# [PERSON]'s Law and Perceptual Features for Face Image Quality Assessment
###### Abstract  
The rapid growth in multimedia, storage systems, and digital computers has resulted in huge repositories of multimedia content and large image datasets in recent years. For instance, biometric databases, which can be used to identify individuals based on fingerprints, facial features, or iris patterns, have gained a lot of attention both from academia and industry. Specifically, face image quality assessment (FIQA) has become a very important part of face recognition systems, since the performance of such systems strongly depends on the quality of input data, such as blur, focus, compression, pose, or illumination. The main contribution of this paper is an analysis of [PERSON]'s law-inspired first digit distribution and perceptual features for FIQA. To be more specific, I investigate the first digit distributions in different domains, such as wavelet or singular values, as quality-aware features for FIQA. My analysis revealed that first digit distributions with perceptual features are able to reach a high performance in the task of FIQA.  
Benford's law; perceptual features; face image quality +
Footnote †: journal: Medical Image Quality Assessment  
20232  
Recent advancements in deep learning have also impacted the field of FIQA. Convolutional neural networks (CNNs) have been employed to automatically learn features and classify face images based on their quality [3]. Overall, the goal of face image quality assessment is to provide reliable measures or algorithms that can assess the visual quality of face images objectively, enabling an improved performance and reliability in applications such as biometrics [4], surveillance [5], and facial recognition systems [6, 7]. Face image quality assessment has also gained importance as a critical component of smart home and smart city applications that rely on facial recognition technology. Namely, it enhances security, access control, public safety, user experience, data accuracy, ethical considerations, and resource efficiency [8, 9].  
The goal of this study is to give a detailed analysis about the performance of [PERSON]'s law-inspired and perceptual features in FIQA. The contributions of this study are as follows:  
1. First, I investigate the first digit distributions (FDDs) of different image domains for FIQA.
2. Second, I empirically corroborate that the FDD of an image domain is a rather mediocre predictor for face image quality. However, taking the fusion of different domains' FDDs results in a strong predictor whose performance can be further increased by considering several simple perceptual features, such as colorfulness, the global contrast factor, the dark channel feature, entropy, and phase congruency.  
[PERSON]'s law, also known in the literature as the law of anomalous numbers or first digit law, refers to an empirical observation that describes the frequency of leading digits in many numerical datasets, such as population numbers [10], geographical data [11], or physical constants [12]. Namely, the leading digits are not uniformly distributed but follow a pattern, which can be given as  
\[P(d)=\log_{10}(d+1)-\log_{10}d=\log_{10}\left(\frac{d+1}{d}\right)=\log_{10} \left(1+\frac{1}{d}\right), \tag{1}\]  
where \(d\in\{1,2,\ldots,9\}\) denotes the leading digit and \(P(d)\) is its relative frequency in a numerical dataset. The distribution defined by Equation (1) is depicted in Figure 1.  
The remaining sections of this research study are organized as follows. Section 2 provides a brief overview about related and previous works. Next, Section 3 includes a detailed explanation of my methodology investigating [PERSON]'s law-inspired and perceptual features for FIQA. Section 4 contains a brief description of the used benchmark database,  
Figure 1: Relative frequency of leading digits based on the prediction of [PERSON]’s law.  
the performance metrics, my experimental results, and my analysis. Section 5 includes the overall conclusion with future work.