Thanh Hai Thai

Rémi Cogranne
email: remi.cogranne@utt.fr

Estimation of Primary Quantization Steps in Double-Compressed JPEG Images Using a Statistical Model of Discrete Cosine Transform

Keywords: Digital image forensics, Double JPEG compression, Quantization step estimation, Statistical image model, DCT coefficient analysis

Double compression of images occurs when one compresses twice, possibly with different quality factors, a digital image. Estimation of the first compression parameter of such a double compression is of a crucial interest for image forensics since it may help revealing, for instance, the software or the source camera. This paper proposes an accurate method for estimating the primary quantization steps in double-compressed JPEG images. This original methodology is based on an accurate statistical model of Discrete Cosine Transform (DCT) coefficients that has been proposed in our previous works. We also present a thorough analysis of the double compression properties, carefully taking into account carefully the effect of round-off noise. This analysis is used to derive an accurate range of possible value for quantization of primary DCT coefficients with respect to the secondary quantization step. Using both the statistical model of quantized DCT coefficients and the range of possible values of first quantization step, a model of the twice quantized DCT coefficients is established. Eventually, it is proposed to estimate the primary quantization value by finding, among a set of possible candidates, the one that best match the proposed statistical model in terms of minimal symmetrized Kullback-Leibler (KL) divergence. Numerical experiments on large databases of real images and comparisons with state-of-the-art approaches emphasize the relevance of the proposed method.

I. INTRODUCTION

T HE evolution of digital imaging and information tech- nologies in the past decades has raised a number of information security challenges. Digital images can be easily edited, altered or falsified due to a large availability of lowcost image editing tools and then transmitted via communication network. The credibility and trustworthiness of digital images have been eroded in consequence. Therefore, the field of digital image forensics has emerged in response to the increasing need to verify the authenticity of digital images, see [START_REF] Stamm | Information forensics: An overview of the first decade[END_REF] and references therein for a detailed introduction.

The JPEG standard is by far the most commonly used image format for image storage due to its rather good tradeoff between efficiency in data compression, quality and low computational costs. Hence, JPEG images are involved in many forensics issues, such as authenticity of JPEG compression history [START_REF] Fan | Identification of bitmap compression history: JPEG detection and quantizer estimation[END_REF], steganalysis [START_REF] Holub | Low-complexity features for JPEG steganalysis using undecimated DCT[END_REF] or image forgery detection [START_REF] Bianchi | Image forgery localization via block-grained analysis of JPEG artifacts[END_REF]. Recently, some research works have been paid more attention on forensics issues related to double JPEG compression [START_REF] Bianchi | Image forgery localization via block-grained analysis of JPEG artifacts[END_REF]- [START_REF] Yang | An effective method for detecting double JPEG Compression with the same quantization matrix[END_REF]. In fact, double JPEG compression occurs when an image that has been previously JPEG compressed goes through a second cycle of JPEG compression. Since JPEG image is submitted to IEEE Access, May, 2019 commonly the output of most digital cameras, if a forger carries out some acts of manipulation (e.g. splicing, data hiding) on the original JPEG image and stores it in JPEG format, then the resulting image is JPEG double-compressed. Therefore, the information about double JPEG compression is of important interest for forensic analysts since it could be a clue indicating that the image in question might have been manipulated.

A. STATE OF THE ART

The problem of double JPEG compression typically includes two tasks: detection of double JPEG compression [START_REF] Bianchi | Image forgery localization via block-grained analysis of JPEG artifacts[END_REF], [START_REF] Pevny | Detection of double compression in JPEG images for applications in steganography[END_REF], [START_REF] Yang | An effective method for detecting double JPEG Compression with the same quantization matrix[END_REF]- [START_REF] Bianchi | Detection of non aligned double JPEG compression based on integer periodicity maps[END_REF] and estimation of primary quantization matrix [START_REF] Bianchi | Image forgery localization via block-grained analysis of JPEG artifacts[END_REF]- [START_REF] Galvan | First quantization matrix estimation from double compressed JPEG images[END_REF], [START_REF] Li | Detecting doubly compressed JPEG images by using mode based first digit features[END_REF]. The estimation of primary quantization matrix can be considered as a subsequent and complementary task of the double compression detection. Indeed, detection of double compressed images being usually much simpler on a computational point of view, performing detection and then, only for images classified as double compressed, carry out an estimation of primary quantization matrix would largely reduce computational time. In addition, it can be useful for forensic analysis, once suspecting that an inspected image has been double-compressed, to estimate the primary quantization matrix as such an estimation can be used to identify origin of the image [START_REF] Kee | Digital image authentication from JPEG headers[END_REF], to detect hidden message [START_REF] Pevny | Detection of double compression in JPEG images for applications in steganography[END_REF] and to help image forgery detection [START_REF] Bianchi | Image forgery localization via block-grained analysis of JPEG artifacts[END_REF]. However, the estimation of primary quantization matrix is generally more difficult and complicated than the double compression detection. The latter problem only focuses on finding discriminative features between single and double compression while the former needs to analyze the effect of double compression on DCT coefficients and carry out an exhaustive search to provide an estimate of primary quantization matrix. Moreover the detection of double JPEG compression can be carried out using together all DCT coefficient modes ; on the opposite, estimation of primary quantization step can only rely on DCT coefficients from the corresponding frequency. Therefore, compared with the detection of double compression, much fewer methods have been proposed for estimation of primary quantization matrix in the literature.

Prior research works dedicated to the estimation of primary quantization matrix could be divided into three categories:

1) Pattern-based methods [START_REF] Lukas | Estimation of primary quantization matrix in double compressed JPEG images[END_REF], [START_REF] Pevny | Detection of double compression in JPEG images for applications in steganography[END_REF] are based on peculiarities in the shape of the histogram of DCT coefficients (e.g. double peaks, zeros) due to double compression.

In [START_REF] Lukas | Estimation of primary quantization matrix in double compressed JPEG images[END_REF], the authors discussed three methods to estimate primary quantization steps for the first three lowfrequencies in zig-zag order. Two of them relied on matching the original histograms of individual DCT coefficients with the histograms obtained by calibration while the third one utilized a collection of neural networks to detect patterns caused by different combinations of primary and secondary quantization steps. Empirical results have shown that the neural network-based method provided best overall perfor-mances among those three. Using a very similar approach also based on the shape of histograms of DCT coefficients the authors in [START_REF] Pevny | Detection of double compression in JPEG images for applications in steganography[END_REF] proposed to employ, instead of neural networks, a soft-margin Support Vector Machine (SVM) to train multi-class classifiers in order to estimate primary quantization steps for the first nine low-frequencies in zig-zag order. However, these methods focused only on some small values of primary and secondary quantization steps. Hence, they could not work for high frequencies or low quality factors. 2) Quantization-based methods [START_REF] Farid | Exposing digital forgeries from jpeg ghosts[END_REF], [START_REF] Galvan | First quantization matrix estimation from double compressed JPEG images[END_REF] rely on properties of successive quantizations to detect local minima in the difference of DCT coefficients before and after successive quantizations, which is called the error function. In [START_REF] Farid | Exposing digital forgeries from jpeg ghosts[END_REF], the author proposed to perform a third JPEG compression on the double-compressed image under investigation. By varying the quantization step of this compression, the error function would lead to two local minima corresponding to primary and secondary quantization steps. Partially inspired by this approach, the authors in [START_REF] Galvan | First quantization matrix estimation from double compressed JPEG images[END_REF] proposed a strategy of filtering to mimic the effect of noise in the histogram of DCT coefficients and designed another error function devoted for good localization of primary quantization steps. However, these approaches only worked when the secondary quality factor is higher than the primary. 3) Model-based methods [START_REF] Bianchi | Image forgery localization via block-grained analysis of JPEG artifacts[END_REF], [START_REF] Li | Detecting doubly compressed JPEG images by using mode based first digit features[END_REF] employ a statistical model of DCT coefficients for estimation of primary quantization steps. In [START_REF] Li | Detecting doubly compressed JPEG images by using mode based first digit features[END_REF], the authors proposed to model the distribution of the first digits of quantized DCT coefficients by the generalized Benford's law and employ probabilities of the first digits as features to train multiclassifiers based on Fisher Linear Discriminant (FLD) in order to estimate the primary quality factor. Alternatively, the authors in [START_REF] Bianchi | Image forgery localization via block-grained analysis of JPEG artifacts[END_REF] proposed a heuristic model of double-quantized coefficients and estimated primary quantization steps using the expectation-maximization algorithm. While the statistical modeling of DCT coefficients has been considerably studied in the literature (see for instance [START_REF] Muller | Distribution shape of two-dimensional DCT coefficients of natural images[END_REF]- [START_REF] Thai | Statistical model of quantized DCT coefficients: Application in the steganalysis of Jsteg algorithm[END_REF] and references therein), those models have not been used in the previously mentioned methods for primary quantization step estimation. As a consequence, those methods fail to capture coefficients statistics with high precision which leads to the degradation of ensuing quantization step estimation accuracy. Moreover, the method proposed in [START_REF] Bianchi | Image forgery localization via block-grained analysis of JPEG artifacts[END_REF] obtained unsatisfactory performance in case of secondary quality factor lower than the primary.

The method proposed in the present paper falls within the third category. It is based on the state-of-the-art statistical model proposed in [START_REF] Thai | Statistical model of quantized DCT coefficients: Application in the steganalysis of Jsteg algorithm[END_REF]. This model is used together with a study of quantization and impact to propose a statistical model of double JPEG-compressed image DCT coefficients. Based on this model and on the impact of quantization a method is proposed for estimating with high accuracy the primary quantization step by, roughly speaking, searching for the quantization step that would result in a distribution of DCT coefficients that best matches the empirical distribution of coefficients from the inspected image.

B. POSITION AND CONTRIBUTIONS OF THE PAPER

The present paper addresses the problem of estimation of primary quantization steps in double-compressed JPEG images assuming that secondary quantization table is available in JPEG header and the double JPEG-compressed image is available. The main contributions of this paper are the following:

• In contrast with prior statistical model-based methods, this paper proposes to exploit the state-of-the-art statistical model of once-quantized DCT coefficients that has been established in our previous works [START_REF] Thai | Statistical model of quantized DCT coefficients: Application in the steganalysis of Jsteg algorithm[END_REF]- [START_REF] Thai | Steganalysis of Jsteg algorithm based on a novel statistical model of quantized DCT coefficients[END_REF]. This model can accurately capture statistics of DCT coefficients, which leads to improve considerably the estimation accuracy. • The effect of quantization in the spatial domain, when converting a JPEG-compressed image back into the original spatial domain, has often been omitted from prior-art methods for simplicity. However the resulting round-off noise can heavily impact statistics of secondary DCT coefficients, hence mislead the estimation algorithm. By contrast, this paper statistically analyze double JPEG compression properties and takes into account the effect of round-off noise, then establishes a proper range of values of primary DCT coefficients with respect to the secondary quantization step. Based on this range, the distribution of the secondary DCT coefficients is derived. • The paper designs a method for estimation of primary quantization steps using the proposed model of secondary DCT coefficients. To do this, we propose a simple, yet efficient strategy to filter candidates of primary quantization step. Reducing the set of possible primary quantization step greatly simplifies the estimation which relies on minimizing the symmetrized Kullback-Leibler divergence between the observed empirical distribution of DCT coefficients and the theoretically established model. The proposed approach achieves high estimation accuracy when the primary quantization step is finer than the secondary, while most of existing methods failed is such a case. Numerical experiments on large real image databases with heterogeneous image content and differently customized compression scheme highlight the relevance of the proposed approach.

C. ORGANIZATION OF THE PAPER

The rest of the paper is organized as follows. Section II briefly describes the double JPEG compression operations. Section III recalls the statistical model of primary DCT coefficients that was established in our previous works, establishes the range of values of primary DCT coefficients with respect to a secondary quantization step and proposes the corresponding model of secondary DCT coefficients. Using this model, Section IV proposes the method for estimation of primary quantization steps. Section V presents numerical results on large real image databases. Finally, Section VI provides some discussions and concludes the paper. Throughout this paper, we will use lower case characters, such as x to represent real and integer values and upper case (except for Greek letters) boldface characters X to represent matrices.

II. DOUBLE JPEG COMPRESSION CHAIN

The JPEG compression pipeline has been already detailed in several publications and books, such as [START_REF] Pennebaker | JPEG Still Image Compression Data[END_REF] ; thus, the present paper only recalls the main parts of the JPEG compression that are essential to understand the proposed method. For clarity, the main steps involved in JPEG compression and decompression are recalled in Figure 1 along with the main notations used in the present paper. Besides, without loss of generality, we will also assume for simplicity that the inspected given image is in grayscale, the extension of color images is straightforward by applying the proposed method on each color channel independently. Let us denote X the original uncompressed image; the JPEG compression algorithm starts by dividing the image X into blocks of fixed size of 8 × 8 pixels1 and performing the DCT operation on each block separately:

U 1 = DCT(X). (1)
In the paper U 1 denotes primary unquantized DCT coefficients. The coefficient at location (0, 0) in 8×8 block is called the DC coefficient (for Direct Current because it represents the mean value of the block), the remaining 63 coefficients are called the AC coefficients (for Alternating Current as they correspond to sine wave oscillations in spatial domain). As explained above, prior steps such as color space transformation and chroma subsampling are not presented for the sake of clarity. The next step is the quantization operation that simply consists in dividing each primary unquantized DCT coefficient by the corresponding quantization step extracted from the primary quantization table Q 1 of size 8×8. The final DCT coefficient is eventually rounded to the nearest integer:

C 1 = round U 1 Q 1 , (2)
where the above division represents the element-wise operation and C 1 denotes primary quantized DCT coefficients. We note that for clarity and readability, the present paper describes the process for a single block because, as already explained, all the blocks are processed in exactly the same manner.

The final step of JPEG standard consists in lossless compression, generally using Huffman and Run Length Encoding (RLE) ; this step of lossless compression is not detailed as it does not affect the decoded value of DCT coefficient and ensuing image pixels.

×Q 1 IDCT(•) round(•) DCT(•) round(•) X U 1 C 1 D 1 X 1 U 2 C 2
The JPEG decompression process is carried out by applying the inverse of previous operations in the reverse order, that is: "dequantization" and then inverse DCT (IDCT). Dequantization involves multiplying a primary quantized DCT coefficient by the corresponding quantization step:

D 1 = C 1 • Q 1 , (3)
where the above multiplication is an element-wise operation. The decompressed-JPEG image X 1 results from applying IDCT operation on the primary dequantized DCT coefficients D 1 to return to the spatial domain, and performing the rounding operation:

X 1 = round (IDCT(D 1 )) . (4)
Though the rounding operation is not fundamentally needed (and, hence, could theoretically be avoided) it is almost always applied because most image manipulation software uses integers for better computational efficiency. Besides, it is always applied prior to the JPEG compression which is the main focus of current paper. It is important to note that quantization is the only source of loss of information during the JPEG compression. In addition, it is also important to remind that the quantization operation is used at two difference stages: first for representing quantized DCT coefficients (2) using integers, for storing purposes, and second when decompressing the image for representing pixels' value (4). Double JPEG compression occurs when the image X 1 suffers further JPEG compression cycle with second quantization table

Q 2 : C 2 = round U 2 Q 2 , with U 2 = DCT(X 1 ), (5)
where U 2 and C 2 denote secondary unquantized and quantized DCT coefficients, respectively. One can observe that the rounding of pixels value, after decompression, is essential in the present analysis. Indeed, without this step one would have, from Equations ( 4) and ( 5): U 2 = DCT(X 1 ) = DCT (IDCT(D 1 )) = D 1 . Hence, should this statement holds true, one could decompress and recompress the same image several times with the quantization step without making any modification. This observation is well-known to be false. As already explained, rounding of pixel values before the second JPEG compression is required for two main reasons: first, because most of the image manipulation and processing softwares use integers for efficiency and ( 2) almost all the JPEG compression library applies the DCT from integer-value pixel to integer-value DCT because it is much more efficient computationally.

It must be noted that the present paper only focuses on the problem of aligned double JPEG compression, i.e. the second JPEG compression adopts a DCT grid aligned with the one used by the first compression [START_REF] Bianchi | Image forgery localization via block-grained analysis of JPEG artifacts[END_REF]. The problem of non-aligned double JPEG compression is beyond the scope of the paper. From the secondary quantized DCT coefficients C 2 , one can apply the decompression process to obtain the decompressed image X 2 .

III. STATISTICAL ANALYSIS AND MODELING OF DOUBLE JPEG COMPRESSION

Images in spatial domain are usually modeled statistically using the well-known additive white Gaussian (AWG) model. However, it is well-known, especially in the field of digital image forensics, that the noise variance is slightly different for each pixel. Such differences in pixels noise variance has been successfully used for instance in source camera identification [START_REF] Chen | Determining image origin and integrity using sensor noise[END_REF], [START_REF] Li | Source camera identification using enhanced sensor pattern noise[END_REF]. The methods presented in the present paper found its root in the heteroscedastic pixel noise model [START_REF] Healey | Radiometric CCD camera calibration and noise estimation[END_REF], [START_REF] Foi | Practical Poissonian-Gaussian noise modeling and fitting for single-image rawdata[END_REF] that, taking into account photon counting shot noise, related the expectation of pixels value with their variance.

The extension of the heteroscedastic noise model of DCT coefficient of JPEG image has been studied, see for instance [START_REF] Thai | Camera model identification based on the heteroscedastic noise model[END_REF]- [START_REF] Qiao | Individual camera device identification from jpeg images[END_REF], and is briefly recalled because the present paper is based on this model. Then, this model is extended to study and model double JPEG-compressions.

A. STATISTICAL MODEL OF PRIMARY DCT COEFFICIENTS

Let u 1 and c 1 be random variables representing respectively primary unquantized and quantized DCT coefficients at a frequency. In this paper, DCT coefficients at each frequency are processed separately. For simplicity, the index of frequency is omitted. Based on the heteroscedastic model of uncompressed images X 1 , the statistical model of primary unquantized and quantized DCT coefficients has been already proposed in our previous works. The reader is thus referred to [START_REF] Thai | Statistical model of quantized DCT coefficients: Application in the steganalysis of Jsteg algorithm[END_REF]- [START_REF] Thai | Steganalysis of Jsteg algorithm based on a novel statistical model of quantized DCT coefficients[END_REF] for details and for more justifications.

In order to account for the non-stationarity and heterogeneity in a natural image, the primary unquantized DCT coefficient u 1 can be characterized by the doubly stochastic model as [START_REF] Thai | Statistical model of quantized DCT coefficients: Application in the steganalysis of Jsteg algorithm[END_REF], [START_REF] Thai | Statistical model of natural images[END_REF]:

u 1 ∼ N 0, σ 2 , with σ 2 ∼ G(α, β), (6)
where N µ, σ 2 ) represents the normal distribution, with µ the expected value and σ 2 the variance of the block of 8 × 8 pixels, G(α, β) denotes Gamma distribution with shape parameter α and scale parameter β. As originally proposed in the well-known paper [START_REF] Lam | A mathematical analysis of the DCT coefficient distributions for images[END_REF], the above doubly stochastic model allows the characterizing of probability density function (pdf) f u1 of the primary unquantized DCT coefficient u 1 using the theorem of total probability:

f u1 (z) = 2 π |z| β 2 α-1 2 β α Γ(α) K α-1 2 |z| 2 β , (7)
where Γ(•) denotes the gamma function and K ν denotes the modified Bessel function of second kind of order ν [START_REF] Ryzhik | Tables of Integrals, Series, and Products[END_REF]. This model includes Laplacian model as special case (as α = 1) and Gaussian model as limiting case (as α → ∞).

The statistical model of the primary quantized DCT coefficient c 1 is established by taking into account the impact of quantization operation with quantization step q 1 [START_REF] Thai | Statistical model of quantized DCT coefficients: Application in the steganalysis of Jsteg algorithm[END_REF], [START_REF] Thai | Steganalysis of Jsteg algorithm based on a novel statistical model of quantized DCT coefficients[END_REF]. The probability mass function (pmf) P c1 of the quantized DCT coefficient c 1 is given by

P c1 (k) = P c 1 = k = F |k| -F |k| -1 ∀k ∈ Z * 2F (0) k = 0, (8)
where P A denotes the probability of occurrence of the event A, and

F (k) = 1 2 g(k) K α-1 2 (g(k))L α-3 2 (g(k)) + K α-3 2 (g(k))L α-1 2 (g(k)) , (9)
with

g(k) = q 1 (k + 1 2 ) 2 β and L ν (•)
is the modified Struve function [START_REF] Ryzhik | Tables of Integrals, Series, and Products[END_REF].

Maximum Likelihood (ML) estimation of the model parameters (α, β) from unquantized and quantized DCT coefficients has been also proposed in [START_REF] Thai | Statistical model of quantized DCT coefficients: Application in the steganalysis of Jsteg algorithm[END_REF], [START_REF] Thai | JPEG quantization step estimation and its applications to digital image forensics[END_REF]. As noted above, the proposed model of DCT coefficients is complicated, establishing analytically ML estimates of the parameters is hardly possible. We have proposed to resolve the problem by using numerical optimization method [START_REF] Nelder | A simplex method for function minimization[END_REF].

B. RANGE OF PRIMARY DCT COEFFICIENTS WITH RESPECT TO SECONDARY QUANTIZATION STEP

As noted in Equations ( 3)-( 4), due to the rounding operation in spatial domain the decompression process adds round-off noise in the image X 1 : [START_REF] Li | Detecting doubly compressed JPEG images by using mode based first digit features[END_REF] where ξ represents the random variables (matrix) that represent the quantization noise: ξ = round(IDCT(C 1 Q 1 )) -IDCT(C 1 Q 1 ). One can easily observe that those random variable are independent because the quantization is an element-wise operation, hence, applied on each and every value independently.

X 1 = round (IDCT(C 1 Q 1 )) = IDCT(C 1 Q 1 ) + ξ,
Hence, when performing the DCT operation on the decompressed image X 1 the resulting DCT coefficient u 2 is only an approximated version of the "unrounded" primary dequantized DCT coefficient d 1 :

U 2 = round DCT[X 1 ] , = round DCT IDCT(C 1 Q 1 ) + ξ , = C 1 Q 1 + round (DCT(ξ)) , = C 1 Q 1 + , (11)
where represents the round-off error in DCT domain. To be comprehensive, we should also remind that the impact of "clipping", which occurs when values exceed the quantizer bounds, is assumed to be negligible in this paper. Indeed, in a vast majority of the case, pixels with values over 255 are saturated, or over-exposed, and hence belong to blocks in which pixels value is constant. In such cases all the AC (Alternative Current) DCT coefficients will be zero. As already noted in prior works, see for instance [START_REF] Fan | Identification of bitmap compression history: JPEG detection and quantizer estimation[END_REF], [START_REF] Luo | JPEG error analysis and its application to digital image forensics[END_REF], excluding blocks, in the image, with saturated pixels allows removing the difficulties related to clipping. In what follows, for clarity of presentation and easy of understanding, we will present the method element-wise, and not block-wise. Hence we will use the same variable in lower case and not-boldface font to represent values of elements from previous matrices. Due to the round-off error , the secondary unquantized DCT coefficient u 2 would cluster around integer multiples of the quantization step q 1 . It is widely admitted that, when the quantization step is rather small, that the rounding noise ξ can be modeled as a uniformly distributed random variables in the interval [- 1 2 , 1 2 ). As a consequence, the rounding error in the DCT domain can be modeled as a linear combination of spatial-domain round-off errors. In virtue of Central Limit Theorem (CLT), it is proposed in the present paper to model the error as random variable following a Gaussian distribution with zero-mean and variance of 1 12 [START_REF] Fan | Identification of bitmap compression history: JPEG detection and quantizer estimation[END_REF], [START_REF] Luo | JPEG error analysis and its application to digital image forensics[END_REF]:

∼ N 0, 1 12 . ( 12
)
Consequently, it follows from Eq. ( 5) and [START_REF] Huang | Detecting double JPEG compression with the same quantization matrix[END_REF], that the secondary quantized DCT coefficient c 2 can be rewritten as:

c 2 = round c 1 q 1 + q 2 . ( 13
)
By using properties of the rounding, ceiling and floor function, we can define the range of values of the primary dequantized DCT coefficient d 1 = c 1 q 1 as in the following Proposition: ) for which the quantization steps for both the quality factors QF 1 and QF 2 ) respectively are given as q 1 and q 2 .

Proposition 1. For a given secondary quantization step q 2 and secondary quantized DCT coefficient c 2 , the value of the corresponding primary dequantized DCT coefficient d 1 falls into the range R q2 (c 2 ), defined as:

R q2 (c 2 ) = c 2 q 2 - q 2 2 + , c 2 q 2 + q 2 2 --1 , (14)
where x and x denote the ceiling and floor function, respectively.

Proof: The proof of Proposition 1 is given in Appendix.

Depending on the parity of q 2 , let us study the range R q2 (c 2 ) in two following cases:

• Case 1: q 2 = 2m, m ∈ Z From ( 14), the range R q2 (c 2 ) can be rewritten as:

R q2 (c 2 ) = c 2 q 2 -m - , c 2 q 2 + m + --1 = c 2 q 2 -m - , c 2 q 2 + m - -1 . ( 15
)
It follows from ( 12) that the probability that the error lies outside the interval [-3 2 , 3 2 ] is negligible 2 . Therefore, depending on the value of , the range

R q2 (c 2 ) is reduced to:            R (1) q2 (c 2 ) = c 2 q 2 -m + 2 , c 2 q 2 + m + 1 if ∈ I (1) R (2) q2 (c 2 ) = c 2 q 2 -m + 1 , c 2 q 2 + m if ∈ I (2) R (3) q2 (c 2 ) = c 2 q 2 -m , c 2 q 2 + m -1 if ∈ I (3) R (4) q2 (c 2 ) = c 2 q 2 -m -1 , c 2 q 2 + m -2 if ∈ I (4) (16
) where 3 2 ].

I (1) = [-3 2 , -1[, I (2) = [-1, 0[, I (3) = [0, 1[, I (4) = [1,
• Case 2: q 2 = 2m + 1, m ∈ Z In this case, the range R q2 (c 2 ) is rewritten as:

R q2 (c 2 ) = c 2 q 2 -m-+ 1 2 , c 2 q 2 +m-- 1 2 -1 . (17)
2

The exact probably that

∈ [-3 2 , 3 2 ] is 2 × Φ(3 √ 3) ≈ 2.10 7 .
Similarly, the range R q2 (c 2 ) can be reduced to:

     R (1) q2 (c 2 ) = c 2 q 2 -m + 1 , c 2 q 2 + m + 1 if ∈ I (1) R (2) q2 (c 2 ) = c 2 q 2 -m , c 2 q 2 + m if ∈ I (2) R (3) q2 (c 2 ) = c 2 q 2 -m -1 , c 2 q 2 + m -1 if ∈ I (3) (18)
where

I (1) = [-3 2 , -1 2 [, I (2) = [-1 2 , 1 2 [, I (3) = [ 1 2 , 3 2
]. In the literature, the round-off error is often omitted to simplify the range R q2 (c 2 ). However, as noted in above analysis, due to its variability, the error impact considerably the range R q2 (c 2 ). Moreover, the probability that a value falls in the sub-range R (j) q2 (c 2 ) equals to the one that the error is in the corresponding interval I (j) . Therefore, the values in the range R q2 (c 2 ) do not occur equally but with a different probability due to the variability of .

Corollary 1. Combining all the sub-ranges R (j) q2 (c 2 ), 1 ≤ j ≤ n, the range R q2 (c 2 ) can be given as:

R q2 (c 2 ) = c 2 q 2 - q 2 2 -1 , c 2 q 2 + q 2 2 + 1 . (19)
Proof: From above analysis, by taking the minimum of left bounds and the maximum of right bounds of all the subranges, the proof of Corollary 1 is straightforward.

Corollary 2. Let p j , 1 ≤ j ≤ n, be the probability that the error is in the interval I (j) , and p(k) represents the probability that the value k occurs in the range R q2 (c 2 ). Then, the probability p(k) can be expressed as:

p(k) = n j=1 1 k ∈ R (j) q2 (c 2 ) • p j , (20)
where 1[A] denotes the indicator function of an event A.

Proof: Since the probability that the value k occurs in the range R q2 (c 2 ) equals to a sum of probabilities of occurrence of the value k in each sub-range, the proof of Corollary 2 follows immediately.

C. STATISTICAL MODEL OF SECONDARY DCT COEFFICIENTS

The analysis of round-off effect on the range of primary DCT coefficients allows us to establish an accurate model of secondary DCT coefficients. Let P C2 denote the pmf of the random variable C 2 representing secondary quantized DCT coefficient. Based on the law of total probability, the pmf P C2 can be given as:

P C2 (c 2 ) = P C 2 = c 2 = k∈Rq 2 (c2) , k=q1l P k C 2 = c 2 • P D 1 = k = k∈Rq 2 (c2) , k=q1l P k C 2 = c 2 • P C 1 = l . (21)
In other words, only multiples of q 1 in the range R q2 (c 2 ) are taken into account in order to calculate the probability P C2 (c 2 ). Finally, we derive that

P C2 (c 2 ) = k∈Rq 2 (c2) , k=q1l p(k) • P C1 (l). (22)
Therefore, from the model of primary quantized DCT coefficient C 1 given in [START_REF] Galvan | First quantization matrix estimation from double compressed JPEG images[END_REF], the model of secondary quantized DCT coefficient C 2 can be established straightforwardly.

In order to show the efficiency of the proposed model of secondary quantized DCT coefficients, an experiment is conducted on the image Lena3 with different pairs of quality factors (QF 1 , QF 2 ) = (85, 75), (75, 60), (65, 80). Figure 2 shows the histogram of secondary quantized DCT coefficients at frequency (1,0) with corresponding quantization steps compared with the proposed model. It is noted that the proposed model can capture accurately the statistics of secondary quantized DCT coefficients, even capture specific artifacts in the histogram such as double peaks or zeros (see [START_REF] Lukas | Estimation of primary quantization matrix in double compressed JPEG images[END_REF], [START_REF] Pevny | Detection of double compression in JPEG images for applications in steganography[END_REF] for details about those artifacts).

IV. ESTIMATION OF PRIMARY QUANTIZATION STEP A. FILTERING OF CANDIDATES OF PRIMARY QUANTIZATION STEP

Since q 1 is a discrete integer parameter, a common philosophy is to carry out an estimation algorithm over a set of integers to provide the best candidate following certain criteria. However, in order to avoid conducting an exhaustive search, we propose a simple technique to filter possible candidates of primary quantization step q 1 . Proposition 2. For every c 2 ∈ C 2 , where C 2 denotes the set of values of secondary quantized DCT coefficients, there always exists a corresponding c 1 whose dequantized value c 1 q 1 is in the range R q2 (c 2 ).

Proof: Since R q2 (c 2 ) is the range containing possible values c 1 q 1 with respect to the given value c 2 , the Proposition 2 is straightforward.

Let κ c2 (q) be the function of q with respect to a coefficient c 2 defined as follows:

κ c2 (q) = 1 ∃k ∈ Z | kq ∈ R q2 (c 2 ) = 1 if ∃k ∈ Z such as kq ∈ R q2 (c 2 ) 0 if not. ( 23
)
Then, we define the function κ(q) as the normalized sum of all functions κ c2 (q) for all coefficients c 2 ∈ C 2 :

κ(q) = c2∈C2 κ c2 (q) N , ( 24
)
where N is the cardinality of the set C 2 , i.e. the number of secondary quantized DCT coefficients at a frequency of the image in question. Let S denote the set of candidates of q 1 . Perfectly, the set S could be given as

S = q ∈ Z * κ(q) = 1 . (25)
The proposed filtering technique is illustrated in Figure 3 for two scenarios q 1 < q 2 and q 1 > q 2 . The DCT coefficients at the frequency (0,1) of the image Lena are used in this experiment. It could be noted that in case of q 1 < q 2 , the true primary quantization step q 1 is included in the values smaller than q 2 . Generally, the proposed technique can narrow down the search on the primary quantization step q 1 , which allows to reduce considerably execution time.

In practice, the function κ(q) at the true primary quantization step can have value slightly lower than 1 due to the small effect of truncation errors and modeling error of [START_REF] Fan | Identification of bitmap compression history: JPEG detection and quantizer estimation[END_REF], [START_REF] Thai | JPEG quantization step estimation and its applications to digital image forensics[END_REF]. Therefore, the set of candidates S is proposed as follows:

S = q ∈ Z * κ(q) ≥ 0.9 . (26)

B. PROPOSED ESTIMATION ALGORITHM

After establishing the candidate set S, this paper provides the optimal candidate based on the criteria of symmetrized KL distance. In probability and information theories, the KL divergence [START_REF] Kullback | On information and sufficiency[END_REF], which is an asymmetric discrepancy measure between two probability distributions P 1 and P 2 , is defined as follows:

D KL P 1 P 2 = log dP 1 dP 2 dP 1 . (27)
In case of discrete probability distributions, the KL divergence D KL P 1 P 2 can be given by:

D KL P 1 P 2 = i P 1 (i) log P 1 (i) P 2 (i) . (28)
However, it is noted that D KL P 1 P 2 is not a strict concept of distance since it does not obey the triangle inequality and it is not symmetric in general, i.e. D KL P 1 P 2 = D KL P 2 P 1 . Therefore, in order to quantify the similarity 2, the quantization steps are given by q 1 and q 2 for quality factors QF 1 and QF 2 ) respectively. between two probability distributions P 1 and P 2 , we rely on the symmetrized KL distance given as:

D KL = D KL P 1 P 2 + D KL P 2 P 1 . (29)
To provide the optimal candidate of q 1 , we calculate the symmetrized KL distance between the normalized observed histogram H of secondary quantized DCT coefficients and the theoretical model P C2 under a candidate q ∈ S

D KL (q) = |i|≥2 H(i) log H(i) P C2 (i | θ) + |i|≥2 P C2 (i | θ) log P C2 (i | θ) H(i) , (30)
where i denotes the histogram bin index and θ = (α, β, q, q 2 ) is the parameter vector specifying the probability distribution P C2 . It is noted that in the calculation of D KL (q), the bin index at 0 and ±1 are excluded. The main difficulty in [START_REF] Luo | JPEG error analysis and its application to digital image forensics[END_REF] is that the nuisance parameters (α, β) are unknown in practice, hence the distance D KL (q) could not be computed.

To overcome this difficulty, it is proposed to transform the decompressed image X 2 to the DCT domain, then estimate the parameters (α, β) from unquantized DCT coefficients by the ML method (see Section III-A). The ML estimates are then replaced in [START_REF] Luo | JPEG error analysis and its application to digital image forensics[END_REF]. Finally, the optimal candidate of primary quantization step q 1 is given as the value minimizing the symmetrized KL distance:

q * = arg min q∈S D KL (q). (31)
It should be noted that one can, in fact, estimate the quantization step by find the value that minimizes a different "metric" than the proposed asymmetric Kullback-Leibler divergence. Indeed by simply using in Eq. ( 31) a quantitive comparison between the theoretical distribution P C2 (i | θ) with the empirical observation H(i). We focus in the present paper on the asymmetric Kullback-Leibler divergence because, as shown in Table 1 it has been empirically observed that it provide the best results.

It is important to keep in mind that the present paper focuses on estimating an individual primary quantization step rather than a primary quality factor. This is of great interest in practice because most quantization tables used in digital cameras are often customized by camera manufacturers using their own compression scheme, despite standard quantization tables have been provided by the Independent JPEG Group [START_REF] Pennebaker | JPEG Still Image Compression Data[END_REF]. Therefore, the proposed approach could partially recover the employed primary quantization table, which helps to identify the source camera model [START_REF] Kee | Digital image authentication from JPEG headers[END_REF].

V. EXPERIMENTS

For estimation of primary quantization step, the proposed approach excludes the secondary coefficients whose absolute value is smaller than 1, see [START_REF] Luo | JPEG error analysis and its application to digital image forensics[END_REF]. When an image is highly compressed, i.e. the secondary quantization step is very coarse, all secondary coefficients could be quantized into the interval [-1, 1]. In this case, the estimation is declared undetermined. To the best of our knowledge, no method (including the present one) can deal with this case.

Moreover, the proposed approach could not provide reliable estimation of primary quantization steps when q 1 is a divisor of q 2 (including 1 and q 2 ) because the histogram in these cases is very close to the one generated from single compression, i.e. the DCT coefficients are not technically double-compressed [START_REF] Lukas | Estimation of primary quantization matrix in double compressed JPEG images[END_REF], [START_REF] Pevny | Detection of double compression in JPEG images for applications in steganography[END_REF]. These cases would be our focus in future research.

In this paper, the performance of the method is evaluated using the accuracy metric, which is calculated as a percentage of the number of correctly estimated quantization steps. TABLE 1: Estimation accuracy of the proposed approach for different metrics when the second quantization step is q 2 = 8 and for various primary quantization step q 1 ∈ [START_REF] Stamm | Information forensics: An overview of the first decade[END_REF][START_REF] Chen | Determining image origin and integrity using sensor noise[END_REF]. The various metrics used in the comparison with the proposed method [START_REF] Kullback | On information and sufficiency[END_REF] are respectively the symmetrized Kullback-Leibler divergence (Sym. KL), the Pearson's chi-sqaured, χ 2 , the Maximum Likelihood (ML) the (asymmetric) Kullback-Leibler divergence denoted Asym. KL, the L 1 -norm and the L 2 -norm respectively.

(a) Simulated data

q 1 Sym. KL χ 2 ML Asym. KL L 1 L 2 1 - - - - - - 2 - - - - - - 3

A. VALIDATION OF THE PROPOSED STATISTICAL FRAMEWORK

The proposed statistical framework has shown a possibility of employing a different model of DCT coefficients (e.g. Laplacian, Generalized Gaussian or Generalized Gamma distribution) or a different metric for similarity measure. In our previous works [START_REF] Thai | Statistical model of quantized DCT coefficients: Application in the steganalysis of Jsteg algorithm[END_REF], the proposed model of DCT coefficients has been already shown to outperform other models. Therefore, in this paper, we conduct experiments to validate the choice of symmetrized KL distance over different metrics. Firstly, we conduct an experiment on simulated data. The simulated data is generated using 10000 uncompressed grayscale images of size 512 × 512 in the BOSSBase [START_REF] Bas | Break our steganographic system -the ins and outs of organizing boss[END_REF] (version 1.01). The simulation process is given as follows:

1) The experiment starts from transforming each uncompressed image in the BOSSBase into DCT domain, then estimate the parameters (α, β) from unquantized DCT coefficients at frequency (1,0). 2) From a given couple (α, β), a vector of 4096 coefficients u 1 is randomly generated following the model as in [START_REF] Farid | Exposing digital forgeries from jpeg ghosts[END_REF].

3) The coefficients u 1 are then quantized and dequantized with primary step q 1 , obtained d 1 . 4) The primary dequantized coefficients d 1 are added by noise following the Gaussian distribution as in [START_REF] Bianchi | Detection of non aligned double JPEG compression based on integer periodicity maps[END_REF] to create u 2 .

5) The coefficients u 2 are finally quantized with secondary step q 2 , obtained c 2 . By doing so, we obtain the coefficients c 2 that fit the theoretical model perfectly. Moreover, the fact of using the parameters (α, β) by estimating from a real image for simulation process could provide DCT coefficients close to reality. In this simulation, the step q 2 is fixed to q 2 = 8 and the step q 1 varies in the interval [START_REF] Stamm | Information forensics: An overview of the first decade[END_REF][START_REF] Chen | Determining image origin and integrity using sensor noise[END_REF]. The above simulation process is applied for 10000 uncompressed images in the BOSSBase. Therefore, the simulated data involves 10000 vectors of coefficients c 2 for each primary quantization step q 1 . The experiment is conducted for different metrics, including the Symmetrized KL distance χ 2 -distance, ML method, asymmetric KL divergence, L 1 -norm and L 2 -norm.

Next, we conduct a second experiment on real data, which also relies on the BOSSBase. The uncompressed images are double-JPEG compressed with constant quantization tables, where all steps in the quantization table take the same value. The primary steps vary in the interval [START_REF] Stamm | Information forensics: An overview of the first decade[END_REF][START_REF] Chen | Determining image origin and integrity using sensor noise[END_REF] and the secondary ones are fixed to 8. Then DCT coefficients at frequency (1,0) are extracted for estimation. By doing so, this experiment is carried out in a similar scenario as the first one on simulated data.

The estimation accuracy of the proposed approach for different metrics on the simulated data and real data is shown in Table 1. It should be noted that the performance DCT frequency (zig-zag order) for all values q 1 divisors of q 2 is not presented in this table. The proposed approach using the symmetrized KL distance achieves mostly perfect performance, which validates this choice over other metrics. The performance for other metrics is unsatisfactory when q 1 < q 2 . Surprisingly, the proposed approach based on ML method and asymmetric KL divergence obtains the poorest performance in that case. The high performance of the proposed approach in those experiments shows that the above statistical framework is relevant to model statistics of secondary DCT coefficients. Furthermore, the proposed method can estimate accurately individual primary quantization steps in the case when the primary quantization step is finer than the secondary one.

B. NUMERICAL RESULTS ON LARGE REAL IMAGE DATABASE

In order to emphasize the effectiveness of the proposed approach, we carry out experiments on large real image databases and comparisons with existing methods. Firstly, an experiment is carried out on the BOSSBase. All 10000 grayscale uncompressed images in this database are JPEGcompressed twice using the Matlab JPEG Toolbox [START_REF] Sallee | Matlab JPEG Toolbox[END_REF] with quality factors QF 1 ∈ {60, 65, 70, 75, 80, 85, 90, 95} and QF 2 ∈ {60, 65, 70, 75, 80, 85, 90, 95, 100}. It should be noted that images from the BOSSBase have an important variability in their content. Conducting the experiment on such a large image database can allow us to assess the robustness of the estimation method. Moreover, to assess the performance for different image sizes, those 512 × 512 original images are center-cropped into images of size 256 × 256 and 128 × 128 before conducting double compression. Thus, the final dataset contains 720000 double-compressed images for each image size.

It is noted that the proposed approach is based on statistics of secondary DCT coefficients at each separate frequency. Due to insufficient statistics at high frequencies, the paper only focuses on studying the performance at low and medium frequencies. Table 2 shows the estimation accuracy of the proposed method at the first 10 frequencies in zig-zag order of double-compressed images of size 512 × 512 in the BOSSBase dataset. The column and row of Table 2 represent respectively primary and secondary quality factor. Similarly, the case in which primary step is a divisor of secondary one is not presented in this Table. These results emphasize the high performance of the proposed method when applying on real large image database. When QF 1 < QF 2 (i.e. q 1 > q 2 ), the estimation accuracy at low frequencies is nearly perfect despite high compression. For higher frequencies, it can be observed that the estimation accuracy decreases with the decline of the quality factor. Especially, when QF 1 > QF 2 (i.e. q 1 < q 2 ), the estimation accuracy tends to increase as the primary quality factor decreases. This is due to the fact that when the primary quality factor is very high, the effect of double compression is minor (i.e. the image is close to the one generated from the single compression of QF 2 ), hence the proposed estimator can be mistaken.

Furthermore, the performance of the proposed method for the images with different sizes from the BOSSBase is shown in Figure 4. The estimation accuracy for various DCT coefficients (scanned using the zig-zag pattern order) in the case of QF 1 < QF 2 and QF 1 > QF 2 is reported respectively. The values in this Figure are averaged estimation accuracy considering all combinations of (QF 1 , QF 2 ) studied in Table 2. It can be clearly observed that the estimation accuracy of the proposed method is strictly correlated with the image size. Moreover, the decline of the performance in case QF 1 > QF 2 tends to be faster than the one in case QF 1 < QF 2 .

To further confirm the effectiveness of the proposed approach, an experiment is conducted on the real JPEG image from the Dresden database [START_REF] Gloe | The Dresden image database for benchmarking digital image forensics[END_REF] that is publicly available for download. This database includes 16958 color JPEG images acquired from 73 different camera devices of different models/brands with the highest available resolution and JPEG quality setting, see more details in [START_REF] Gloe | The Dresden image database for benchmarking digital image forensics[END_REF]. The color JPEG compression introduces some additional steps such as color space transformation (e.g. RGB to YCbCr and inverse) and chrominance component downsampling. These steps generate noise into the color components. By conducting experiments on real color JPEG images, we do not only verify the robustness of the proposed method to color noise although it is not yet incorporated in the proposed model, but also show its effectiveness in the real JPEG compression scheme designed by different manufacturers. In this experiment, the color JPEG images are converted into grayscale and then gone through a second JPEG compression using the Matlab JPEG Toolbox [START_REF] Sallee | Matlab JPEG Toolbox[END_REF] with quality factors QF 2 ∈ {90, 80, 70}. The estimation accuracy of the proposed method on those double-compressed JPEG images is shown in Figure 5. It is noted that there is a drop in performance at frequency (0,0). This is due to the fact that the DC coefficient represents the average of pixel values in a block, which causes a difficulty to provide an accurate model due to high heterogeneity in a natural image. Nevertheless, the proposed method can estimate accurately the primary quantization steps at low frequencies. This highlights its practicability at recovering a part of employed quantization matrix in natural JPEG images taken by a digital camera, which could help to identify the image origin.

Considering the scenario of estimating individual primary quantization steps, the paper proposes to carry out a comparison with two recent methods, including the quantizationbased method proposed by Galvan et al. [START_REF] Galvan | First quantization matrix estimation from double compressed JPEG images[END_REF] and the modelbased method proposed by Bianchi et al. [START_REF] Bianchi | Image forgery localization via block-grained analysis of JPEG artifacts[END_REF] 4 . The comparison is conducted on the double-compressed JPEG images with size of 512 × 512 in the BOSSBase. The performance of all the methods is shown in Figure 6. Because Galvan's method cannot be performed in the case QF 1 > QF 2 , its performance is not reported in Figure 6b. As noted, the proposed method outperforms the others in terms of estimation accuracy, especially when the primary quantization step is finer than the secondary.

VI. CONCLUSION, DISCUSSION AND FUTURE WORKS

This paper proposes an accurate method for estimation of primary quantization step in a double-compressed JPEG image. The approach is based on the statistical model of DCT coefficients that has been given in our previous works. By analyzing the double quantization effect and taking into account the round-off noise, the paper establishes an appropriate model of the secondary DCT coefficients. Moreover, by using a simple and efficient filtering technique, a set of possible candidates of primary quantization step is provided. From this set of candidates, the optimal candidate is given following the criteria of symmetrized KL distance.

Numerical experiments on large image databases highlight the relevance of the proposed approach.

The strength of the proposed method is the high estimation accuracy for a wide variety of images with different image contents, image sizes, and quality factors. Moreover, the proposed method can provide reliable estimation of primary quantization steps in the case that the primary quantization step is finer than the secondary while all the prior methods have failed. The accuracy of the proposed method is emphasized when applying on real color JPEG images acquired from different camera models/brands, which shows the robustness of the proposed method to color noise introduced during JPEG compression pipeline. However, the proposed method is more time-consuming than prior-art ones. This drawback is due to the fact that the ML estimation of DCT model parameters is accomplished using a numerical optimization method.

The approach proposed in this paper explore several directions in future researches. Firstly, based on the study of double compression effect in this paper, we could design a statistical test following hypothesis testing theory in order to detect the presence of double-JPEG compression. This test can allow the guaranteeing of a prescribed false-alarm rate, which is crucial in an operational context. It should be noted that such statistical test has not yet studied in the literature. The second research is the estimation of primary quantization steps from non-aligned double-compressed JPEG images. Last but not least, it is of crucial interest to detect and localize copy-paste forgery using inconsistencies in JPEG compression history among different parts in the image under investigation.

APPENDIX. PROOF OF PROPOSITION 1

The goal of this appendix is to provide a proof for the Proposition 1 and, more precisely, to prove Equation [START_REF] Muller | Distribution shape of two-dimensional DCT coefficients of natural images[END_REF].

Let us recall that using that the image compressed only once is decompressed in spatial domain, via multiplication of coefficients c 1 with quantization step q 1 , inverse DCT transformation and rounding of pixels' value. This decompressed image is then compressed a second time with a different quantization step q 2 . Modeling the rounding error of pixels' value with a uniformly distributed random variable allows the writing of Equation ( 13):

c 2 = round c 1 q 1 + q 2 .
Based on the definition of the rounding function and on the previous relation, it immediately follows from that:

c 2 - 1 2 ≤ c 1 q 1 + q 2 < c 2 + 1 2 , (32)
which is equivalent to:

c 2 q 2 - q 2 2 -≤ c 1 q 1 < c 2 q 2 + q 2 2 -, (33)
Taking into account that c 1 and q 1 are both integers, one has:

c 2 q 2 - q 2 2 -≤ c 1 q 1 ≤ c 2 q 2 + q 2 2 --1. ( 34
)
Therefore, we derive

c 2 q 2 + - q 2 2 + ≤ c 1 q 1 ≤ c 2 q 2 + q 2 2 --1. (35)
Since -x = -x , we obtain:

c 2 q 2 - q 2 2 + ≤ c 1 q 1 ≤ c 2 q 2 + q 2 2 --1, (36)
Finally, observing that the product c 1 q 1 is the same quantity as the corresponding primary dequantized DCT coefficient d 1 , it is obvious that the previous Equation (36) corresponds to Equation ( 14) which concludes the proof of Proposition 1.

FIGURE 1 :

 1 FIGURE 1: Illustration of the main steps involved in JPEG compression and decompression and used in the present work for modeling and analyzing the effect of double compression.

5 FIGURE 2 :

 52 FIGURE 2: Illustration of the proposed model of secondary quantized DCT coefficients on the image Lena with different pairs of quality factors (QF 1 , QF 2 ) = (85, 75), (75, 60), (65, 80).The example presented in those figures is the distribution of DCT coefficient at location (2, 2) for which the quantization steps for both the quality factors QF 1 and QF 2 ) respectively are given as q 1 and q 2 .

  q 1 = 11, q 2 = 5

FIGURE 3 :

 3 FIGURE 3: Illustration of the proposed technique for filtering of candidates of primary quantization step using the DCT coefficient at location (2, 2) and with two examples for quality factors 1 , QF 2 ) = (85, 75), (55, 80). As in Figure2, the quantization steps are given by q 1 and q 2 for quality factors QF 1 and QF 2 ) respectively.

  QF 1 > QF 2

FIGURE 4 :

 4 FIGURE 4: Estimation accuracy of the proposed method using one of the first 15 frequencies (in zig-zag order) of images with different sizes from the BOSSBase. The values are obtained by averaging over all (QF 1 , QF 2 ).

FIGURE 5 :

 5 FIGURE 5: Estimation accuracy of the proposed method on the Dresden database.

  QF 1 > QF 2

FIGURE 6 :

 6 FIGURE 6: Comparison in terms of estimation accuracy between the proposed methods and prior work. The comparison has been carried out on all images from the BOSSBase with the original size of 512 × 512 pixels.

  THANH HAI THAI received the Diploma, M.S. and Ph.D degrees in systems optimization and safety from the University of Technology of Troyes, in 2011 and 2014, respectively. He is currently a Researcher with the Institute of Research and Development, Duy Tan University, Da Nang, Viet Nam. His research focuses on statistical image processing, hypothesis testing theory, steganography and steganalysis, and digital image forensics. RÉMI COGRANNE is an Associate Professor at Troyes University of Technology (UTT), France, since 2013. He has regularly been a visiting scholar at Binghamton University between 2014 and 2017. He received his PhD in Systems Safety and Optimization from UTT in 2011, since on, his research focuses on hypothesis testing applied to image forensics, steganalysis, steganography and computer network anomaly detection which lead to more than 60 papers and 3 International patents. He has been the general Chair of ACM Workshop on Information Hiding and Multimedia Security (2019), main organizer of ALASKA steganalysis challenge (https://alaska.utt.fr), Member of IEEE and Elected Member of IEEE Technical Committee Information Forensics and Security (2019â Ȃ Ş2021) submitted to IEEE Access, May, 2019

TABLE 2 :

 2 Estimation accuracy of the proposed method using each and every first ten DCT coefficients (according to the zig-zag pattern order) and over all the images in the BOSSBase with the original size of 512 × 512 pixels.

				(a) Frequency (0,0)										(b) Frequency (0,1)
		95	90	85	80		75		70		65		60		95	90	85	80	75	70	65	60
	100	100.00	100.00	100.00	100.00	99.99	99.99	99.99	99.97	100	-	100.00	100.00	100.00	99.99	99.99	99.99	99.97
	95	-	100.00	100.00	100.00	99.99	99.98	99.98	99.97	95	-	100.00	100.00	100.00	99.99	99.98	99.98	99.97
	90	99.98	-	99.99	99.98		99.98	99.98	99.98	99.97	90	-	-	99.99	99.98	99.98	99.98	99.98	99.97
	85	99.95	99.95	-	99.99		99.98	99.98	99.95	99.95	85	-	99.85	-	99.99	99.98	99.98	99.95	99.95
	80	-	-	99.88	-		99.98	99.95	99.95	99.92	80	-	-	99.23	-	99.98	99.95	99.95	99.92
	75	-	97.77	98.72	99.69		-		99.92	99.89	99.89	75	-	-	-	99.65	-	99.92	99.89	99.89
	70	-	96.75	-	99.40		99.68	-		99.89	99.89	70	-	93.42	94.52	95.74	93.58	-	99.89	99.89
	65	94.26	96.19	96.22	98.07		99.65	99.65	-		99.86	65	-	-	91.12	-	95.16	95.68	-	98.85
	60	92.30	92.31	98.23	92.33		99.21	99.28	99.30	-	60	-	90.71	-	92.21	99.50	97.52	97.17	-
				(c) Frequency (1,0)										(d) Frequency (2,0)
		95	90	85	80	75	70	65	60		95	90	85	80	75	70	65	60
	100	-	100.00	100.00	100.00	99.99	99.99	99.99	99.97	100	-	100.00	99.99	99.97	99.97	99.95	99.95	99.93
	95	-	100.00	100.00	100.00	99.99	99.98	99.98	99.97	95	-	99.99	99.93	99.90	99.90	99.90	99.84	99.84
	90	-	-	99.99	99.98	99.98	99.98	99.98	99.97	90	-	-	99.82	99.80	99.81	99.75	99.76	99.77
	85	-	-	-	99.99	99.98	99.98	99.95	99.95	85	-	99.37	-	99.79	99.75	99.75	99.70	99.70
	80	-	97.57	99.20	-	99.98	99.95	99.95	99.92	80	-	-	99.34	-	99.69	99.62	99.62	99.60
	75	-	-	99.62	96.30	-	99.92	99.89	99.89	75	-	96.77	99.25	98.20	-	99.60	99.42	99.40
	70	-	93.33	94.52	99.58	99.05	-	99.89	99.89	70	-	91.12	-	96.99	95.31	-	99.42	99.40
	65	-	-	-	95.83	96.93	95.06	-	98.86	65	-	87.13	89.72	92.81	95.85	93.17	-	97.69
	60	-	-	92.43	-	96.84	96.77	97.73	-	60	-	85.22	85.89	90.33	90.21	96.23	95.22	-
				(e) Frequency (1,1)										(f) Frequency (0,2)
		95	90	85	80	75		70		65		60			95	90	85	80	75	70	65	60
	100	-	100.00	99.99	99.97	99.97	99.95	99.95	99.95	100	-	100.00	99.99	99.97	99.97	99.95	99.95	99.95
	95	-	99.99	99.93	99.90	99.90	99.90	99.89	99.89	95	-	99.99	99.93	99.90	99.90	99.90	99.86	99.86
	90	-	-	99.85	99.84	99.84	99.82	99.80	99.77	90	-	-	99.82	99.81	99.81	99.77	99.76	99.77
	85	-	-	-	99.84	99.84	99.82	99.80	99.77	85	-	99.59	-	99.79	99.79	99.75	99.72	99.72
	80	-	99.77	99.75	-	99.83	99.82	99.80	99.78	80	-	-	98.94	-	99.75	99.70	99.55	99.58
	75	-	-	99.32	98.11	-		99.79	99.69	99.63	75	-	96.66	98.60	98.87	-	99.70	99.43	99.45
	70	-	91.56	91.77	98.99	94.15	-		99.70	99.56	70	-	-	-	98.76	97.42	-	99.42	99.40
	65	-	-	-	92.62	95.91	95.43	-		97.99	65	-	86.65	90.33	93.21	98.54	93.83	-	98.98
	60	-	-	87.28	-	91.79	95.48	95.03	-		60	-	-	85.57	-	91.16	95.64	94.39	-
				(g) Frequency (0,3)										(h) Frequency (1,2)
		95	90	85	80	75	70	65	60		95	90	85	80	75	70	65	60
	100	99.99	99.96	99.95	99.90	99.88	99.82	99.80	99.65	100	-	100.00	99.97	99.95	99.90	99.88	99.83	99.78
	95	-	99.63	99.60	99.51	99.41	99.45	99.32	99.40	95	-	99.94	99.86	99.85	99.74	99.67	99.62	99.45
	90	98.68	-	99.38	99.31	99.25	99.29	99.30	99.29	90	-	-	99.86	99.85	99.70	99.58	99.49	99.40
	85	97.57	98.06	-	99.11	99.04	98.72	98.17	98.23	85	-	98.23	-	99.12	99.03	98.98	98.82	98.63
	80	-	-	94.53	-	98.37	96.92	97.44	96.31	80	-	-	97.78	-	98.42	98.11	97.90	97.21
	75	-	89.37	93.12	95.22	-		97.08	97.13	96.26	75	-	90.39	94.47	96.12	-	97.99	97.85	96.84
	70	-	85.32	-	89.18	90.29	-		94.62	92.38	70	-	86.89	-	90.41	92.05	-	96.79	96.66
	65	79.83	84.19	83.54	86.74	88.35	90.03	-		92.10	65	-	83.18	84.24	88.55	90.02	90.84	-	94.83
	60	77.87	79.99	82.61	83.13	83.48	84.76	87.72	-		60	-	80.67	82.46	83.86	85.18	88.51	88.92	-
				(i) Frequency (2,1)										(j) Frequency (3,0)
		95	90	85	80	75		70		65		60			95	90	85	80	75	70	65	60
	100	-	100.00	99.97	99.95	99.92	99.90	99.87	99.82	100	-	100.00	99.97	99.95	99.92	99.90	99.87	99.82
	95	-	99.97	99.90	99.90	99.73	99.73	99.68	99.61	95	-	99.98	99.90	99.86	99.80	99.73	99.68	99.61
	90	-	-	99.72	99.70	99.58	99.41	99.23	99.17	90	-	-	99.72	99.70	99.63	99.66	99.37	99.33
	85	-	97.95	-	99.20	99.03	99.01	98.72	98.65	85	-	98.54	-	99.24	99.18	99.15	98.88	98.65
	80	-	95.34	97.52	-	98.92	98.40	97.77	97.75	80	-	-	98.07	-	98.79	98.46	98.45	98.31
	75	-	90.35	92.39	97.22	-		98.59	97.70	97.53	75	-	90.04	91.96	95.54	-	96.75	95.64	95.30
	70	-	87.46	-	92.40	93.73	-		96.41	95.82	70	-	86.23	-	90.25	91.65	-	95.59	95.19
	65	-	-	84.28	87.28	90.82	91.38	-		94.47	65	-	83.46	84.28	86.88	91.59	92.06	-	93.43
	60	-	80.89	83.33	-	88.57	90.89	92.32	-		60	-	80.32	81.72	83.77	86.83	90.94	90.99	-

submitted to IEEE Access, May, 2019

In this paper, it is assumed that the size of the given image is a multiple of 8. Otherwise, the JPEG compression standard uses the padding by duplicating the very last pixels.submitted to IEEE Access, May,

submitted to IEEE Access, May, 2019

submitted to IEEE Access, May, 2019

submitted to IEEE Access, May, 2019 T.H.Thai & R.Cogranne: Estimation of Primary Quantization Steps in Double-Compressed JPEG Images

The reference image Lena can be downloaded, for instance, from https: //www.ece.rice.edu/~wakin/images/.

submitted to IEEE Access, May, 2019 T.H.Thai & R.Cogranne: Estimation of Primary Quantization Steps in Double-Compressed JPEG Images

submitted to IEEE Access, May, 2019

The Matlab code of this method has been provided at https://iapp.dinfo.unifi.it/.submitted to IEEE Access, May, 2019

This work was supported by the Conseil Régional de Champagne-Ardenne, the Grand Troyes County and the European Regional Development Fund under the program Cybersec of University of Technology of Troyes.


Jacques Haiech
email: haiech@unistra.fri.

Marie-Claude Kilhoffer

Using data mining to explore calmodulin bibliography

Keywords: data mining, scientometry, calcium signal, calmodulin

In this chapter, we present a strategy and the technics to approach a scientific field from a set of articles gathered from the bibliographic database, Web of Science. The strategy is based on methods developed to analyze social networks. We illustrate the use of such strategy in studying the calmodulin field. Such method allows to structure a huge number of articles when writing a review, to detect the key opinion leaders in a given field and to locate his own research topic in the landscape of the themes deciphered by our own community.

We show that the free software VosViewer may be used without knowledge in computing science and with a short learning period.

Introduction

The number of publications has increased exponentially in the last 40 years. It is becoming almost impossible to perform a bibliographic analysis using a traditional approach by gathering all new articles and analyzing them. We need to amplify our intelligences to be able to tackle the tsunami of new publications and to get a coherent landscape describing the evolution of a given scientific field.

Such an approach has been developed in scientometry, a discipline that aims to analyze science by means of quantitative and qualitative indicators. This discipline is linked to the humanities and therefore, the methods and strategies developed by scientometrists have not yet perfused in Biology. The strategy has been used to analyze the emergence of synthetic biology but by a team of sociologists [START_REF] Raimbault | Mapping the Emergence of Synthetic Biology[END_REF].

In this article, we aim to present some of the strategies and tools used in scientometry and as an example, to show how such an approach may help to understand the function of calmodulin.

The number of articles containing calmodulin in the title is higher than 10000 and over 40000 when seeking for calmodulin in the abstract or the keywords. As a human being, it is almost impossible to scan this full set of articles and to extract a coherent and logical set of knowledge. Data mining is a set of technics allowing to analyze data from different points of views in order to extract useful knowledge, mainly by analyzing the relations between the different data or by detecting common patterns. Such analysis may be performed on a huge set of data and data mining is therefore often associated with Big Data.

In our case, the number of papers published on calmodulin cannot be considered as enormous, but it is big enough to escape our human abilities. We will use the study of calmodulin to illustrate the capabilities of data mining and the strategies used to extract knowledge from an unstructured set of publications. Such strategies and technics may be used in any scientific field.

We have realized that the use of this strategy by students allows them to better approach a new field of knowledge. Moreover, they can take into account the temporal evolution of the field and to tackle it via the most significant publications.

Materials and definitions

Corpus:

A corpus is a set of articles and is defined by three parameters: the name of a data base, a request and a date. In our case, we will use either PUBMED [START_REF] Vellay | Interactive text mining with Pipeline Pilot: a bibliographic web-based tool for PubMed[END_REF] or Web of Science [START_REF] Li | Web of Science use in published research and review papers 1997-2017: a selective, dynamic, cross-domain, content-based analysis[END_REF] as bibliographic data base. Both data bases allow to download the result of a request under several possible formats (Figure 1).

Analyzing and visualizing:

In this paper, we will use Vosviewer version 1.6.8 as the tool to analyze and visualize our data [START_REF] Van Eck | Software survey: VOSviewer, a computer program for bibliometric mapping[END_REF].

The use of PUBMED files allows to create maps of researchers based on coauthorship links and maps of terms based on co-occurrences in the same publications of keywords either provided by the authors or by the curators of the bibliographic databases.

On top of the previous maps, the use of Web of Science files allows one to easily create maps of scientific publications, scientific journals, researchers, or research organizations based on either cocitation relations (i.e., multiple items being cited by the same publication) or bibliographic coupling relations (i.e., multiple items citing the same publication).

VosViewer contains an algorithm able to extract terms from titles and abstracts using natural language processing tools. Basically, a term is a suite of adjective(s) and noun(s) finishing by a noun. The algorithm is working only on English text. A map of term is built by linking two terms when they are present in the same publication. As we will see in methods and results, we will see that each available tool allows to explore an aspect of the gathered corpus.

Methods and Results

3.1. The general strategy in data mining is composed of three steps (Figure 2):

1.
Building the corpus by extracting the data from any source, by loading and by transforming the data including the possibility to cure the data 1) Analyzing the corpus by using tools allowing either to detect common patterns in the corpus or between subsets of the corpus or to detect relationships between specific properties of items of the corpus. In a bibliographic corpus, an item is an article and properties of an article are authors, title, abstract, year of publication, …. 2) Visualizing the results from the previous analyzes using either tools to represent clusters or graphs. To illustrate our method, we will tackle the study of calmodulin, the stereotypical calcium binding protein involved in the modulation of numerous cellular events [START_REF] Berchtold | The many faces of calmodulin in cell proliferation, programmed cell death, autophagy, and cancer[END_REF][START_REF] Haiech | Mutant analysis approaches to understanding calcium signal transduction through calmodulin and calmodulin regulated enzymes[END_REF].

To build the corpus, we will use at first a very simple request, namely looking for calmodulin in the title of the publications. We obtain in PUBMED 10641 publications and in Web of Science 14789 publications.

The difference between the two corpus is in part due to the presence of meeting abstracts and notes in the Web of Science database. We obtain 10389 items when we select only articles and reviews in the Web of Science corpus.

In the following part of the paper, we will use the complete corpus obtained from the Web of Science database (14789 items) onJune12 th ,2018.

What are the main themes and their temporal evolutions in the calmodulin field?

To answer this question, after loading the corpus in the Vosviewer software, we use a tool allowing to create a map based on the text from titles and abstracts of the publications. Vosviewer may tag adjectives and nouns, extract terms composed of adjectives and nouns finishing by a noun and compute a score of relevance for each term. However, to improve the significance of the term set, it is needed to iteratively build a thesaurus file to remove non-significant terms in our studied field and to pinpoint the putative synonyms (see the thesaurus file as supplemental material S1). Such file is a text file with two columns separated by a tabulation. The two columns are with the heading "label" and "replace by".

Two terms are linked if they appear in the same publication. The strength of the link is the number of publications where the two terms co-occur. It is a parameter that we set to 50. Then, we choose to select the 200 most relevant terms. Vosviewer may draw the network named the term map. The default parameters given by Vosviewer were used to draw this map (Figure 3). To perform the clustering, VosViewer uses a smart local moving [START_REF] Waltman | A smart local moving algorithm for large-scale modularitybased community detection[END_REF] algorithm that we may optimize using two parameters (resolution and min.cluster size under the analysis signet). Using resolution equal to 1 and min.cluster size to 10, we obtain 4 clusters (Figure 3), one nucleated around calmodulin antagonist (blue color), one around calmodulin dependent protein kinase II (green color), one around calmodulin binding site (red color) and one around heart and modulation of calcium channel (yellow color). The cluster one is the most ancient and the cluster 4 the most recent. By increasing the resolution parameter, the initial clusters are subdivided into smaller ones. For instance, the cluster "calmoduline binding site" may be split into a cluster dealing with the characterization of calcium sites and another focusing on calmodulin binding peptides/targets.

At this point, we have one alternative:

1) Modifying the corpus to focus on one specific cluster or sub-cluster by adding terms to the initial request 2) Continuing with the initial corpus to explore other questions.

We will go along with the second part of the alternative.

How to find the main authors in the different topics and the key articles to read?

There are three maps that allow to answer those two questions of the section [START_REF] Klavans | Which Type of Citation Analysis Generates the Most Accurate Taxonomy of Scientific and Technical Knowledge[END_REF]:

1) The citation map creates a link between two publications when one of them cites the other, 2) The co-citation map links two publications when they are cited by the same article of the corpus, 3) The bibliographic coupling map creates a link between two articles of the corpus when both cites the same document [START_REF] Weinberg | Bibliographic coupling: A review[END_REF].

Each of those maps may be build with publications or authors as nodes (Figure 4). The citation map using authors as nodes (Figure 4A) has been built keeping the most significant 100 authors. This map pinpoints the five main scientific communities working in the calmodulin field. The first one (in red) does not show a central author nucleating the community although CB Klee is playing a central role, the second one (in green) appears to be organized around fives authors (E Miyamoto, K. Fukunaga, T Yamauchi, TR Soderling and H. Schulman) the third one (in blue) is clearly built around A Means, the fourth one appears to be centralized around H Fujisawa and I. Kameshita and finally, the fifth one is a small cluster of two authors, Anderson and Wu.

This first map allows to discriminate five scientific communities also pinpointing five main topics and their evolutions:

1) Cluster 1: This cluster gathers scientists that worked on calmodulin per se, namely the structure and the calcium binding properties of calmodulin. 2) Cluster 2: Scientists in this cluster mainly worked on the role of calmodulin kinases and specifically calmodulin kinase II in brain. The small cluster 5 is a spinoff of this cluster 2. The authors are working on calmodulin kinase II but in heart 3) Cluster 3: This cluster is structured around the work of AR Means on calmodulin kinase kinases and, recently, mainly on CaMKK2 4) Cluster 4: The work of the scientists in this cluster mainly deals with the couple calmodulin kinases and the phosphatases that reverse the phosphorylation of the calmodulin kinases.

The co-citation map has been constructed by keeping the 100 most significant articles (Figure 4B). In this map, we distinguished three clusters.

1) The first cluster presents papers published between 1970 up to 1982 and dealing with the primary characterization of calmodulin (purification, calcium binding properties, primary sequence, cloning the gene of calmodulin).

2) The second cluster gathers the publications describing mainly the 3D structure of calmodulin and calmodulin complexed with target proteins or peptides, for the period ranging from 1985 to 2003. 3) Finally, the third cluster deals with the calmodulin kinases.

Using this tool, it is possible to select for each cluster two publications that are the most cited. We may consider that the selected six papers constitute the fundamental publications to read (Table I). We may see that those six papers may be considered as an excellent starting point for a newcomer to enter the calmodulin field. Let notice that this choice of publications has been realized without any "à priori" or implication of knowledge from experts.

Cluster Publication Reference 1

Calmodulin plays a pivotal role in cellular regulation. Cheung WY.

[10] 1 Calmodulin. Klee CB, Crouch TH, Richman PG.

[11]

2 Solution structure of a calmodulin-target peptide complex by multidimensional NMR. Ikura M, Clore GM, Gronenborn AM, Zhu G, Klee CB, Bax A.

[12]

2 Target enzyme recognition by calmodulin: 2.4 A structure of a calmodulin-peptide complex.

Meador WE, Means AR, Quiocho FA.

[13]

3 Neuronal Ca2+/calmodulin-dependent protein kinases.

Hanson PI, Schulman H.

[14]

3 Regulation of brain type II Ca2+/calmodulin-dependent protein kinase by autophosphorylation: a Ca2+-triggered molecular switch. Miller SG, Kennedy MB.

[15]

Table 1: For each cluster from figure 4B, the two most cited reference were selected.

The third map is the bibliographic coupling map (Figure 4C). From a technical point of view, it is the exact opposite of the co-citation map. Co-citation map is a way to represent the "current paradigm" of the domain. It is a retrospective analysis. On the contrary, the bibliographic coupling map seems to be a prospective analysis and to detect the trends of the scientific domain. To build this map, we have selected the 200 most cited publications. Keeping the default parameters of Vosviewer for the clustering, we obtained 5 clusters:

1. The first cluster (in red) integrate work on the structure of calmodulin-complexes with target proteins or peptides and work on new calcium dyes using the properties of such complexes. The papers in this cluster were published around the year 2000. 2. The second cluster (in green) gathers works dealing with the biochemical purification and characterization of calmodulin and calmodulin interacting proteins. Small molecules interacting with calmodulin are used to perturb the system to decipher the molecular mechanism. The work in this cluster peaks around 1980. 3. The third cluster (in blue) is focusing on the CaM kinase II and specifically its role in the brain.

Most of the work is done around 1990. 4. The fourth cluster (in yellow) is mainly dealing with CaM kinases in the cardiac tissue and/or the modulation of ion channels. The publications in this cluster are the most recent ones, peaking around 2005. 5. Finally, the fifth cluster (in violet) links most of the work on calmodulin dependent nitric oxide synthase mainly in endothelial cells, starting around 1990.

These clusters appear to describe the trends in the calmodulin field.

Combining our different maps, we can summarize the state of the art in the calmodulin field, the six publications to read for the beginners in the field and the putative trends (Figure 5). Calmodulin has been discovered in 1968 as a calcium dependent regulator of phosphodiesterase and the name calmodulin was coined to the protein in 1977. At first, the scientific community aimed at purifying and characterizing the protein and cloning its genes. In parallel, calmodulin interacting proteins started to be identified and the molecular mechanism of interaction was investigated. Simultaneously, the interaction of calmodulin with small molecules was studied, mainly in Japan and triggered a subdomain of investigation on calmodulin antagonists.

The number of calmodulin interacting proteins rapidly increased up to several hundred. The 3D structure of calmodulin was solved first by X ray crystallography and then by NMR.These results were rapidly followed by the structure of calmodulin complexed to target proteins domains or synthetic calmodulin interacting peptidess.

Among the large family of calmodulin interacting proteins, the kinase family attracted many scientists. Calmodulin dependent kinase II became one of the most studied calmodulin target and its role was mainly investigated in brain function.

More recently, the discovery of mutations in calmodulin linked to cardiac pathologies, prompted the field exploring the role of calmodulin dependent enzymes in the regulation of cardiac physiology.

Finally, as the oxidative stress appeared to play a specific action in oncogenesis, calmodulin dependent nitric oxide synthase became a hot topic in the field.

Nowadays, as shown in the Figure 6, the number of publications with the term "calmodulin" in the title is decreasing. The number of publication was at the acme between 1985 and 1995. We may assume that nowadays, the scientific questions are more global and does not tackle the role of one given protein, as important as it can be.

In this chapter, we have shown how to use tools first developed in the field of social network analysis to tackle questions as diverse as:

1) What are the different subtopics of a given scientific field? 2) Who are the key opinion leaders in a given topic?

3) What are the main publications in a given subtopics? 4) How a scientific field has evolved upon time?

We have illustrated the use of such method in the frame of the evolution of the calmodulin field but this approach may be used in any field and may help to answer other questions such the ranking of different scientific institutes for a given scientific domains.

Notes

In this section, we will cite some limits of the strategy.

The Corpus:

It is important to work with large corpus (few hundred publications at least) in order to get meaningful and robust result.

On the other hand, it is important to polish the request to get a corpus specific of the studied domain. For instance, using a request seeking calmodulin in the abstract and not only in the title, will gather papers with all kind of calcium binding proteins and will not focus on calmodulin.

The thesaurus file:

The thesaurus file is important to get a significant network of terms when analyzing the different subtopics of a scientific field.

The aim is to remove all the terms that are often present in the abstract but does not bring any meaningful information. For instance, the terms of the request have to be removed or general term such as biology.

Building the thesaurus file is an iterative and subjective process. We have noticed that during this iterative process, removing one term may change the geometry of the network. However, the clustering of the terms are not profoundly modified.

The co-citation analysis

The co-citation analysis allows to retrospectively analyze the field. However, when the same type of technics are used in the field, we end up with technical papers that are not pertinent for a specific field.

For instance, the paper describing the technics to make a western blot or to measure the protein concentration must be removed when doing a co-citation analysis.

Figure 1 :

 1 Figure 1: To create the corpus, we use either the PUBMED database (left panel) or the Web of Science database (right panel) using the simple request "calmodulin" in the title field. The download of the corpus is done in PUBMED under the format MEDLINE and in WoS under the format text.

Figure 2 :

 2 Figure 2: This scheme illustrates the strategy of data mining when using bibliographic data.

Figure 3 :

 3 Figure 3: Term map created with Vosviewer version 1.6.8 by extracting significant terms from abstract and titles. The links are drawn when two terms appear together in more than 50 items.The 200 most significant terms were kept using a thesaurus file to remove general terms in the field or by regrouping synonyms (Thesaurus file as supplemental materials)

Figure 4 :

 4 Figure 4: This figure presents the citation map (panel A), the co-citation map (panel B) and the bibliographic coupling map (panel C) obtained with Vosviewer version 1.6.8 and the parameters discussed in the section Methods and Results.

Figure 5 :

 5 Figure 5: Combining the results from the different maps. Colors for the term map and the bibliographic coupling map are the same as those used in the figures 3 and 4 for the visualization of the different clusters.

Figure 6 :

 6 Figure 6: Number of publications per year with the term "calmodulin" in the title.