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A generalization of Benford's law and its application to images
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We present a generalization of Benford's law for the first sig-nificant digit. This generalization is based on keeping two terms of the Fourier expansion of the probability density func-tion of the data in the modular logarithmic domain. We prove that images in the Discrete Cosine Transform domain closely follow this generalization. We use this property to propose an application in image forensics, namely, detecting that a given image carries a hidden message.
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... This property has been largely exploited in multimedia forensics to detect image tampering. In fact, natural image DCT coefficients can be typically modeled by a Laplacianlike distribution [21], which naturally follow Benford's law, and for this reason, the mentioned rule can be successfully used in image forensic applications [22]. ...
... This processing chain is used by the JPEG coding standards and proves to be tailored to the spectral characteristics of images. Some of the past works highlight that, in the frequency domain, the quantized DCT coefficient statistics of natural images must follow Benford's law [22]. ...
The advent of Generative Adversarial Network (GAN) architectures has given anyone the ability of generating incredibly realistic synthetic imagery. The malicious diffusion of GAN-generated images may lead to serious social and political consequences (e.g., fake news spreading, opinion formation, etc.). It is therefore important to regulate the widespread distribution of synthetic imagery by developing solutions able to detect them. In this paper, we study the possibility of using Benford's law to discriminate GAN-generated images from natural photographs. Benford's law describes the distribution of the most significant digit for quantized Discrete Cosine Transform (DCT) coefficients. Extending and generalizing this property, we show that it is possible to extract a compact feature vector from an image. This feature vector can be fed to an extremely simple classifier for GAN-generated image detection purpose.
... Também conhecida como Lei do primeiro dígito, possui aplicaçaplicaç˜aplicação prática em diversasáreasdiversas´diversasáreas do conhecimento e ´ e utilizada para diferentes finalidades. Por exemplo, na computaçcomputaç˜computação especificamente, pode ser empregada na detecçdetecç˜detecção de anomalias na rede [Arshadi and Jahangir 2014] ou identificaçidentificaç˜identificação de mensagens escondidas em imagens [Pérez-González et al. 2007]. ...
The exploration of human behavior patterns is a central question for the development of new applications and technological solutions. However, few studies investigate how user habits can improve the performance of Information-Centric Network architectures. This work presents an analysis of the behavioral profiles of music users and how different profiles influence the performance of cache replacement policies. The results of an experimental study using ndnSIM with real traces from several users show that user habits are determining factors in choosing an optimized cache replacement policy. This research also reveals that the popularity distribution of the songs follows an approximation of Benford's Law, and it is possible to differentiate profiles for different users according to the behavior of Benford curve for these accessed songs.
We provide general expressions for the joint distributions of the k most significant b -ary digits and of the k leading continued fraction (CF) coefficients of outcomes of arbitrary continuous random variables. Our analysis highlights the connections between the two problems. In particular, we give the general convergence law of the distribution of the j th significant digit, which is the counterpart of the general convergence law of the distribution of the j th CF coefficient (Gauss-Kuz’min law). We also particularise our general results for Benford and Pareto random variables. The former particularisation allows us to show the central role played by Benford variables in the asymptotics of the general expressions, among several other results, including the analogue of Benford’s law for CFs. The particularisation for Pareto variables—which include Benford variables as a special case—is especially relevant in the context of pervasive scale-invariant phenomena, where Pareto variables occur much more frequently than Benford variables. This suggests that the Pareto expressions that we produce have wider applicability than their Benford counterparts in modelling most significant digits and leading CF coefficients of real data. Our results may find practical application in all areas where Benford’s law has been previously used.
Benford’s law states that the frequency of lower first significant digits(FSD) is higher than that of upper FSD in many naturally produced numbers. The law can be applied to many various fields, so it is important to know which common probability distributions obey Benford’s law. We revisit whether the Log-normal probability distribution obeys the law by using the method of Fourier analysis and numerical simulation. Moreover, we use simulation method to judge whether the Weibull distribution and the Inverse Gamma distribution are close to Benford’s law under some conditions. Our work give some reasons to support why Benford’s law is universal in real world.
In forensic analysis, such as forensic auditing, multimedia forensic, and financial fraud detection, the auditor needs to detect data tempering to find clue for possible fraud. First digit distribution such as Benford’s law is proved to be an efficient tool and is used by many auditing companies to preprocess the data before the actual auditing. However, when the range of the data is limited, the first digit distribution usually does not conform to Benford’s law. Using temperature data from a sensor network, we show that if the data can be modeled by a graph signal model, then after the graph Fourier transformation, the distribution of first digits conforms to a generalized Benford’s law. In addition, a graphic model based on historical data provides better fit to the Benford’s model than that based on geodesic distance. This model is evaluated for simulated data and temperature sensor network. This finding may help to build models for forensic analysis of accounting data and sensor network data for fraud detection.
We address the known problem of detecting a previous compression in JPEG images, focusing on the challenging case of high and very high quality factors (>= 90) as well as repeated compression with identical or nearly identical quality factors. We first revisit the approaches based on Benford--Fourier analysis in the DCT domain and block convergence analysis in the spatial domain. Both were originally conceived for specific scenarios. Leveraging decision tree theory, we design a combined approach complementing the discriminatory capabilities. We obtain a set of novel detectors targeted to high quality grayscale JPEG images.
Benford's law states that many data sets have a bias towards lower leading digits (about 30% are 1s). There are numerous applications, from designing efficient computers to detecting tax, voter and image fraud. It's important to know which common probability distributions are almost Benford. We show the Weibull distribution, for many values of its parameters, is close to Benford's law, quantifying the deviations. As the Weibull distribution arises in many problems, especially survival analysis, our results provide additional arguments for the prevalence of Benford behavior. The proof is by Poisson summation, a powerful technique to attack such problems.
Benford's law states that the leading digits of many data sets are not uniformly distributed from one through nine, but rather exhibit a profound bias. This bias is evident in everything from electricity bills and street addresses to stock prices, population numbers, mortality rates, and the lengths of rivers. Here, Steven Miller brings together many of the world's leading experts on Benford's law to demonstrate the many useful techniques that arise from the law, show how truly multidisciplinary it is, and encourage collaboration. Beginning with the general theory, the contributors explain the prevalence of the bias, highlighting explanations for when systems should and should not follow Benford's law and how quickly such behavior sets in. They go on to discuss important applications in disciplines ranging from accounting and economics to psychology and the natural sciences. The contributors describe how Benford's law has been successfully used to expose fraud in elections, medical tests, tax filings, and financial reports. Additionally, numerous problems, background materials, and technical details are available online to help instructors create courses around the book. Emphasizing common challenges and techniques across the disciplines, this accessible book shows how Benford's law can serve as a productive meeting ground for researchers and practitioners in diverse fields.
As the advent and growing popularity of image rendering software, photorealistic computer graphics are becoming more and more
perceptually indistinguishable from photographic images. If the faked images are abused, it may lead to potential social,
legal or private consequences. To this end, it is very necessary and also challenging to find effective methods to differentiate
between them. In this paper, a novel leading digit law, also called Benford’s law, based method to identify computer graphics
is proposed. More specifically, statistics of the most significant digits are extracted from image’s Discrete Cosine Transform
(DCT) coefficients and magnitudes of image’s gradient, and then the Support Vector Machine (SVM) based classifiers are built.
Results of experiments on the image datasets indicate that the proposed method is comparable to prior works. Besides, it possesses
low dimensional features and low computational complexity.
Key wordsLeading digit law–Benford’s law–Digital image forensic–Computer graphics
Techniques for information hiding have become increasingly more sophisticated and widespread. With high-resolution digital images as carriers, detecting hidden messages has become considerably more difficult. This paper describes an approach to detecting hidden messages in images that uses a wavelet-like decomposition to build higher-order statistical models of natural images. Support vector machines are then used to discriminate between untouched and adulterated images.
Near a stable fixed point at 0 or ∞, many real-valued dynamical systems follow Benford’s law: under iteration of a map T the proportion of values in {x,T(x),T 2 (x),⋯,T n (x)} with mantissa (base b) less than t tends to log b t for all t in [1,b) as n→∞, for all integer bases b>1. In particular, the orbits under most power, exponential, and rational functions (or any successive combination thereof), follow Benford’s law for almost all sufficiently large initial values. For linearly-dominated systems, convergence to Benford’s distribution occurs for every x, but for essentially nonlinear systems, exceptional sets may exist. Extensions to nonautonomous dynamical systems are given, and the results are applied to show that many differential equations such as x ˙=F(x), where F is C 2 with F(0)=0>F ' (0), also follow Benford’s law. Besides generalizing many well-known results for sequences such as (n!) or the Fibonacci numbers, these findings supplement recent observations in physical experiments and numerical simulations of dynamical systems.
Benford's law states that the first digits of a large body of naturally occurring numerical data in decimal form are not uniformly distributed but follow a logarithmic probability distribution. The values of radioactive decay half lives, which have been accumulated throughout the present century and vary over many orders of magnitude, afford an excellent opportunity to test the predictions of this law. To this end, we examine the frequency of occurrence of the first digits of both measured and calculated values of the half lives of 477 unhindered alpha decays and compare them with the predictions of Benford's law. Good agreement is found, and a similar distribution law for second digits is also considered.