Elsevier

Pattern Recognition Letters

Volume 36, 15 January 2014, Pages 89-99
Pattern Recognition Letters

Estimation of linear transformation by analyzing the periodicity of interpolation

https://doi.org/10.1016/j.patrec.2013.09.028Get rights and content

Highlights

  • The proposed method estimates the linear transformation of an investigated image.

  • The second derivatives for row and column directions exploit the periodicities.

  • Both the magnitude and phase information of the derived signals are analyzed.

Abstract

Linear transformation, such as rotation, scaling, or any combinations of geometric attacks, is among the most common forms of image manipulation. This letter proposes a forensic technique that estimates the linear transformation of an investigated image. We exploited the periodic properties of interpolation by the second-derivative of the transformed image in both the row and column directions. Both the magnitude and phase information of the derived signals were analyzed to estimate the transformation matrix accurately. Empirical evidence from a large database of manipulated images indicates the superior performance of the proposed method.

Introduction

With the development of highly sophisticated forms of information technology, digital images are commonly used in our life. The benefits of digital images, such as the ease with which they can be edited, shared, and stored, have led people to use digital images in place of analog images. At the same time, the very nature of digital images, which can be easily manipulated, brings into question many of the positive aspects associated with these types of images.

Digital image forensics, which detects the modification of digital images, has been actively researched over the last decade (Sencar and Memon, 2013). Forensics techniques are based on the assumption that any post-processing of an image leaves inherent traces in the resulting image. Therefore, a suspicious image can be passively investigated without any prior knowledge so that the corresponding statistical properties of the manipulations are uncovered in the forensic analysis. Such traces, which are revealed in the form of post-processing artifacts, can allow one to deduce whether or not an image has been processed.

Among the numerous post-processing techniques, linear transformation is one of the most frequently used types. Because the transformation inherently incorporates a resampling process, several detectors exploiting the periodicity of interpolation through a resampled image have been researched. Gallagher first uncovered periodicity in the second-derivative signal of an interpolated image (Gallagher, 2005). A similar method was proposed by Prasad and Ramakrishnan (2006). They detected a resampling procedure by analyzing a binary signal obtained by the zero crossings of the second-derivative of the interpolated signal, after which they localized the region which had been altered. However, these methods are not able to detect other types of linear transformation apart from resizing. Mahdian and Saic analytically extended Gallaher’s method (Mahdian and Saic, 2008). They proved that the variance of the nth order derivative of an interpolated signal has a periodicity equal to the sampling rate of the original signal if the given signal satisfies the stationary signal requirements. Because their detector only found trace of linear transformation, it cannot be used to approximate the transformation matrix. Wei et al. proposed a rotation angle estimator based on Gallagher’s method (Wei et al., 2010). They thoroughly analyzed the relationship between the rotation angle and the periodicity in the image’s edge map from the interpolated image. Although the detector nearly estimated the rotation angle, it was still difficult to estimate the complex linear transformation. Aside from analyzing the nth order derivative of the interpolated image, Popescu and Farid identified correlations among neighboring pixels by a local linear predictor that adopts an expectation/maximization (EM) algorithm (Popescu and Farid, 2005). In another study (Kirchner, 2008), Kirchner analytically derived the relationship between the derivative-based detector and that by Popescu and Farid. From his analysis, he proposed an improved resampling detector. He also showed the specific periodicities that can be detected in a series of tailored row and column predictors instead of relying on a single predictor (Kirchner, 2010). Feng et al. extracted 19 features from the normalized energy density of second-derivative images in the frequency domain, after which the feature vectors were applied to train and test a support vector machine (SVM) classifier (Feng et al., 2011). To determine whether an investigated image was upscaled or downscaled, Pfennig and Kirchner combined the approach by Feng et al. with the linear predictor of Pfennig and Kirchner (2012). However, to the best of our knowledge, all of these detectors still have difficulties when used to estimate the exact linear transformation matrix of an investigated image. Therefore, in this letter, we propose a novel forensic detector that approximates the linear transformation of an image by analyzing the second-order derivative signals of the image in both the row and column directions.

The rest of this letter is structured as follows. We first introduce the interpolation artifacts of a resampled image in Section 2. Based on the information given in this section, the proposed scheme to estimate the linear transformation of a suspicious image is explained in Section 3. Subsequently, Section 4 reports the experimental results from a massive test setup before Section 5 concludes the letter.

Section snippets

Analysis of periodicity of interpolation

Interpolation is the process of obtaining new values located between the given samples of an original signal. Images generated by linear transformation inherently include a specific interpolation artifact. Thus, as mentioned in Section 1, various detectors that exploit the periodicity of an interpolated signal have been proposed. In particular, most of them are based on the variance of the nth order derivative of an interpolated signal having a periodicity equal to the sampling rate of the

Estimation of linear transformation from periodic peaks

In order to estimate the linear transformation of an investigated image, the interpolation procedure caused by the transformation should be considered. Suppose a linear transformation ofxy=Mxy,whereM=abcd.

Here, (x,y) and (x,y) are the pixel coordinates of the original and transformed image, respectively. Because the original image is mapped onto the rectangular grid of the transformed image (i.e., the integer coordinates of x and y), interpolation among the neighboring pixels of the

Experimental environments

This section describes the performance of the proposed linear transformation estimator. The setup includes our method, as described in Section 3, and two alternative estimators, which also exploit the periodicity of interpolation to detect resizing artifacts (Gallagher, 2005) and rotation angles (Wei et al., 2010). With the detectors, we conducted numerous experiments to detect the linear transformation of investigated images. For this purpose, 100 RAW images were randomly chosen from the BOSS

Conclusion

With the growth of the digital imaging industry, image forensic examination faces more complicated and visually convincing manipulations. Given that prior studies concentrated on “single” manipulation such as scaling or rotation, in this letter we focused on how to estimate general linear transformations by revealing the periodicity of the interpolation. For this purpose, both row and column direction second-derivative signals were analyzed. From our study, we proved that two peaks from the

Acknowledgment

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2012R1A2A1A05026327).

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