EP3123398A1

EP3123398A1 - System for identifying a photographic camera model associated with a jpeg-compressed image, and associated method performed in such a system, uses and applications

Info

Publication number: EP3123398A1
Application number: EP15718526.5A
Authority: EP
Inventors: Rémi COGRANNE; Florent RETRAINT; Thanh Hai THAI
Original assignee: Universite de Technologie de Troyes
Current assignee: Universite de Technologie de Troyes
Priority date: 2014-03-28
Filing date: 2015-03-27
Publication date: 2017-02-01
Also published as: FR3019350A1; FR3019350B1; WO2015145092A1; JP2017517222A; US20180173993A1

Abstract

The invention relates to a system and method for identifying a photographic camera model from a photograph taking the form of a compressed image that has undergone post-acquisition processing and fulfilling a linear relationship between the expectation and the variance of the pixels, such that σ²y_m,n= aµ_ym,n+ b. The system is characterised in that it comprises an image processing device that can supply an analytical relationship between parameters (α, ß ) from the model of the discrete cosine transform (DCT) coefficients and the parameters (a, b) of the camera in the form (I), such that the parameters (c, d) determine fingerprints that characterise a photographic camera model and depend on both the frequency (p, q) and parameters a and b. The system also comprises a device for carrying out statistical hypothesis tests on the DCT coefficients and a statistical analysis device for determining if the photograph was taken by the photographic camera model or by another photographic camera model. The statistical properties of these tests have been established, allowing error probabilities to be controlled a priori. These results can be used, in particular, to guarantee compliance with a constraint on the probability of one of the two possible errors in the test (false positive or false negative). The invention further relates to the use of the method and to the application thereof.

Description

SYSTEM FOR IDENTIFYING A MODEL OF PHOTOGRAPHIC APPARATUS ASSOCIATED WITH A JPEG COMPRESSED IMAGE AND METHOD IMPLEMENTED IN SUCH A SYSTEM, USES AND APPLICATIONS THEREOF 1 - TECHNICAL FIELD OF THE INVENTION

[001] The invention relates to the identification of a camera model, more particularly, the invention relates to a system for determining the identification of a camera model from a digital photograph. having followed all the processing of the acquisition chain, or even compressed according to the JPEG standard. The invention also relates to a method for implementing such a system. These systems find important applications for determining the provenance of a photograph.

[002] Digital forensics or the search for evidence in a digital medium has grown significantly over the past decade. In this field, the proposed methods are distinguished into two categories depending on whether one wishes to identify the camera model or the device itself (an instance of a certain model).

[003] In general, the identification methods are passive or active. In the case of active methods, the digital data representing the content of the image are modified in order to insert an identifier (method called tattooing or watermarking). When the inspected image does not contain a tattoo, the identification of the acquisition device must be done from the image data.

2- STATE OF THE PRIOR ART

[004] Concerning forensic science, two key problems are identified: identification of the origin of the image and detection of false images (see [1] - [3] and the references incorporated in this document). The identification of the origin of the image is intended to verify whether a given digital image is acquired by a specific camera (ie an instance) and / or to determine its model. The detection of false images is intended to detect any act of manipulation such as splicing, removing or adding to an image. To solve these problems, there are two approaches, active and passive. Digital tattooing is considered an active approach. There are nevertheless some limitations cf. [3] because the mechanism incorporation needs to be available, and the credibility of the information embedded in the image remains debatable. The passive approach has been increasingly studied in the last decade. The watermark, or the prior information of the image, including the availability of the original image, is not required in its mode of operation. Passive forensic methods rely on the fingerprints of the camera, left in the image, to identify its origin and verify its authenticity. These prints are extracted by the image acquisition processing chain; see references [4] - [6], for an overview of the different stages and structure of treatments within a digital camera.

[005] The passive forensic methods proposed for the problem of identifying the origin of the image can be divided into the following two basic categories. The first category method is based on the assumption that there are differences between device models, both for image processing techniques and for technological components. Indeed, the aberration of the objective cf. [7], the "Color Filter Array" (CFA), the interpolation algorithm, demosaicing cf. [8] - [1 1], and JPEG compression see references [12], [13] are considered influential factors for camera model identification when white balance algorithms, " white balancing ", [14] are used to identify the source device. Based on these factors, a set of features is provided and used in the machine learning algorithm. The main challenge is that image processing techniques remain the same or similar and the components, produced by some manufacturers, are shared between camera models. In addition, as in all machine learning applications, it is difficult to select a set of specific features. In addition, the analysis of the implementation of the detection performance remains an open problem cf. [15].

[006] The method of the second category is to identify the unique characteristics, or fingerprints, of the acquisition device. The "Sensor Pattern Noise" (SPN) or noise characteristic of a sensor, is based on the imperfections resulting from the process of manufacturing the photographic sensor and on the non Uniformity in electronic photo conversion due to lack of homogeneity of silicon wafers (aka "Photo-Response Non-Uniformity" or PRNU). This is a unique fingerprint, see References [17] - [21]. In addition, the methods based on the presence of a non-uniformity noise (PRNU) are also used in reference [22] for the identification of the model of the camera. These methods are based on the assumption that the fingerprint obtained from an image in TIFF or JPEG format contains traces of the intrinsic watermark that is the SPN containing information about the model of the camera. [007] It should be noted that the two main components of the SPN Sensor Pattern Noise are the "Fixed Noise Pattern" (FPN) and the "Photo-Response Non-Uniformity" PRNU. The FPN, or "fixed form noise structure", which is used in reference [23] for the identification of the apparatus, is generally compensated in a camera by subtracting a dark image on the image Release. As a result, the Fixed Noise Pattern (FPN) is not a robust fingerprint and can not be used in subsequent work. The UNPN is directly used in some works, see references [17], [18], [21]. The ability to reliably extract this image noise is the main challenge in this category. Another challenge is the falsification of the origin of the image due to "counter-analysis" activities, see reference [24]. However, existing methods are designed with very limited exploitation of hypothesis test theory and statistical image models. As a result, their performance remains analytically undefined.

[008] Most of the forensic image methods rely on sensor noise, see references [22], [25], or feature-oriented camera features, see reference [10]. Most digital cameras export images in JPEG format. The ability to extract image features is in doubt because JPEG compression can severely damage these features. In the patent application referenced in [25] we proposed to use the parameters (a, b) to passively identify a camera model. This method is based on the heteroscedasticity of the present noise in a RAW image. A RAW image is one that has not undergone any post-processing treatment chain processing). The patent application referenced in [25] shows a perfect detection performance for the identification of a camera model from RAW images, uncompressed images or lossless compressed images. However, it is useful to extend this method to compressed TIFF or JPEG images.

[009] The problem addressed in the present invention is that of the passive identification of an acquisition camera model from a given compressed image. By passive identification, it is intended to make a decision in the case where the image is not supposed to contain information identifying the source. The issues that are being addressed are (1) ensuring that a photograph has not been taken by a given device when the latter is compromising, or (2) conversely, ensuring that an inspected photograph has well been taken by one device rather than another. The examples that can be given are numerous among which: this camera is at the origin of the photograph of the confidential document (such as those available on wikileaks); could a given child pornography photograph have been acquired with the camera of a suspect; has a copy of a contract been scanned with the customer's device; the photograph of a document makes it possible to copy and mark the document etc.

3- DESCRIPTION OF THE INVENTION

The object of the invention is to provide a system for identifying a camera model from images acquired with a known camera model and a photograph in the form of a compressed image, said photograph having followed a post-acquisition processing and responding to the linear relation between the expectation and the variance of the pixels such that: o ² _ym , n = a ym.n + b where a and b are two parameters characterizing said model photographic apparatus, _ym , n and a ² _ym , _n are respectively the expectation and the mathematical variance of the pixel y _m , _n in position (m, n) having followed post acquisition processing, the system is characterized in that it comprises an image processing device capable of providing an analytic relationship between parameters (α, β), the statistical distribution model of the DCT coefficients, Discrete Cosine Transformation, and the parameters ( a, b) of the camera in the form: Çj = 4- d _p ^

so that the parameters (c, d) determine fingerprints characterizing a camera model and depend both on the frequency (p, q) and the parameters a and b, in that the system further comprises a device for performing statistical hypothesis tests of the DCT coefficient and a statistical analysis device for determining whether said photograph was taken by said camera model or other model of camera. Basically, the images for which the acquisition model is known are the source of the identification of the camera model.

Advantageously, the analysis device provides an indication of the identification of said camera model by certifying the accuracy of the identification with a previously defined accuracy.

The invention also relates to a method implemented in said system, which comprises the following steps: - prior analysis of images acquired with a known camera model,

reading a compressed image Z in order to determine the matrices representing the value of the pixels,

- estimation of the fingerprint parameters: (c _p , _q , d _p , _q ), - deletion of the content of the compressed image Z in order to obtain a residual image,

estimating the DCT coefficients of said residual image, - performing comparative statistical tests to identify a camera.

A prior analysis of images acquired with a known camera model is carried out, with a view to estimating imprint parameters (c _p , _q , d _p , _q ), according to the same method as for the analysis of images. an unknown image.

Advantageously, the statistical tests are performed according to a prescribed constraint on the probability of error.

According to the invention, the photograph is a compressed image, according to the JPEG compression standard.

[0015] Advantageously, the photograph is a reference image, or interframe, belonging to a video stream, and compressed according to the MPEG compression standard.

The invention also relates to the use of the above method for unsupervised detection of the falsification of an area of a photograph.

Furthermore, the invention relates to the use of the above method for detecting, in a supervised manner, the falsification of an area of a photograph. This detection is performed by testing whether a prior known area comes from the same camera as the rest of the inspected image.

The invention also relates to the use of the method above in the search for evidence from a compromising image.

The invention relates to the application of the above method in specialized software, in the search for evidence from digital media.

4- BRIEF DESCRIPTION OF THE FIGURES Other features, details and advantages of the invention will emerge on reading the description which follows, with reference to the appended figures, which illustrate:

- Figure 1 shows a system for determining the identification of a camera model according to the invention;

Figure 2 illustrates the entire chain of acquisition of a natural image and the treatments performed in digital cameras;

- Figure 3 shows the comparison between the exact model of DCT coefficient equation (7) and the DCT coefficient equation approximation model (8); - Figure 4 shows the parameters (, ^" βθ4) estimated from each image of the Dresden image database captured with two different camera models: Canon Ixus 70 and Nikon D200;

- Figure 5 shows the parameters (C6 ₄ , ^" d6 ₄ ) estimated from each image of the Dresden image database captured with two separate camera models: Canon Ixus 70 and Nikon D200;

FIG. 6 shows the detection performance of the δ ^* test for

simulated data with parameters a = 3, cO = 1 1, 5, dO = -4, c1 = 13, d1 = -5.5;

FIG. 7 shows the detection performance of the ^" δ * ι and ^" δ * 2 tests for data simulated with parameters a = 3, cO = 1 1, 5, dO = -4, c1 =

13, d1 = -5.5;

FIG. 8 shows the detection performance of the δ tests for 500

images from the Dresden image database, obtained by two Canon Ixus 70 and Nikon D200 cameras.

For clarity, identical or similar elements are identified by identical reference signs throughout the figures.

5- DETAILED DESCRIPTION OF AN EMBODIMENT

FIG. 1 represents a system for determining the identification of a photographic camera. Reference 1 indicates the system and reference 2 the model of camera that took a photograph 3.

It is from this photograph 3 that the system 1 will determine the model of camera that captured this photograph. This system consists of a photo analyzer that will examine this photograph 3. The photograph 3 is in the form of a compressed file suitable for the treatment that follows. JPEG formats or any image resulting from the decompression of an image previously compressed in JPEG format, compressed according to the JPEG standard, is suitable for this processing.

The system 1 can be implemented on a PC type computer. This system 1 is provided with an input member 10 to accommodate the data of the photograph 3. These data are processed by a processing member 12 which implements a treatment which will be explained below. A device for performing statistical hypothesis tests of the DCT coefficients and a statistical analysis device 14 will provide an indication of the identification of the camera model at the origin of said photograph.

According to the method according to the invention, in the first step, the digital photograph 2 is seen as one or more matrices whose elements represent the value of each of the pixels. In the case of a grayscale image, the photograph can be represented by a single matrix:

Z = Zj with 1 <i <L

For color images, three distinct colors are usually used: red, green and blue. In this case, an image can be likened to 3 distinct matrices, one matrix per color channel:

Z = _Zi ^k with l≤ ≤

The second step of the method consists in separating the different color channels, when the analyzed image is in color. The sequence of operations being carried out identically with each of the matrices representing the color channels, we consider that the image is represented by a single matrix (the index k is omitted).

The noise present in digital photographs, has the property of being heteroscedastic: The stochastic properties (random) of noise are not constant on all the pixels of the image.

Due to the large number of photons incident on the sensors, it is possible to closely estimate the counting process by a Gaussian random variable. Figure 2 illustrates the entire chain of acquisition of a natural image and the treatments performed. This acquisition chain comprises several processing steps (demosaicing, white balance, and gamma correction) after which a polychromatic image is obtained from the light intensity measured by each photosensitive cell of the sensor. Depending on the method of each step, the quality of the final image may vary significantly. Each step affects the final output image. It should be noted that the sequence of operations differs from one manufacturer to another.

Today, the JPEG image format appears more and more as a standard in digital images. Most digital cameras and software encode images in this format. The use of JPEG compression is a matter of balance between storage size and image quality. An image that is compressed with a high compression factor requires little storage space to the detriment of lower visual quality.

The Discrete Cosine Transformation (DCT) operation is one of the key steps in JPEG compression. Given an arbitrary image Z, the DCT operation is applied to each block of 8 χ 8 pixels of Z as follows

where z _m , n represents a pixel within an 8 x 8 block of Z, 0 <m <7, 0 <n <7 and Ip q denotes the two-dimensional coefficient of DCT and

The term Tq can be easily derived by analogy with the term Tp. The DCT coefficient at the (0, 0) position, called the direct current coefficient (DC), represents the average value of the pixels in the 8 ^ 8 pixel block. The remaining 63 coefficients are called the AC coefficient. Two The main advantages of the DCT operation are suboptimal decorrelation and energy compaction. After the DCT operation, the energy is located mainly in the low frequencies while the high frequencies mainly contain noise components.

The modeling of the DCT coefficient distributions has been widely studied in the literature. The Laplacian models in reference [27], generalized Gaussian (cf [28]) and generalized Gamma (Gr) (cf [29]) have been proposed for DC coefficients. But Laplacian distribution remains a dominant choice in image processing because of its simplicity and relative accuracy. In our previous work, we established a rigorous mathematical framework to model the AC coefficients, see more details in reference [6]. Since the coefficient DC represents the average value of the pixels in each 8 8 pixel block, the DC coefficient distribution can not be directly derived because of the heterogeneity in a natural image. Let I be a coefficient AC, for the sake of clarity, the index of said coefficient AC is omitted. Due to the variability of the block variance, the probability density function (pdf) of the I is given as a bi-stochastic model (see [27]) as follows where fx (x) denotes the probability density function (pdf) of a random variable denoted X and where σ ² denotes the variance of the block considered. It is assumed that the pixels inside a block are independently and identically distributed, cf. [27]. Given σ ² a constant variance of the block, the distribution of the AC coefficients of I can be approximated by a Gaussian distribution of zero mean, by virtue of the Lindeberg central limit theorem (CLT) for the correlated random variables cf. [27], [30]

(4) In addition, the distribution of the variance o ² over all the blocks constituting an image can be modeled in an approximated way by the distribution law of gamma ⁽ g (α, β) cf [6] defined by the density function of probability (pdf): where a is a positive form parameter, β is a positive scale parameter, and Γ ( ^■ ) represents the gamma function. Starting from (3), (4) and (5), the statistical distribution model of the AC coefficient of I is given as:

Starting from [31] and using the integral representation of the modified Bessel function, written Kv (), the integral (6) can be written

where Kv (x) represents the modified Bessel function [31, chapter. 5.5]. The model of AC coefficients proposed in (7) includes the particular cases of Laplacian distribution and Gaussian distribution (cf [6]). As shown in [6], this model surpasses the Laplace and Gauss models, but at the expense of more complex expressions. Using the Laplace approximation [32] (see more details in Appendix A), an approximation of the function fi (x) can be given as: It should be noted that other polynomial expansions of the modified Bessel function Kv (x) are also given in [31], so a polynomial approximation of fi (x) can be derived. However, these approximations are not considered in the present application. The precision of the approximation given in (8) depends on the choice of the functions g (t) and h (t) (see later in relation (56)). The main advantage of relation (8) is to provide an approximation in the form of an exponential function. This approximate model is used to simplify the calculation of the likelihood functions as well as the calculation of the threshold for the likelihood ratio used in the proposed tests. The estimation of the parameters is performed based on the exact model defined in relation (7).

It may be noted that this approximate model is a special case of the generalized gamma distribution model when y = 1 (the variable y is given in reference [29, Eq (6).]). An example is given in FIG. 3 to illustrate the accuracy of the exact model of the distribution of the DCT coefficients of a JPEG image and the precision of the approximate model of the distribution of the DCT coefficients. The empirical data is extracted from a real image of the Dresden image database (see [33]) previously compressed using the JPEG compression standard. The parameters are estimated from the empirical data based on the exact model of the DCT coefficient distribution. The exact and approximate models of the DCT coefficient are represented with the parameters estimated by maximum likelihood. The main disadvantage of the approximation model is that when x tends to 0, the function (8) tends to 0 when a> 1, and it tends to infinity when a <1. This leads to inaccuracy in the vicinity of 0, as shown in FIG. 3. Nevertheless, this does not cause a loss in the detection performance, when designing the LRT test, as much in terms of ability to guarantee a probability of false alarm as in terms of probability of correct detection. As the model of coefficients AC defined by the relation (7) is symmetrical, the odd moments disappear. On the basis of the law of total expectancy, the calculation shows that ^'< f U 3a •: 0 ^' f

(10) where E _x denotes the expectation of a random variable X. Therefore, the method of moments (MM) allows the estimation of parameters (α, β) as follows: a = d

M Al 3m¾

'(1 1) (12) where ^A m2 and ^A m4 are respectively the second and fourth empirical moments of I. The maximum likelihood (ML) estimate of the parameters (a, β) are defined as the solution of the problem of maximization

where N is the number of coefficients at the same frequency. It should be noted that the likelihood function is differentiable, but calculating the derivative seems extremely difficult. Since there is no exact or analytic form for maximum likelihood (13), it is proposed to solve the maximization problem numerically using the Nelder-Mead optimization method [34]. The MM estimates ( ^Λ α ^MM , ^Λ β ^ΜΜ ) are taken as initial solution in the optimization algorithm.

As discussed above, one of the main advantages of the (8) approximation is the compaction of energy. Energy tends to be located mainly in low frequencies while high frequencies mainly contain noise components. Therefore, there is a difference in scale between the DCT coefficients. The DCT coefficients do not share the same parameters (α, β). Since the estimation of the parameters (α, β) is carried out separately on each frequency, we must designate the parameters (a _p , _q , β _{ρ ς} ) with respect to the DCT coefficient of l _p , _q . 5.A The intrinsic footprints of the camera

In our previous work references: [25] and [35], the parameters (a, b) have been exploited to passively identify a camera model, where a and b represent characteristic parameters of a of a camera. This method is based on the heteroscedasticity of the noise present in a RAW image. Stochastic (random) noise properties are not constant across all pixels in the image. More precisely, the value of each pixel depends linearly on the number of incident photons. This model, which represents all the noise contaminating the RAW image, gives the variance of the noise as a linear function of the mathematical expectation of the pixels and responds to the following relation:

¾ ¾ " ^: " Λ ^: f! ½. _^ ,,, " ^■ ÷ ^■ h]

^'" ~ (14) where Sm, n is the measured value of the RAW pixel at position (m, n) and Sm, n is its mathematical expectation Although this method shows almost perfect detection performance, there are two main limitations First, it focuses on RAW images, which may not be available in practice, because the most difficult part when extending this method to other image formats, for example TIFF and JPEG, is the impact of the post-acquisition process (demosaicing, white balancing and gamma correction) as well as the compression process, since demosaicing causes spatial correlation between pixels and non-linear operations destroy the linear relationship between the expectation and variance of the pixel Secondly, the proposed fingerprint, defined by the parameters (a, b), depends on the ISO sensitivity, although this is not crucial in practice because there is not much sensitivity ISO and only a small number of images is sufficient to estimate the reference parameters (a, b) for each ISO sensitivity, It is desirable to rely on a footprint that is invariant with respect to the content of the image and which is robust for nonlinear transformation operations (eg gamma correction factor). To make a complete color image at the output and improve its visual quality, a RAW image requires a post-acquisition process, for example, demosaicing, white balancing, and gamma correction. To extend the process referenced in [21] to the compressed images, suppose that the effect of the demosaicing and white balancing algorithm is negligible on the heteroscedastic relationship of pixel expectancy and variance. Then the compressed image corresponds to the relationship between the mean and variance of pixels such q ue ^1: σ ² ym, n = ¹ to Pm, n + b where a and b are both ¹ p arameters characterizing a camera model , μ and o ² y _m , _n are respectively the expectation and the mathematical variance of the pixel y _m , _n at position m, n.

We will provide an analytical relationship between the parameters (α, β) of the model of the DCT coefficient (discrete cosine transform) and the parameters of the camera (a, b). It is assumed that the pixels in each block of

8 x 8 are independent and identically distributed. In addition, we assume that the effect of the demosaicing and white balancing algorithm is negligible on the heteroscedastic relationship of pixel expectancy and variance. The gamma correction is defined by the transformation, applied to each pixel independently, defined as follows:

(15) where | | denotes the absolute value and y is the correction factor (typically, y = 2.2). Here, is referenced as the balanced white pixel and η is a zero average signal, representing the pixel noise at the position (m, n) of variance: σ ² η = Var [n] = / u + b. The first order of serial development

<m, n m, n ym, n

Taylor's (1 + x) V ^Y for x = 0 leads to:

Taking expectation and variance on both sides of equation (16), we will obtain:

where z _m , _n and o ² z _m , _n represent, respectively, the value of the expectation and the variance of the pixel z _m , _n . The relation (17) is justified under the assumption of ⁿ _m , _n "

^ y _m, n -By Moreover, it is noted that the variance of the gamma corrected pixel, o ² z _m, _n is directly proportional to the noise variance n ^2, _m, n, and inversely proportional to the value of the expectation ^ z _m , _n . Taking the variance on both sides of equation (1), it follows that Var [l _P , _q ] = Var [z _m , _n ] = o ² z _m , _n . This is justified under the assumption that the pixels are independent and identically distributed in block 8. It follows from equation (9) that

Moreover, starting from (9) and (10), we obtain 9)

where it follows from (16) that (22)

Therefore, from (19) - (22) we obtain:

For the sake of simplification, by solving equations (18) and (23) and using the Taylor series again, we obtain a roughly linear relationship

where the parameters (c _p , _q , d _p , _q ) depend on both the frequency (p, q) and the parameters (a, b) of the camera. As a result, the parameters

(c _p , _q , dp, _q ) can be used as a fingerprint for identifying the model of the camera.

5.B- Estimation of the impression of the camera

It is important to remember that JPEG compression involves basic steps, namely: the DCT operation, the uniform quantization of the DCT coefficients with a quantization matrix and an entropy coding of the quantized values. Therefore, it appears that the fingerprints of the camera (c _p , _q , d _p , _q ) given by the mathematical analysis above could work with the quantized original DCT coefficients extracted from the JPEG file. However, because of the effect of quantization, information that is not visually significant is ignored. High frequency contains most zeros. Thus, there are not enough statistics to estimate the parameters (c _p , _q , d _p , _q ). In addition, the parameters (c _p , _q , d _p , _q ) estimated at low frequency can be contaminated because their DCT coefficients are strongly influenced by the content of the image.

Let Z be a color image with three components Z = {Z ^c } where c {R, G, B} denotes the red, green and blue channel index. In order to mitigate the impact of the image content, we propose to remove the content of the image from the given Z image in the spatial domain by using a denoising filter T ⁾ on each color channel to to obtain a residual image W ^c (25)

The residual image with 3 components {W ^c } is converted into a monochrome image as

W = 0.298 W ^R + CIS87W ° - CU14W ^S

(26)

The residual image ^~ W is then transformed into the domain DCT = DCT (W)

(27) where Γ is the image of the DCT coefficients of the residual image W.

For the sake of clarity, the image Γ is arranged in 64 DCT coefficient vectors following the order of "zig -zag" used in the JPEG compression standard. Let l ^w _K = (I _M ^w , ..., l ^w _k , _N ) T, k <≡ {1, ..., 64}, the vector of length N representing the k-th coefficient DCT where l, , 1 <i <N, denotes the value of the k-th DCT coefficient of the block i and U ^T denotes the transpose of the matrix U. Similarly, the parameters characterizing the distribution of l ^w k are designated (αι _< , βκ ) and the fingerprints of the camera are also noted (Ck, d _k ).

The mathematical analysis performed above is based on the strong assumption that the pixels are identically distributed within a small block of 8 x 8. This assumption may not be realistic because of the presence of edges or details in a natural image. Therefore, to approximate the mathematical framework, we only work on the homogeneous blocks 8 8 and we estimate the parameters on the selected data. Before selecting a block, saturated blocks must be excluded because they distort the mathematical framework above. A pixel is designated as saturated if its gray level value is in the area corresponding to the lower or upper 10% of the dynamic range (typically the dynamic range is [0255] for an 8-bit image, the saturated areas are therefore [0.25] and [220 255]). A block is designated as saturated if there is at least one saturated pixel in the block. The block selection is performed by a simple non-adaptive method based on the standard deviation of the block. The idea is that in a homogeneous block without edge or in detail, the standard deviation of the block must be small. The standard deviation of the block is estimated by a robust estimator: the median of the absolute value of the deviations from the mean (MAD). Therefore, a block i is selected if both of the following conditions are met where ke {2, ..., 64} and I _k , i denote the k-th coefficient DCT in the block ί of the image ^~ Z in grayscale using the same transformation as in (26). Here, instead of calculating the median of the absolute value of the deviations from the mean (MAD) of each block in the spatial domain, we calculate the median of the absolute value of the deviations from the mean (MAD) in the DCT domain to take advantage of take advantage of the decorrelation and compaction properties of the DCT and thus obtain a better estimate of the standard deviation. DC coefficients are excluded from the calculation. The thresholds Ti and T ₂ are set at Ti = 1, 5 and T ₂ = 0.8. It should be noted that the first condition is to eliminate blocks with strong edges and the second is to remove blocks where a disturbance may exist because of the denoising filter.

In practice, the print of the camera (Ck, d _k ) must be estimated from a single image. Therefore, several pairs of (<¾, β ^ from a single image are required.) SoifN the number of blocks selected, for each frequency, samples are randomly extracted with N≤ ^~ N from Ι,, i { 1, ..., N} and C {1, ..., L}, from N samples, the maximum likelihood ML of the parameters (<¾, β ^ - Parameters (Ck, dk) are estimated by considering L pairs of ( ^" <¾, _c , ^" k), C {1, ..., L}, and applying ordinary least squares (OLS) estimates, see [36]. from the relation (24) we obtain:

The MCO estimates (δ _k , ^" d _k ) are unbiased and are asymptotically equivalent to ML estimates because the ML estimates (<¾, ι, βκ, ι) follow the asymptotic Gaussian distribution cf [37] Therefore, the estimates (ô _k , ^" d _k ) also follow the distribution. To ensure that there are enough statistics for the estimate, we use L = 200 and N = 4096 (about 10% of ^~ N).

Even if the content of the image is removed, the DCT coefficients of l ^w _k at low frequency can still be slightly affected. Therefore, it is more relevant to use the DCT coefficients of l ^w _k at high frequency. An example is given in FIG. 4 to illustrate the linear relationship between <¾ and _k ^"1. In addition, Figure 5 shows a scatter plot of the parameters (c, ^" d) estimated from each image by following the above method. The images used for Figure 4 and Figure 5 cover different scenes and different settings of the camera. The parameters (c, d) are invariant and robust for the processing of nonlinear algorithms. In addition, because the parameters (c, d) are related to parameters (a, B) that have been proposed for the identification of the camera model in [25] and [35], they are also discriminative for different models of the camera and they can also be exploited for the identification of the model of the camera.

6- Formulation of the statistical hypothesis test

The present invention aims to identify camera models based on DCT statistics. This part allows to analyze two models of devices 0 and 1. Each model of camera j, j <≡ {0,1}, is characterized by parameters (Ck, dk) where k designates the frequency with k <≡ {2, ..., 64}. In a binary hypothesis test, the inspected Z image is either acquired by a camera model 0, or by a camera model 1. The test object is to decide between two hypotheses defined by:

where P _Q k _, kj- represents the statistical distribution of the DCT coefficients of l ^w _k , i under the assumption i of the parameters (<¾, P _k j) - As explained above, the emphasis is placed on the prescribed guarantee of a probability of false alarm. Therefore, we define the class of tests whose probability of false alarm is less than the limit prescribed by αθ. Here, P _r (E) represents the probability of event E under the assumption 3 ^~ C with j {0,1}, and the supremum above Θ must be understood as any value for the parameters of the model . Among all the tests of the Καο class, we aim to find a test δ that maximizes the power function, defined by the probability of correct detection:

The problem defined in (30) highlights three fundamental difficulties in identifying the model of the camera. First, even if all the model parameters (<¾, Ck, j, _dk , j) are known, the most powerful test, namely the LRT, has never been studied for this problem. The second difficulty concerns the unknown nuisance parameters _k it in practice. A possible approach to deal with parameters of unknown nuisance consists in eliminating them by using the principle of invariance cf. [38]. This approach has been discussed in references [39], [40] and has performed well in some applications cf. [41]. However, this approach can not be applied here because the hypotheses tested do not possess the "symmetry" properties required for the application of the invariance principle in statistics. Another approach is to design a GLRT test by replacing the unknown parameters with ML estimates. [42]. Finally, the two hypotheses J ^~ C _o and are composite because the parameters of the camera (Ck j, dk j) are unknown.

For the sake of clarity, we assume that the parameters of the camera o, d _k , o) are known and we only solve the problem in which the alternative hypothesis is composite, that is, the camera settings (Ck, i; d _k , i) are not known. It should be noted that a test that maximizes the detection power of any kind (Ck, i; d _k , i) could exist. Our main goal is to study the LRT test and design the GLRT test to answer the second and third difficulties.

In addition, it should be noted that the GLRT test treated with unknown nuisance parameters <¾, when the camera parameters are known, can be interpreted as a closed hypothesis test where a given image is is acquired by the camera model 0, or by the camera model 1. While, the GLRT test processed with the parameters of the unknown camera (Ck, i; d _k , i) becomes an open hypothesis test in which a given image is acquired by a camera model 0 or not . Indeed, the given image can be acquired by an unknown camera model. Therefore, the two proposed tests can be applied, depending on the context requirement.

6.A- Likelihood ratio test for two simple hypotheses.

When all the parameters of the model are known, by virtue of the Neyman- Pearson lemma [31, Theorem 3.2.1], the most powerful test, δ which solves the problem (30), is the LRT test proposed by the following decision rule: decision threshold 7 is the solution of the following equation:

¾0 A (Z>> T = f ^' .)

(32) to ensure that the LRT test is in the Καο class, using Γ approximation of the DCT coefficient model, defined in (8), the likelihood ratio (LR) of an observation Ι, is defined by

The statistical distribution of the likelihood ratio, LR denoted Λ (Z), under each assumption 3 ^~ (j is given by

Π

A (Z) A (ir V _j )

(34) where the notation denotes the convergence towards the distribution and (ιη, Vj) respectively denote the mean and the variance of LR Λ (Z) defined later in (66) and (67). Since a natural RAW image is heterogeneous, it is proposed to normalize the LR Λ (Z) in order to set the decision threshold independently of the content of the image. Normalized LR is defined by

As a result, the corresponding test δ * is rewritten as follows:

The normalization of the LR of Λ (Z) allows the test δ * to be applicable to any natural RAW image because the normalized LR Λ ^* (Z) follows the Gaussian standard distribution under the assumption 3- vs. The decision threshold 7 * and the power function β _δ * are obtained by the following theorem:

Theorem 1: Assuming that all the parameters of the model (<¾, Ck, dkj) are exactly known, the decision threshold and the power function β _δ * of the test δ * is given by

* = φ ^· - ^{· 1} (1 - «)

(37)

where Φ () and Φ ^"1 () respectively denote the distribution function of the standard Gaussian distribution and its inverse function.

This decision threshold guarantees that the probability of false alarm will be equal to <¾, that is to say to decide that the photograph does not come from the camera 0 when it is indeed the case.

As we propose to design the LRT test using the approximation function of the DCT coefficient model defined in (8), it is desirable to evaluate the power loss between the theoretical LRT test and the approximate LRT. In the theoretical LRT test, we use the exact DCT coefficient model, defined in (7), and the mean as well as the vahance of the LR are numerically calculated by integral calculations. The power function of the two LRTs tests is shown in Figure 6 for simulated data with parameters a = 3, Co = 1 1, 5, do = -4, Ci = 13, di = -5.5. These parameters correspond respectively to parameters at frequency 64 of Canon Ixus 70 and Nikon D200 devices of the Dresden image database cf. [33]. (see Fig. 5). They are used to generate two vectors of coefficients 2 ¹⁰ and 2 ¹² . The simulation is performed with 5000 repetitions.

Figure 6 clearly shows that the power loss between the theoretical LRT test and the approximate LRT is negligible. The detection power βδ * serves as the upper limit of a statistical test for the problem of the identification of the camera. The δ * test makes it possible to justify a prescribed false alarm rate and also maximizes the probability of detection. Since its statistical performance is analytically established, it can provide a predictable analytic result for any false alarm probability <¾.

In fact, the LRT test aims to make a decision using the ratio of the maximum likelihood function of a given image under the alternative hypothesis / i characterized by the parameters of the camera (Ck, 1, dk, 1) and the maximum likelihood function under the null hypothesis 3 ^~ (o characterized by the parameters of the camera (Ck, o, dk, o) ■ If this ratio is less than a threshold, the null hypothesis ^3- (o is accepted. in contrast, the alternative hypothesis / i is accepted. therefore, the greater the distance between the two points (Ck, o, dk, o) and (Ck, 1, dk, 1) is small, the more difficult the identification of the camera.

6.B- Generalized likelihood ratio test

In this part, two GLRTs tests are designed to deal with unknown parameters. It is proposed to replace the unknown parameters by their maximum likelihood ML estimates defined in (33). Assuming that the camera parameters (Ck, o, dk, o) and (Ck, 1, dk, 1) are known, the first GLRT test is designed as follows

(39) where, again to ensure ^" δι to be in the class ΚαΟ, ^" T? is the solution of the equation

and the generalized likelihood ratio (GLR) of Ύ \ ι (Ι,) is given by

Here, the maximum likelihood ML estimate of ^" <¾" is given by the method previously proposed and

^" βΛ, ί = ck, j ^" cik + dk, j, j ^e {0, 1}. The ML estimate of ^" <¾ converges asymptotically to its true value: ^" a _k ^fS - a _k . Therefore, according to the Slutsky theorem [38, Theorem 1 1 .2.1 1], we obtain the asymptotic statistical distribution of the GLR of ^" Λ1 (Z) under each assumption 3 ^~ C

A: (Z) - i

(42)

As in the case of the LRT test, it is proposed to standardize the GLR ^" Λ1 (Z), however, the expectation m ₀ and the variance v ₀ are not defined because the parameter ci _k is unknown in practice. ¾ by ^" a _k defined in (66) and (67) to obtain the estimates of m ₀ and v ₀ denoting respectively ^" m ₀ ⁽¹⁾ and ^" v ₀ ⁽¹⁾ ■ Therefore, the normalized GLR Ύ \ ^* ι (Z) is defined by: and ^"δ * ι matching is rewritten as follows

By using again Slutsky's theorem [38, Theorem 11.2.11], we obtain the decision threshold and the power function of the test ^" δ * ι.

Theorem 2. When the parameters of the camera (Ck, o, dk, o) and (Ck, i, dk, i) are known, the decision threshold and the power function of the test ^" δ * ι are given by

When the camera settings (Ck, 1, dk, 1) are not known, we can also design the GLRT test following the procedure above. The unknown parameters (Ck, i, dk, 1) are replaced by the MCO estimates of (¾, ι, ^" dk, i) defined in (33)

where ^" βΛ, ο = ^" Ck, o ^" a _k + d _k , o and ^" βΛ ,, = ^" c _k , i ^" a _k + ^" d _k , i

As the MCO estimates of (¾, ι, ^" dk, i) conform, the GLR

7 \ 2 (Z) = i = i ^" Λ2 ( _k , i) also converges to the Gaussian distribution with mean rrij and variance Vj under each assumption 3 ^~ Similarly, the normalized GLR 7 \ 2 (Z) is defined by where ^" mo ^" and ^" vo ^" are the estimates of the expectation m ₀ and the variance v ₀ by replacing a _k by ^" a _k and (Ck, i, dk, i) by (¾, ι, ^" dk, i). The corresponding ^" 5 * 2 ^" test is rewritten as follows

According to Slutsky's theorem, the decision threshold and the power function of the ^" 5 * 2 ^" test are given by the following theorem: Theorem 3. When the parameters of the camera ((Ck, o, dk, o) are known and the parameters (C _k , i, d _k , ₁ ) are not known, the decision threshold and the power function of the 5 * 2 test are given by:

The two proposed GLRTs tests can be applied in the practical context. The first GLRT ^" 5 * i test is designed to detect any given image acquired by camera model 0 or camera model 1 while the GLRT ^" 5 * 2 test aims to verify whether the given image is acquired by the model of the camera 0.

This decision threshold ensures that the probability of false alarm will be equal to cio, that is to say, to decide that the photograph does not come from the camera 0 when it is indeed the case. FIG. 7 represents the power function of two GLRTs tests ^" δ * ι and ^" 5 * 2 compared with the LRT test δ * for the coefficients 2 ¹⁰ and 2 ¹² . The loss of power is obviously revealed because of the error of estimation of the parameters of the model. On the other hand, it can be noted that the power loss decreases as the number of coefficients increases. For coefficients 2 ¹⁴ used, all the tests are perfect, β _δ * = β -5 * 1 = β ~ _δ * ι = 1, ie there is no detection error on 5000 simulated images from the camera model 0 and over 5000 simulated images from the camera model 1.

The proposed method can be extended to images from a video stream. A video stream is composed of a succession of images that scroll at a fixed rate. Video compression is a method of data compression that involves reducing the amount of data, minimizing the impact on the visual quality of the video. The advantage of video compression is to reduce the cost of storing and transmitting video files. The video sequences contain a very high statistical redundancy, both in the time domain and in the spatial domain. The fundamental statistical property on which compression techniques are based is the correlation between pixels. This correlation is both spatial, the adjacent pixels of a current image are similar, and temporally, the pixels of past and future images are also very close to the current pixel. MPEG-based video compression algorithms use discrete cosine transform (DCT) transformations on 8x8 pixel blocks to effectively analyze spatial correlations between neighboring pixels in the same image. Thus, in the method according to the invention the photograph may be an image from a video stream and compressed according to the MPEG standard. It is possible to envisage another case of use of the identification system of a camera model in order to determine whether an area of the image has not been falsified, by copying / pasting. from another photograph or by deleting an element.

Finally, it is possible to envisage another case of use of the identification system of a camera in order to determine, in a supervised manner, whether an area of the image has not been falsified ( copy / paste from another photograph or by deleting an element). Here, the term "supervised" means that the user wishes to ensure the integrity of a previously defined area. The principle is then to apply the identification method to the two "sub-images" respectively from the area targeted by the user and its complementary (the rest of the image). If the item inspected comes from another photograph and has been added by copy / paste, the noise properties will be different, which the proposed system will be able to identify (assuming the photographs were not taken in the same acquisition conditions and with the same model of camera, which seems reasonable). The method of identification of the proposed camera model meets the two weaknesses of the methods briefly presented in the state of the art: 1) their performance is not established and, 2) these methods can be implemented. failure by the calibration of the camera. Since the proposed method is based on the noise properties inherent to the acquisition of compressed photographs, it is applicable regardless of the pot-acquisition treatments applied by a user (in particular to improve the visual quality). In addition, the parametric modeling of the statistical distribution of the value of the pixels in the frequency domain makes it possible to provide analytically the performances of the proposed test. This advantage allows in particular to ensure compliance with a prescribed constraint on the probability of error.

The main fields of application of the invention are on the one hand, the search for evidence from a "compromising" image and, on the other hand, the guarantee that a photograph has been acquired by a given device. The proposed method can be extended to the control of the integrity of a photograph. The goal is to guarantee that a photograph has not been modified / falsified since its acquisition. This allows for example to detect photographs with elements from a different camera, ie imported after the acquisition, or to ensure the integrity of a document scanned or photographed (a legal document for example). The method of the invention may be developed in software manufacturers specialized software, in the search for evidence from digital media. The method according to the invention can be used with the courts to provide a decision support tool. 7- Digital experiments

7.A- Results on a large database

ANNEX A

Laplace approximation of DCT coefficient model

We will briefly describe the Laplace approximation [43]. The Laplace method aims to provide a comparison of the integrals of the form J exp (- g (t)) dt when the function g (t) reaches the global minimum at t ^* . Consider the integral

Taylor's development of the function g (t) at t ^* gives: where g '(t) and g "(t) represent respectively the first and the second derivatives of the function g (t), since the function g (t) reaches a minimum at t ^* , g' (t ^* ) = 0. Therefore, the integral I can be approximated as

This integral takes the form of the Gaussian integral. we obtain:

A generalization was made in [32] with an arbitrary function h (t)

From relation (6), the DCT coefficient model of f | (x) is rewritten as follows

Q (Î) = - 4- ^~ m h (t) = if * ^~ s

^" ' ^■ " H vt ""

(58)

The function g (t) reaches the minimum at t ^* = | x | V / 2 and its second derivative is defined by g "(t) = x ² / t ^3. Therefore, the function fi (x) can be approximated as following :

ANNEX B

Statistical distribution of the LR test Λ (Ζ)

Note that from the relation (33) it is necessary to define the first two moments of the variable | I | . Given a known variance σ ² , the random variable I is the Gaussian variable of zero mean with the variance σ ² . So, the random variable | I | follows the half-normal distribution cf.

[44]. Therefore, we get

On the basis of the law of total hope, the mathematical expectation of | I | is given by

[V (. ± i

V ^" Γ (α) (61)

In addition, the variance of | I | is given by

Var _/ [| / |] = E _/ [| / | ² ] -E [\ I

ri \ 'in ^■ ¾)

····· o

2 (α) (62)

As a result, it follows from (33) that the first two moments of LR Λ (Ι,) under each hypothesis 3 ^~ C are given by:

(63)

,. [Λ (¾)] = 2 (- _ν ¾Γΐ ->,.,. ',) ^'''

X ";, ^. .- (64) where [ ^■ ] and Var [ ^■ ] respectively denote the expectation and the mathematical variance under assumption 3 ^~ (j) Because of a large number of coefficients in a natural image, by virtue of the central limit theorem of Lindeberg '(CLT) [38, Theorem 1 1 .2.5], the statistical distribution of LR Λ (Z) is given by

(65) where the notation denotes convergence in distribution and

with EA [(l ^w _k, i)] and Var [A (l ^w _k, i)] defined respectively in (63) and (64).

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Claims

1. System for identifying a camera model (2) from images acquired with a known camera model and a photograph (3) in the form of a compressed image, said photograph (3) having followed a post-acquisition processing and responding to the linear relationship between the expectation and the variance of the pixels such that: a ² _ym , _n = a _ym , n + b where a and b are two parameters characterizing said camera model, _ym , n and a ² _ym , n are, respectively, the expectation and the mathematical variance of the pixel y _m , _n in position (m, n) having followed the post acquisition processing, the system is characterized in that it comprises a device image processing apparatus (12) capable of providing an analytic relationship between parameters (α, β), the statistical distribution model of the DCT coefficients, Discrete Cosine Transformation, and the parameters (a, b) of the camera Under the form :

so that the parameters (c, d) determine fingerprints characterizing a camera model and depend both on the frequency (p, q) and the parameters a and b, in that the system further comprises a apparatus for performing statistical hypothesis tests of the DCT coefficients and a statistical analysis device (14) for determining whether said photograph was taken by said camera model, or by another model of a camera .

2. System according to claim 1, characterized in that the statistical analysis device (14) provides an indication of the identification of said camera model by certifying the accuracy of the identification with a previously defined accuracy.

3. Method implemented in the system according to claims 1 and 2 characterized in that it comprises the following steps: prior analysis of images acquired with a known camera model, - reading of a compressed image Z in order to determine the matrices representing the value of the pixels,

- estimation of the print parameters: (c _p , _q , d _p , _q ),

removing the content of the compressed image Z in order to obtain a residual image,

estimating the DCT coefficients of said residual image,

- performing statistical hypothesis tests to identify a camera.

4. Method according to claim 3, characterized in that the statistical hypothesis tests are performed according to a prescribed constraint on the probability of error.

5. Method according to claim 3 characterized in that the photograph is a compressed image, according to the JPEG compression standard.

6. Method according to claim 3 characterized in that the photograph is a reference image, or inter-frame, belonging to a video stream and compressed according to the MPEG compression standard.

7. Use of the method according to one of claims 3 to 6 for the unsupervised detection of the falsification of an area of a photograph.

8. Use of the method according to one of claims 3 to 6 for the detection, in a supervised manner, of the falsification of an area of a photograph.

9. Use of the method according to one of claims 3 to 6 for the search for evidence from a compromising photograph.

10. Application of the method according to one of claims 3 to 6 in specialized software for the search of evidence from digital media.