# [PERSON]'s Law and Perceptual Features for Face Image Quality Assessment
## 3 Methodology
#### Features  
In this subsection, the proposed quality-aware Benford's law-inspired and perceptual features are introduced. The applied features and their descriptions are summarized in Table 1 and discussed in detail in the following paragraphs. As one can observe based on the information provided in Table 1, FDDs are extracted from the wavelet domain, discrete cosine transform coefficients, singular values, and shearlet domain. Further, FDDs are boosted with several popular perceptual features, i.e., colorfulness, the global contrast factor, the dark channel feature, entropy, and the mean of the phase congruency image, which are consistent with human quality judgment [30, 31]. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Feature Number** & **Description** & **Number of Features** \\ \hline f1–f9 & FDD of horizontal wavelet coefficients & 9 \\ f10–f18 & FDD of vertical wavelet coefficients & 9 \\ f19–f27 & FDD of diagonal wavelet coefficients & 9 \\ f28–f36 & FDD of DCT coefficients & 9 \\ f37–f45 & FDD of singular values & 9 \\ f46–f54 & FDD of absolute shearlet coefficients & 9 \\ f55–f59 & Perceptual features (colorfulness, global contrast factor, dark channel feature, entropy, mean of phase congruency) & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Quality-aware features applied in the proposed FIQA method. Figure 2: Process for investigating the effectiveness of [PERSON]’s law-inspired and perceptual features for face image quality assessment. Two-dimensional wavelets and filter banks are commonly applied in image processing [38]. Namely, wavelets are a general way to represent and analyze multiresolution images. In the case of one-dimensional discrete wavelet transform (DWT), a one-dimensional input signal \(S_{0}(n)\) is recursively decomposed into approximation and detail at the next lower resolution. The computation of 1D DWT [39] can be described as  
\[S_{i+1}(n)=\sum_{k}g(k)S_{i}(2n-k), \tag{2}\]  
\[W_{i+1}(n)=\sum_{k}h(k)S_{i}(2n-k), \tag{3}\]  
where \(S_{i}(n)\) and \(W_{i}(n)\) are the approximation and detail at level \(i\), respectively. In contrast to 1D DWT, the 2D DWT applies 2D filters, which can be separable or non-separable. Specifically, in the case of separable filters the 2D DWT decomposes an image into an approximation and three detailed images (horizontal, vertical, and diagonal features), as depicted in Figure 3. In this study, Daubechies mother wavelets were applied. Finally, FDDs in horizontal, vertical, and diagonal coefficients were utilized as quality-aware features. The two-dimensional discrete cosine transform (DCT) is a commonly used tool to compress images [40]. Namely, the two-dimensional DCT can interpreted as a representation for an image in terms of a weighted sum of basis images, since 2D DCT expresses an \(M\times N\) image \(I\) as  
\[B_{pq}=\alpha_{p}\alpha_{q}\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}l_{mn}\cos\frac{ \pi(2m+1)p}{2M}\cos\frac{\pi(2n+1)q}{2N},\;0\leq p\leq M-1,\;0\leq q\leq N-1 \tag{4}\]  
where  
\[\alpha_{p}= \begin{cases}\frac{1}{\sqrt{M}},\;p=0\\ \sqrt{\frac{2}{M}},\;1\leq p\leq M-1,\end{cases}  \tag{5}\]  
\[\alpha_{q}= \begin{cases}\frac{1}{\sqrt{N}},\;q=0\\ \sqrt{\frac{2}{N}},\;1\leq q\leq N-1,\end{cases}  \tag{6}\]  
and \(B_{pq}\) represent the 2D DCT coefficients. The FDDs of 2D DCT coefficients were used as quality-aware features. The singular value decomposition (SVD) is a factorization method that decomposes a matrix into three matrices. Formally, it can be written for every real-valued matrix \(A\in\mathbb{R}^{N\times M}\)[41]:  
\[A=U\Sigma V^{T}, \tag{7}\]  
where \(U\in\mathbb{R}^{N\times N}\) and \(V\in\mathbb{R}^{M\times M}\) are unitary matrices with orthonormal columns and \(\Sigma\in\mathbb{R}^{N\times M}\) is a diagonal matrix. Figure 3: Illustration of two-dimensional discrete wavelet transform. The shearlets proposed in 2006 [42] are a mathematical framework and a type of multiscale analysis tool used in signal and image processing.