# [PERSON]'s Law and Perceptual Features for Face Image Quality Assessment
## 3 Methodology
#### Features
They were introduced as an extension of wavelets to overcome some of their limitations in capturing and representing complex geometrical structures, such as edges, corners, and textures, at different scales and orientations. Shearlets employ a system of shear operations, which involve affine transformations that stretch and rotate a basic analyzing function, called a mother shearlet, to capture the local geometric features of a signal or an image. These shear operations allow shearlets to adapt to the local geometry of the data, enabling them to efficiently represent highly anisotropic features. To define a continuous shearlet system, you need a parabolic scale matrix  
\[A_{c}= \begin{pmatrix}c&0\\ 0&\sqrt{c}\end{pmatrix} ,\ c\in\mathbb{R}^{+} \tag{8}\]  
and a shear matrix  
\[S_{s}= \begin{pmatrix}1&s\\ 0&1\end{pmatrix} ,\ s\in\mathbb{R}. \tag{9}\]  
A continuous shearlet system can be given for \(\Psi\in L^{2}(\mathbb{R}^{2})\) as  
\[SH_{cont}(\Psi)=\{\Psi_{a,s,t}=a^{\frac{3}{4}}\Psi(S_{s}A_{a}(\cdot-t))|a>0,s \in\mathbb{R},t\in\mathbb{R}^{2}\}, \tag{10}\]  
and the following mapping is the corresponding continuous shearlet transform  
\[f\to\mathcal{SH}_{\Psi}f(a,s,t)=\langle f,\Psi_{a,s,t}\rangle,\ f\in L^{2}( \mathbb{R}^{2}),\ (a,s,t)\in\mathbb{R}_{>0}\times\mathbb{R}\times\mathbb{R}^{2}. \tag{11}\]  
In the literature, various ways can be found for discretezing a shearlet system [43]. In this study, one of the most common ones was utilized and can be given as  
\[\{(2^{j},k,A_{2^{j}}^{-1}S_{k}^{-1}m)|j\in\mathbb{Z},k\in\mathbb{Z},m\in \mathbb{Z}^{2}\}\subseteq\mathbb{R}_{>0}\times\mathbb{R}\times\mathbb{R}^{2}. \tag{12}\]  
Subsequently, the discrete shearlet system generated by \(\Psi\) can be given as  
\[SH(\Psi)=\{\Psi_{j,k,m}=2^{3j/4}\Psi(S_{k}A_{2^{j}}\cdot-m)|j\in\mathbb{Z},k \in\mathbb{Z},m\in\mathbb{Z}^{2}\} \tag{13}\]  
and the following mapping is the corresponding discrete shearlet transform  
\[f\to\mathcal{SH}_{\Psi}f(j,k,m)=\langle f,\Psi_{j,k,m}\rangle,\ f\in L^{2}( \mathbb{R}^{2}),\ (j,k,m)\in\mathbb{Z}\times\mathbb{Z}\times\mathbb{Z}^{2}. \tag{14}\]  
The FDD of the absolute shearlet coefficients was used as a quality-aware feature. Colorfulness (CF) refers to the perceived intensity or vibrancy of colors in an image [44]. In other words, it measures how rich and diverse the colors appear in a digital image. A colorful image generally has a wide range of distinct, vivid, and saturated colors. In general, humans tend to perceive more colorful images as visually attractive and engaging [45,46]. For measuring the colorfulness of an image, the model of [PERSON] and [PERSON] was adopted in this paper, which can be expressed as  
\[CF=\sqrt{\sigma_{rg}^{2}+\sigma_{yb}^{2}}+\frac{3}{10}\sqrt{\mu_{rg}^{2}+\mu_ {yb}^{2}}, \tag{15}\]  
where \(\mu\) and \(\sigma\) denote the mean and standard deviation of matrices in the subscripts. Further, \(rg=R-G\) and \(yb=\frac{1}{2}(R+G)-B\) where \(R\), \(G\), and \(B\) denote the red, green, and blue color channels, respectively. The contrast of an image refers to the difference in brightness, color, or intensity between different elements or regions within the image. It measures the degree of variation and separation between light and dark areas or between colors in the image. In other words, contrast quantifies the distinguishing ability and visual separation of different image features [47]. In this paper, [PERSON] et al. 's [48] model--called the global contrast factor (GCF)--was used to quantify the contrast of face images. Formally, it can be expressed as  
\[GCF=\sum_{i=1}^{9}w_{i}\cdot C_{i}, \tag{16}\]  
where  
\[w_{i}=(-0.406385\cdot\frac{i}{9}+0.334573)\cdot\frac{i}{9}+0.0877526,\;i\in\{1,2,\ldots,9\}, \tag{17}\]  
and  
\[C_{i}=\frac{1}{w\cdot h}\sum_{i=1}^{w\cdot h}l_{C_{i}}. \tag{18}\]  
Further,  
\[l_{C_{i}}=\frac{|L_{i}-L_{i-1}|+|L_{i}-L_{i+1}|+|L_{i}-L_{i-w}|+|L_{i}+L_{i+w}|}{ 4} \tag{19}\]  
where \(L\) denotes the pixel values after a gamma correction \((\gamma=2.2)\).