On the robustness of JPEG post-compression to resampling factor estimation

Highlights

• We first find the coupling effect between post-JPEG compression and resampling in form of frequency mixing, named as octant symmetric aliasing peaks(OSAP).

• We show that OSAP is non-negligible to the factor estimation of post-JPEG resampling.

• The relationship between the location of OSAP and resampling factor and is proved by an approximate model, which is consistent with statistical results on test image sets.

• We proposed a method based on the resampling feature enhanced by OSAP to estimate the factor of post-JPEG resampling.

• Experimental results demonstrate that the method is effective and efficient.

Abstract

The research on image resampling detection is urgently called for digital image forensics recently. In this paper, the forensics of resampling operation with JPEG compression as post-process is addressed, namely post-JPEG resampling. We first discover the coupling effect between resampling and post-JPEG compression, which is shown as features of frequency mixing in spectrum and named as octant symmetric aliasing peaks(OSAP). The frequency location of OSAP depends on the resampling factor. We verify that the feature of OSAP is prone to be more prominent than resampling when quality factor (QF) of post-JPEG is low enough. By addressing the frequency response of post-JPEG compression, an approximate model of OSAP is derived. The prior knowledge of OSAP can help to enhance the resampling feature, based on which a new method of resampling factor estimation is proposed. In the proposed method, the location of OSAP is estimated from local maximum of spectrum with strong symmetry, and a candidate set of resampling factor is derived from OSAP. The optimal estimation among candidates is chosen in a heuristic way, which weakens the feature of OSAP appropriately. Experiments implemented on natural image set show the effectiveness of proposed method.

Introduction

Digital image has become the most popular kind of data in our daily life. Yet the integrity and authenticity of digital image are heavily affected by the various editing softwares distributed on the internet. The forged images can bring harmful impacts on our daily life and society in some situations, e.g., journalism and criminal investigation. There is a great demand for automatic detector of forged image, where the field of digital image forensics concentrate on [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. Among them, resampling forensics is one of the most important topic, because malicious tampering is usually performed by geometrically adapting the new image elements to the original scene [13], [14], [15], [16]. Such adaption may require the employment of spatial transformations(e.g., scaling, rotation and shearing), which is actually achieved by numerical interpolation and resampling. Hence the detection of resampling can be a clue to reveal malicious tampering.

In recent years, resampling forensics has become a hot topic, with many papers published. Most of them are based on spectrum analysis of periodic resampling feature. Popescu and Farid [1] gave the first blind detector of resampling, along with strict mathematical proof. In their method, the inspected image is processed by a complicate EM estimator to get a p-map, and peaks in the Fourier spectrum of p-map imply a periodic trace left by resampling. In Gallagher’s research [5] the EM estimator was replaced by two-ordered difference, and peaks refer to resampling also exist in the variance spectrum of differenced signal. Kirchner [7] proved the equivalence between [1] and [5], and an optimal filter was also proposed as the substitute of two-order difference. Except the detection of resampling feature, these methods could also estimate the parameter of resampling, but were limited to resize resampling. Mahdian and Saic [8] first considered the parameter estimation of rotation resampling, using radon transformation of the two-dimensional differenced signal. The transformed one-dimensional signal with radon angle equal to rotation angle will show the strongest periodic property. Wei et al. [10] took the advantage of phase differences in each row of rotation image to differentiate rotation from resizing, and the rotation angle could be directly calculated from the one-dimensional spectrum. In the aspect of parameter estimation, the methods based on the location of spectrum peaks inevitably couldn’t distinguish the aliasing causing by undersampling. The estimation of resampling parameter usually has more than one possible values. To deal with this problem, some methods were not based on Fourier spectrum but other mathematical approach, for example, SVD [17], [18] and cyclic correlation spectrum [12], [19]. Some other methods [20], [21] considered not only the location of peaks but also the energy distribution of hole spectrum, and SVM is used to distinguish the aliasing of parameter estimation. Furthermore, the utilization of machine learning could help to estimate the kind of kernel [22] and enhance the robustness of detector [23], [24].

JPEG is the most popular image format in the world, and tampered image is often stored in JPEG format [25], [26]. Besides, common digital camera has two kinds of output formats including JPEG. So it is reasonable to consider the operation chain consist of resampling operation and JPEG compression, which have three kinds of combination:

• post-JPEG resampling, where the untampered image is in lossless-compression format and the tampered one is in JPEG format.

• pre-JPEG resampling, where the untampered image is in JPEG format and the tampered one is in lossless-compression format.

• double-JPEG resampling, where both the untampered and tampered image are in JPEG format.

Gallagher [5] first discussed the case of post-JEPG resampling. He proposed that when no resampling is involved, the noise introduced by JPEG have periodic variance, which consists with the block size of lossy compression. The spectrum of JPEG image would show spectral peaks in fixed frequency ${\omega }_{\text{jp}}^{\left(n\right)}=\frac{n}{8}$. And the amplitude of peaks in ${\omega }_{\text{jp}}^{\left(n\right)}$ depends on the quality factor(QF) of JPEG and the power spectrum of the uncompressed image, which have on closed-form expression. For post-JPEG resampling, the spectrum is simply considered as the linear superposition of spectral lines corresponding to JPEG and resampling, respectively. Kirchner and Gloe [27] proposed that the post-JPEG would weaken the feature of resampling. They also study the case of pre-JPEG resampling, which shows that the spectral peaks refer to pre-JPEG would be shifted by the following resampling operation. By detecting the shift JPEG peaks, the resampling factor can be estimated. This theory was inherited by Bianchi and Piva [28] in the study of double-JPEG resampling, which ignored the peak of resampling and considered the spectrum as the linear superposition of pre-JPEG and post-JPEG. Nevertheless, there is a problem that the peak of pre-JPEG may couldn’t be differentiated from resampling only by the amplitude. This problem is well solved by Liu et al. [2] with the help of rank statistics. There are two difficulties of post-JPEG resampling. Firstly, the compression process will weaken the trace of resampling and even disappear. Secondly, some suspicious peaks will appear in the spectrum of post-JPEG image, which could not be distinguished from the resampling peaks.

In this paper, we concentrate on forensics of post-JPEG resampling. We verify that the confusion between these suspicious peaks and resampling peaks is the main difficulty for resample factor estimation. For fixed parameters of post-JPEG resampling, the location of suspicious peaks for different images is the same, and any two suspicious peaks are symmetric about the JPEG peaks. Because the JPEG peaks locate in frequences of 8/k, these suspicious peaks are named as octant symmetric aliasing peaks(OSAP). The symmetric property of OSAP implies a coupling effect between JPEG compression and resampling operation, shown as frequency mixing in the spectrum. Using some mathematical approximation, we derive a model of OSAP in the general framework of stochastic signal processing. This model help us to understand the nonlinear behavior of the amplitude of OSAP when resampling factor changes. Finally, the knowledge of OSAP is introduced into spectrum based method to boost the resampling feature. The confusion between OSAP and resampling peaks is solved in a heuristic way, which changes the phase for each row of inspect image in proper degree.

The main contributions of this work include:

• We first find the coupling effect between post-JPEG compression and resampling in form of frequency mixing, named as octant symmetric aliasing peaks(OSAP). We show that OSAP is nonnegligible to the factor estimation of post-JPEG resampling. The relationship between the location of OSAP and resampling factor and is proved by a approximate model, which is consistent with statistical results on test image sets.

• We proposed a method based on the resampling feature enhanced by OSAP to estimate the factor of post-JPEG resampling. The proposed method performs better than all state-of-the-art methods.

The remainder of this paper is organized as follows. Section 2 reviews the general model of resampling forensics, then in Section 3 we give the exact formula of OSAP and tries to prove it with appropriate mathematical approximation. With the help of OSAP, an automatic factor estimation method is proposed in Section 4. Section 5 shows the experiment results of the proposed method and discusses some implementation issues. Eventually, the concluding remarks and future works are given in Section 6.