,0)}F_{x}(n,m)^{2}}{F _{x,0}(0,0)^{2}}\right\rangle\) \\ AC–DC Reduced Ratio [72] & \(\phi^{\text{E}}_{\text{RR}}=\left\langle\frac{\sum_{i,m\in\text{R}}F_{x}(n,m)^ {2}}{F_{x,0}(0,0)^{2}}\right\rangle\) \\ Absolute Gradient [73] & \(\phi^{\text{G}}_{\text{DMA}}=\left\langle\max_{D\in\{X,Y\}}\left|I^{D}_{D}(x,y )\right|\right\rangle\) \\ Squared Gradient [73] & \(\phi^{\text{G}}_{\text{DMS}}=\left\langle\max_{D\in\{X,Y\}}\left|I^{D}_{D}(x,y )\right|^{2}\right\rangle\) \\ Brenner Gradient [60; 73; 74; 75] & \(\phi^{\text{G}}_{\text{BMS}}=\left\langle\max_{D\in\{X,Y\}}\left|I^{B}_{D}(x,y )\right|^{2}\right\rangle\) \\ Gradient Energy [66; 77; 77] & \(\phi^{\text{G}}_{\text{DSS}}=\left\langle I^{D}_{X}(x,y)^{2}+I^{Y}_{Y}(x,y)^ {2}\right\rangle\) \\ Gaussian Energy [78; 79] & \(\phi^{\text{G}}_{\text{SSS}}=\left\langle I^{G}_{X}(x,y)^{2}+I^{G}_{Y}(x,y)\right\rangle\) \\ Tenengrad Mean [69; 74; 77; 80; 81] & \(\phi^{\text{G}}_{\text{TM}}=\left\langle\max_{D\in\{X,Y\}}I^{T}_{D}(x,y)^{2}\right\rangle\) \\ Tenengrad Variance [80] & \(\phi^{\text{G}}_{\text{TV}}=\operatorname{var}\left[\max_{D\in\{X,Y\}}I^{T}_{D}\right]\) \\ Laplacian Energy [63; 69; 70; 82] & \(\phi^{\text{E}}_{\text{E}}=\left\langle\left|L*I(x,y)\right|\right\rangle\) \\ Modified Laplacian [83] & \(\phi^{\text{L}}_{\text{M}}=\left\langle\left|L_{x}*I(x,y)\right|+\left|L_{y}*I (x,y)\right|\right\rangle\) \\ Diagonal Laplacian [84] & \(\phi^{\text{L}}_{\text{D}}=\left\langle\phi^{\text{L}}_{\text{M}}(x,y)+\sum \limits_{n\in\{1,2\}}\left|L_{dn}*I(x,y)\right|\right\rangle\) \\ Laplacian Variance [80] & \(\phi^{\text{L}}_{\text{V}}=\operatorname{var}\left[L*I\right]\) \\ Variance [65; 66; 67; 68; 73; 74; 75; 77; 81; 85] & \(\phi^{\text{S}}_{\text{VN}}=\operatorname{var}\left[I(x,y)\right]\) \\ Normalised Variance [72; 73; 74] & \(\phi^{\text{S}}_{\text{VN}}=\frac{1}{\left\langle\left|\left\langle xy \right\rangle\right\rangle}\operatorname{var}\left[I(x,y)\right]\) \\ Localised Variance [80] & \(\phi^{\text{S}}_{\text{VL}}=\operatorname{var}\left[L_{v}(x,y)\right]\) \\ Contrast [86] & \(\phi^{\text{S}}_{\text{C}}=\left\langle\sum\limits_{i\in\text{W}}\sum \limits_{j\in\text{W}}I(x,y)-I(x+i,y+j)\right\rangle\) \\ Mean Ratio [81] & \(\phi^{\text{S}}_{\text{R}}=\left\langle\max\left\{\frac{\mu(x,y)}{\mu(x,y)} \frac{I(x,y)}{\mu(x,y)}\right\}\right\rangle\) \\ Entropy [65; 73; 74; 75; 87] & \(\phi^{\text{S}}_{\text{HE}}=H(I_{H})\) \\ Central Moment [41; 88] & \(\phi^{\text{S}}_{\text{ACM}}=\sum\limits_{k}\left|k-\left\langle I\right| \right|P_{k}\) \\ Absolute Detail Sum [41; 87; 89] & \(\phi^{\text{W}}_{\text{AS}}=\left\langle\sum\limits_{n\in\{\text{LH,HLH}\}} \left|W^{1}_{n}(x,y)\right|\right\rangle\) \\ Detail Variance [87; 89] & \(\phi^{\text{W}}_{\text{V}}=\left\langle\sum\limits_{n\in\{\text{LH,HLH}\}} \operatorname{var}\left[W^{1}_{n}\right]\right\rangle\) \\ Detail–Coarse Ratio [41; 87] & \(\phi^{\text{W}}_{\text{R}}=\left\langle\frac{W^{1}_{\text{IM}}(x,y)^{2}+W^{1}_ {\text{IM}}(x,y)^{2}+W^{1}_{\text{IM}}(x,y)^{2}}{W^{1}_{\text{LL}}(x,y)^{2} +W^{1}_{\text{KL}}(x,y)^{2}+W^{1}_{\text{KL}}(x,y)^{2}}\right\rangle\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of the names, abbreviations and methods of action. \(\operatorname{var}\left[X\right]\) represents the variance of \(X\) across the imaging area, and \(\left\langle X\right\rangle\) represents the mean of \(X\) across the imaging area. energy is more homologous [71]. Different window sizes have also been used, either eight pixels square [41; 68; 72] or four pixels square [71]. Gradient-based FQMs, \(\phi^{\mathrm{G}}\), use approximations of the gradient or the first-derivative of the image to estimate the high-frequency content and hence the image quality. Differentiation is considered analogous to high-pass filtering, so these methods reward high-frequency content in the image.