Eric Benoit

Keywords: uncertainty, fuzzy scale, measurement science

come

INTRODUCTION

The concept of fuzzy scale was introduced in order to add properties to nominal scales and ordinal scales, and by the way, to establish a link between strongly defined measurements used for example in physical sciences and weakly defined measurements used in behavioural sciences as for example in psychophysics [START_REF] Finkelstein | Widely, strongly and weakly defined measurement[END_REF]. Actually, except in Physics, the Measurement Science is more and more concerned by weakly defined measurements throw for example Psychophysics or Economy but also with quantities derived from heterogeneous sources.

Since their first introduction, fuzzy scales were exclusively used to perform a precise representation of a unique property manifestation. In a measurement context, it is now admitted that the representation of a property manifestation needs to expose the imprecision and the uncertainty related to the definition of the property and to the measurement process. A first stage of a measuring process using a ratio scale or an interval scale is able to produce a set of numerical values. This set of values is usually processed to perform a type A measurement, or to characterize the measuring system in order to perform a posterior type B measurements. Within the context of fuzzy scales, a second stage using a fuzzy scale performs a translation from a numerical representation to a fuzzy lexical one. The purpose of this paper is to present how this representation can be used to manage uncertainty. It first introduces the concept of fuzzy scale, and the existing approaches to manage uncertainty. The general approach of evidence theory is recalled then applied to build a probabilistic and an possibilistic representation of the same uncertainty.

FUZZY SCALES

Fuzzy scale are characterized by their capability to represent a similarity relation between manifestations with a fuzzy equivalence relation between quantity values [START_REF] Benoit | Properties of fuzzy nominal scales[END_REF]. The quantity values are expressed as Lexical Fuzzy Subset (LFS), i.e. as fuzzy subsets of lexical terms.

This new family presently includes the weakest fuzzy scales: the fuzzy nominal scales, and metrical fuzzy scales [START_REF] Benoit | An ordinal metrical scale built on a fuzzy nominal scale[END_REF]. The last ones are fuzzy nominal scale enhanced with the definition of a distance. The fuzzy ordinal scales are still under studies. Expressing quantity values with LFSs can be performed by measurements with physical sensors, or with psychophysics experiments. For the first case, quantity values are first expressed on a conventional numerical space with a ratio scale or at least with an interval scale. At this step, empirical relations are mapped into numerical relations. Then a second translation maps the numerical expression of quantity values into a fuzzy lexical expression. The second mapping, based on a fuzzy scale, transfers only the fuzzy equivalence relation from the empirical relational set to the representational relational set in the case of a fuzzy nominal scale, and transfer a distance operator in case of a metrical scale.

Fuzzy scales are defined by fuzzy symbolisms <E,T,R> where: • E is the set of the manifestations of a quantity, • T is a set of terms used to qualify measurements, • R is a fuzzy relation on ExT.

Let denote FS(T) the set of fuzzy subsets of T, and D the injective mapping defined as [START_REF] Finkelstein | Widely, strongly and weakly defined measurement[END_REF] From a mathematical point of view, the measurement values, expressed on FS(T), are elements of a subset of a |T|dimensional cube where |T| is the cardinality of T.

MODELLING UNCERTAINTY

Model defined on a numerical space

In a first trivial approach, the set of the numerical expression of the observations, i.e. the set of quantity values expressed in a numerical space, is used to produce a measurement result defined on this numerical set. The usual way is to define a probability density function (PDF) that models the set of quantity values (see GUM). An approximation of the probability density function by a parametric function allows analytical computation on PDFs. Another way also compatible with GUM is to model the set of quantity values with a possibility function (PF) that supports

D: E FS T ( ) → e E ∈ ∀ D e ( ) t T ∈ eRt { } =

UNCERTAINTY IN FUZZY SCALES BASED MEASUREMENTS

Eric Benoit

LISTIC, Universite de Savoie, Annecy, France urn:nbn:de:gbv:ilm-1-2011imeko-020:2

Joint International IMEKO TC1+TC7+TC13 Symposium

August 31 st -September 2 nd, 2011, Jena, Germany urn:nbn:de:gbv:ilm-1-2011imeko:2 also the approximation with parametric functions [START_REF] Dubois | Probability-Possibility Transformations, Triangular Fuzzy Sets, and Probabilistic Inequalities[END_REF].

The second stage is to perform a projection of this function on the set of LFSs. The result is a PDF (or a possibility function) on LFSs. The advantage of this approach is the possibility to use the existing mathematical tools to create the PDFs or the possibility functions. The disadvantage is that the fuzzy scales need a lot of parameters to be defined [START_REF] Benoit | An ordinal metrical scale built on a fuzzy nominal scale[END_REF]. Then a function with few parameters is mapped into a function with a lots of parameters, and the parametrization becomes useless. Another disadvantage is the impossibility to use this method for psychophysical measurements due to the fact that the numerical stage is avoided.

Models defined on LFSs

In a second approach, each observation is first expressed as a Lexical Fuzzy Subset of terms. Then a parametric function is created on LFSs to represent the set of quantity values. As mentioned in previous studies, the relations and operators that can be used to define the parametric functions are limited by the nature of the scale. For fuzzy nominal scales, the available relation is a fuzzy equivalence relation ~. The metrical fuzzy scale defines a distance between LFSs. Let first define a general family of parametric functions on LFSs when the measurement is performed with a fuzzy nominal scale. If f h,m is such that [START_REF] Benoit | Properties of fuzzy nominal scales[END_REF] then f h,m is equivalent to a probability density function on FS(T) with mean m.

If

h is such that h(0) = 1, then f h,m verifies (3)
and is f h,m is equivalent to a possibility function which kernel includes m.

We can simply verify that d(a,b) = (1 -(~)(a,b)) is actually a distance with a saturation effect. At this step ,we may introduce the next assumption:

σ << 1

where σ is a standard deviation defined on FS(T).

With this assumption, the saturation of d doesn't concern our problem. But the standard deviation on FS(T) need to be defined.

The definition 1 can be extended to the case of metrical fuzzy scales: If f h,m verifies Eq. ( 2) or Eq. ( 3), f h,m is respectively equivalent to a probability distribution or a possibility distribution.

At this step, we can conclude that with a distance, or a fuzzy equivalence between LFSs, it is possible to define parametric PDFs or PFs centred on a LFS.

The available methods to define the parameters are similar to the well known methods used on numerical spaces. Concerning the translations between PDFs and possibility functions, a general approach based on the Dempter-Shafer evidence theory [START_REF] Yamada | Probability-possibility transformation based on evidence theory[END_REF] is directly applicable.

Evidence theory

The Dempster-Shafer evidence theory, also named the transferable belief model (TBM) is a general approach for uncertainty management based on the assignment of probabilities to sets. It generalises both the probability theory and the possibility theory [START_REF] Smets | Data fusion in the transferable belief model[END_REF]. The main element of the TBM is the frame of discernment denoted Ω and presented as a set of elementary events, or as a discourse set for the expression of the belief. The knowledge is then expressed with a mapping denoted m and called belief function from the set of subsets of Ω, denoted 2 Ω , to the set [0,1]. This mapping represents the distribution of a weight of belief over the elements of 2 Ω . It verifies: [START_REF] Dubois | Probability-Possibility Transformations, Triangular Fuzzy Sets, and Probabilistic Inequalities[END_REF] The set {A| m(A) > 0} is called the set of focal elements of m, and the couple ({A| m(A) > 0}, m ) is called a body of evidence of a variable with values on Ω. Two measures Bel and Pl are defined on the subsets of Ω.

( When the focal elements are singletons, then the belief function is qualified as Bayesian. In this case, the two measures Bel and Pl are equal to a probabilistic measure P. The belief function m is then equivalent to an assignment of probabilities to individual values i.e. equivalent to a PDF.

) (6) m FS T ( ) ∈ f h m , s ( ) s FS T ( ) ∈ ∑ 1 = max f h m , s ( ) s FS T ( ) ∈ ( ) 1 = m FS T ( ) ∈ m ∅ ( ) 0 = m A ( ) A 2 Ω ∈ ∑ 1 = Bel A ( ) m B ( ) B A ⊆ ∑ = Pl A ( ) m B ( ) B A ∩ ∅ ≠ ∑ = urn 5
When the focal elements are sets ordered by the inclusion operator, called nested sets or consonant sets, Bel and Pl are respectively the necessity measure N and the possibility measure Π of a possibility distribution derived from m. As a mass function, the belief function is represented by a surface

In [START_REF] Yamada | Probability-possibility transformation based on evidence theory[END_REF] Yamada proposed a probability-possibility transformation based on the evidence theory. The principle is to identify a set of disjoint subsets of Ω computed from a PDF, then to compute a new body of evidence made of consonant subsets. We propose in this paper to build a parametric set of disjoint subsets in order to model a PDF within the framework of evidence theory. Then the Yamada's approach is applied to build the corresponding possibility distribution.

Within the context of metrology, the frame of discernment is the set all individual measurement values, and a measurement result is interpreted as a body of evidence of a quantity. It expresses the known representational information about a quantity.

Build method for PDFs and PFs

Let S ⊂ FS(T) a finite set of measurement values, being the frame of discernment. Let s 0 ∈ S being a measurement value that will play the role of mean value.

Let F = {S i | 1 ≤ i ≤ n} being an ordered family of n non-empty subsets of S defined by

S i = {s | d i ≤ d(s,s 0 ) < d i+1 }.
Where d is a distance on S as defined in Chap. 3.2. and d 1 = 0.

A analogous definition on an euclidian plane will define a family of rings, all centred on s 0 . Fig. 2. Example of an ordered set family centred on s 0 and built with a non-euclidian distance. Fig. 3. Another example on a mono-dimentional space Let F P a set of singletons defined by

F P = {{s}| s ∈ S 1 ∪ S 2 ∪ ... ∪ S n }.
Let m p denotes a Bayesian belief function defined by: m p ({s}| s ∈ S i ) = h(d i ) where h is a monotonic decreasing function with domain [0,+∞] and codomain [0, +∞[ that verifies ,

and h(d n+1 )=0, where |S i | denotes the cardinality of S i .

We can remark that (F P ,m p ) is a body of evidence defining a probability distribution. The value s 0 has the same semantic than the mean in usual PDFs defined on interval scales or ratio scales. Due to the scales used in our case, the addition operator is not allowed for computations in the set of LFSs. So the mean cannot be computed with the usual formula. The value s 0 is then obtain by minimization of the cumulative distance to the data.

d 6 S 4 S 1 h d i ( ) S i ⋅ i 1 = n ∑ 1 = s 0 d 6 m p ({s}) F π F π k 1 k n ≤ ≤ { } = F π k S h
The set family F is also characterized by a distance d on LFSs and a set {d i | 1 ≤ i ≤ n+1}. The distance d is induced by the scale and cannot be considered as a parameter except for a calibration process.

The set {d i | 1 ≤ i ≤ n+1} is related to discrete aspect of the method. The influence of this information the PDF or the PF is small and, in a first approximation, it can be reduced to the parameters Δd and n. The set become {d

i = (i -1)Δd | 1 ≤ i ≤ n+1}.
Finally, h is the main source of parametrization. It can be defined as a continuous parametric function, or as a discrete set of n values. The usual statistical methods are then applied to estimate h parameters.

DISCUSSION

This paper shows that uncertainty can be managed in the case of measurements performed with fuzzy nominal scales and metrical fuzzy scales. It provide a method to represent uncertainty on fuzzy lexical sets with the two main framework that are the probability theory and the possibility theory. The Dempster-Shafer theory is used mainly to build a possibility distribution from a probability distribution.

The capability to manage uncertainty directly on fuzzy lexical sets is an important advance for behavioural sciences that cannot express their measurement result on numerical spaces.

Author: BENOIT Eric, LISTIC, Polytech Annecy-Chambery, University of Savoie, B.P. 80439, 74944 Annecy le Vieux Cedex, France, eric.benoit@univ-savoie.fr

m π F π k ( ) F π k h d k ( ) h d k 1 + ( ) - ( ) = S h h 1 = k ∑ ⎝ ⎠ ⎜ ⎟ ⎜ ⎟ ⎛ ⎞ h d k ( ) h d k 1 + ( ) - ( ) = s 0 d 6 S F π 4 m π F π 4 ( ) h(d 4 ) -h(d 5 ) π s ( ) Π s { } ( ) Pl s { } ( ) m π B ( ) B s { } ∩ ∅ ≠ ∑ = = = s S k ∈ ∀ πs ( ) , m π F π h ( ) h k = n ∑ = F π k h d k ( ) S j h d j ( j k 1 + = n ∑ + =

  . • Let h be a monotonic decreasing function with domain [0,1] and codomain [0, +∞[. • Let ~ be the fuzzy equivalence relation on FS(T) that characterize a fuzzy nominal scale • Let f h,m be a parametric function with domain FS(T), codomain [0, +∞[ and defined by f h,m (x) = h(1 -(~)(x,m)), where (~)(a,b) gives the membership function of the relation a ~ b. (for any a in FS(T), (~)(a,a) = 1 ).

  . • Let h be a monotonic decreasing function with domain [0,+∞] and codomain [0, +∞[. • Let d be the distance on FS(T) that characterizes a metrical fuzzy scale. • Let f h,m be a parametric function with domain FS(T), codomain [0, +∞[ and defined by f h,m (x) = h(d(x,m))

Fig. 1 .

 1 Fig. 1. Sematic use in this paper to represent belief functions.As a mass function, the belief function is represented by a surface

Fig. 4 .

 4 Fig. 4. Belief function m p defined on the singleton set F P Let now follow the Yamada's approach. A new family of sets is built from the family F.

  By definition these sets are nested. It is then possible to define a belief function m π which body

Fig. 5 . 5 .

 55 Fig. 5. Belief function m π defined on the singleton set F πThe possibility distribution is directly deduced from the belief function m π.


Amel Tuama

Frédéric Comby

Marc Chaumont

Frederic Comby

Source Camera Model Identification Using Features from contaminated Sensor Noise

Keywords: Camera Model Identification, POL-PRNU, CFA, Co-occurrences matrix, Feature Extraction, Rich model

This paper presents a new approach of camera model identification. It is based on using the noise residual extracted from an image by applying a wavelet-based denoising filter in a machine learning framework. We refer to this noise residual as the polluted noise (POL-PRNU), because it contains a PRNU signal contaminated with other types of noise such as the image content. Our proposition consists of extracting high order statistics from POL-PRNU by computing co-occurrences matrix. Additionally, we enrich the set of features with those related to CFA demosaicing artifacts. These two sets of features feed a classifier to perform a camera model identification. The experimental results illustrate the fact that machine learning techniques with discriminant features are efficient for camera model identification purposes.

Introduction

Source camera identification is one of the major interests in image forensics. It is the process of deciding which camera has been used to capture a particular image. The problem of establishing the origin of digital media obtained through an imaging device is important whenever digital content is presented and is used as evidence in the court. The general structure of a digital camera consists of lens system, filters, color filter array (CFA), imaging sensor, and digital image processor. The sensor is an array of rows and columns of photo-diode elements, or pixels. To produce a color image, a color filter array (CFA) is used in front of the sensor so that each pixel records the light intensity for a single color only. An interpolation algorithm is used to generate the missing colors values from adjacent pixels. All these elements can be used in extracting features in order to identify a camera device.

There are two families of methods for camera identification. The first one is based on producing a model, for example a PRNU, and then compute the correlation between a given image and the model of a specified camera. The second one is based on features extraction on a machine learning approach.

From the first family of camera identification methods, a reliable one for identifying source camera based on sensor pattern noise is proposed by Lukas et al [START_REF] Lukas | Digital camera identification from sensor pattern noise[END_REF]. Due to imperfections in sensor manufacturing process, the Photo Response Non-Uniformity (PRNU) is a major source of pattern noise. This makes the PRNU a natural feature for uniquely identifying sensors.

Choi et al [START_REF] Choi | Source camera identification using footprints from lens aberration[END_REF] proposed to use the lens radial distortion as a fingerprint to identify source camera model. Each camera model expresses a unique radial distortion pattern that helps to identify it.

Dirik et al [START_REF] Dirik | Source camera identification based on sensor dust characteristics[END_REF] proposed the device identification from sensor dust in digital single lens reflex cameras (DSLR). This problem arises due to the dust particles attracted to the sensor. When the interchangeable lens is removed, a dust pattern is created in front of the imaging sensor. Sensor dust patterns are used as artifacts on the captured images to identify the camera device.

On the other hand, we have the second family of camera identification methods related to features extraction and machine learning . Bayram et al [START_REF] Bayram | Improvements on source camera model identification based on cfa interpolation[END_REF] explored the CFA interpolation process to determine the correlation structure presented in each color band which can be used for image classification. The main assumption is that the interpolation algorithm and the design of the CFA filter pattern of each manufacturer (or even each camera model) are somewhat different from others, which will result in distinguishable correlation structures in the captured images.

Kharrazi et al [START_REF] Kharrazi | Blind source camera identification[END_REF] identified a set of 34 image features that can be used to uniquely classify a camera model. The proposed features are color features, Image Quality Metrics (IQM), and wavelet domain statistics. Celiktutan et al [START_REF] Celiktutan | Blind identification of source cell-phone model[END_REF] used a set of binary similarity measures and a set of Image Quality Metrics to identify the source cell-phone.

Our approach is a mix of the two families of methods since we use a polluted PRNU in a machine learning framework. The polluted PRNU, that we called POL-PRNU, is the sensor noise but also some residual linked to the content of the image. In our approach, extracting the polluted PRNU from a single image leads to an easy way to extract statistics from an image (co-occurrences and color features from polluted PRNU). Indeed, the set of images used to train the classifier will be lightly scattered thus limiting the overfitting effect. Additionally we propose to use a bigger set of features (compared to the classical machine learning approaches) in order to better describe the statistics.

This paper is structured as follows. Section 2 explains the classical approach to compute PRNU. Section 3 presents all the details of our approach, from POL-PRNU extraction to the features computed from co-occurrences and CFA interpolation. In section 4, we describe the experiments, the results, and the database used for experiments. Finally, we conclude in Section 5.

Preliminaries

Camera sensor consists of a large number of photo detectors called pixels which convert photons to electrons. Each pixel in a digital camera's sensor records the Fig. 1. Sample image and its residual noise amount of incident light that strikes it. Slight imperfections in manufacturing introduce small amounts of noise in the recorded image. This noise is spatially varying and consistent over the time and can therefore be used for forensic purposes. It has a stochastic nature and is unique for each sensor. This makes it an ideal candidate for forensic applications, such as camera identification [START_REF] Fridrich | Digital image forensic using sensor noise[END_REF]. Generally, most PRNU-based image forensic techniques extract the residual noise from image by subtracting the denoised version of the image from the image itself as in equation( 1):

N = I -F (I), (1)
where I is the image, F (I) is the denoised image, and F is a denoised filter. Wavelet based denoising filter is recommended and it is used in most cases because it provides the least amount of traces of the scene [START_REF] Fridrich | Digital image forensic using sensor noise[END_REF].

In order to extract the PRNU of a camera, multiple images are averaged. At least 50 images are used to calculate the reference pattern K c [START_REF] Fridrich | Digital image forensic using sensor noise[END_REF] of a known camera C as in equation [START_REF] Choi | Source camera identification using footprints from lens aberration[END_REF].

K c = n i=1 (N i I i ) n i=1 I 2 i . (2)
A common approach to perform a comparison is to compute the Normalized Cross-Correlation which measures the similarity between the reference pattern K c and the estimated noise N of an image under test which is of unknown source [START_REF] Fridrich | Digital image forensic using sensor noise[END_REF]. Normalized Cross-Correlation is defined as:

ρ(N, K c ) = (N -N ).(K c -K c ) N -N . K c -K c . (3)
Where N and K c are the means of N and K c , respectively. By applying equation( 1) on an image I, we obtain the residual noise. The residual noise is a sum of different noise. One of them is the sensor pattern noise PRNU. Other types of noise, such as image content, may pollute the PRNU and are part of the residual noise given in equation ( 1). An example is provided in Figure [START_REF] Lukas | Digital camera identification from sensor pattern noise[END_REF] shows an image and its residual noise which contains some clear parts of the scene.

In this paper, we will consider only the residual noise and call it polluted PRNU (POL-PRNU). This POL-PRNU will then be used for extracting discriminant features. In a machine learning framework, the polluted PRNU is beneficial for the learning process. Indeed, the set of images for a given camera better fill the space and make the obtained cloud in the feature space more spread. Finally, this paper shows that extracting features of high dimension achieves very good results even if the learning database is small.

Proposed method

Camera model identification approach based on machine learning is used to classify the camera based on discriminant features extracted from images. In our approach we extract the features directly from what we called the POL-PRNU. The scheme presented in Figure 2 shows the functional diagram of our proposal. In general, the image is decomposed into its three color channels (r, g, b) considering the central 1024x1024 pixel image block. The POL-PRNU of the image is obtained by subtracting from the original image its filtered version by a wavelet denoising filter. Two sets of features are extracted from POL-PRNU for classification. The following two sub-sections describe the theoretical aspects of the major parts of our approach.

POL-PRNU Extraction

First of all after decomposing the image into its three color channels, the central block 1024x1024 is extracted. Using a small block from the original size reduces the computational complexity, and speeds up the matching process. In [START_REF] Li | Empirical investigation into the correlation between vignetting effect and the quality of sensor pattern noise[END_REF], the authors prove that the false-positive rate FPR decreases as the size of the image block is greater, which reaches the minimum when the block size is 1024x1024 pixels.

Our POL-PRNU N is extracted by subtracting the denoised version of the image from the image itself I [START_REF] Lukas | Digital camera identification from sensor pattern noise[END_REF] as in equation [START_REF] Lukas | Digital camera identification from sensor pattern noise[END_REF]. For the denoising process, a wavelet based denoising filter, F (I), is used based on a Wiener filtering of each wavelet sub-band for each channel as in [START_REF] Lukas | Digital camera identification from sensor pattern noise[END_REF].

In order to suppress all artifacts introduced by color interpolation and JPEG compression, a periodic signal of pattern noise, called the linear pattern L, is extracted by subtracting the average row (respectively average column) from each row (respectively column) of N from each color channel separately [START_REF] Fridrich | Digital image forensic using sensor noise[END_REF]. This leads to the three linear patterns corresponding to each color channel, noted L r for red channel, L g for the green channel, and L b for the blue channel.

Finally, the three linear patterns are combined into one pattern, noted L by using the conversion formula from RGB to gray-scale as in equation( 4). Extracting the features from the recombined linear patterns will be more reliable due to the fact that the three linear patterns are highly correlated and provide a compact information for the classifier [START_REF] Fridrich | Digital image forensic using sensor noise[END_REF].

L = 0.3.L r + 0.6.L g + 0.1.L b . (4)

Description of Features

Co-occurrences matrix The promising aspects of rich models approach [START_REF] Fridrich | Rich models for steganalysis of digital images[END_REF] can be adapted to extract co-occurrences of a POL-PRNU image. Rich models can play a potential role to provide a good model for forensics applications, especially, in forgery detection and localization [START_REF] Qiu | A universal image forensic strategy based on steganalytic model[END_REF][START_REF] Cozzolino | Image forgery detection through residual-based local descriptors and block-matching[END_REF]. Indeed co-occurrences are a very good way to describe the statistics of some data owning neighborhood relations, which is the case for POL-PRNU images. Calculating the co-occurrences of the POL-PRNU allows a reduction of the dimension and gives a good representation of the statistical properties of fingerprint. The co-occurrences feature vector is made of joint probability distributions of neighboring residual samples. In our case, the residual is the POL-PRNU image which is explained in section 3.1. We use four-dimensional co-occurrences matrices formed by groups of four horizontally and vertically adjacent residual samples after they were quantized and truncated as follows: R ← trunc T (round( L/q)), [START_REF] Kharrazi | Blind source camera identification[END_REF] where trunc T is a function to minimize the residual range with T ∈ {-T, ..., T }, round(x) gives the nearest integer value of x, L is the linear pattern of the POL-PRNU given in equation ( 4), and q ∈ {1, 1.5, 2} is the quantization step.

The final co-occurrences matrix will be constructed from horizontal and vertical co-occurrences of four consecutive values from R of equation ( 5). The horizontal co-occurrence matrix C h d is computed as follows:

C h d = 1 Z |{(i,j) | R i,j = d 1 , R i,j+1 = d 2 , R i,j+2 = d 3 , R i,j+3 = d 4 }|, ( 6
)
where Z is the normalization factor, with R i,j ∈ N is the coefficient from the matrix R at position (i, j) ∈ {1, ..., n} 2 , d = (d1, ..., d4) ∈ {-T, ..., T } 4 with T = 2.

Equivalently, we can compute the vertical co-occurrences matrix.

Color Dependencies The underlying assumption is that, the CFA interpolation algorithms leave correlations across adjacent pixels of an image. In digital cameras, the color filter array is placed before sensor to produce the colored image. The CFA is usually periodic and forms a certain pattern. The missing color components are interpolated using existing neighbor color components. The CFA pattern and the way of colors interpolation are important characteristics of the camera model and can be used in the camera identification process [START_REF] Bayram | Improvements on source camera model identification based on cfa interpolation[END_REF].

In this section we will describe the features extracted from the L r , L g , and L b by computing local dependencies or periodicity among neighboring samples. The normalized cross-correlation is computed between the estimated linear pattern from POL-PRNU of color channels and their shifted version as in [START_REF] Filler | Using sensor pattern noise for camera model identification[END_REF]. For each color channel pair (C1, C2), C1, C2 ∈ {L r , L g , L b } and shift 1 ∈ {0, ..., 3}, 2 ∈ {0, ..., 3}. The normalized cross correlation between two matrices is defined as:

ρ(C1, C2, ) = i,j (C1 i,j -C1)(C2 i-1,j-2 -C2) i,j (C1 i,j -C1) 2 i,j (C2 i-1,j-2 -C2) 2 , ( 7
)
where ρ is the normalized cross correlation, = [ 1 2 ] T is the 2D shift, C1 and C2 are sample means calculated from matrices C1 and C2 respectively. This step results in 96 features which are the result of six combinations of color channels by 4 × 4 shifts of 1 and 2 .

Classification

A Support Vector Machine (SVM) constructs a hyperplane, or a set of hyperplanes, in a high dimensional space which can be used for classification. The effectiveness of the SVM depends on the selection of kernel function, and the kernel's parameters [START_REF] Hsu | A practical guide to support vector classification[END_REF].

Using a kernel function provides a single point for the separation among classes. The radial basis function (RBF), which is commonly used, maps samples into a higher dimensional space that can handle the case when the relation between class labels and attributes is nonlinear.

Projecting into high-dimensional spaces can be problematic due to the socalled curse of dimensionality. As the number of variables under consideration increases, the number of possible solutions also increases exponentially. The result is that the boundary between the classes is very specific to the examples in the training data set. The classifier has to handle the overfitting problem, so as it has to manage the curse of dimensionality [START_REF] Bengio | The curse of dimensionality for local kernel machines[END_REF].

In our case, the training and testing sets have 100 instances each, and the number of features is 10860 which is considered much larger than the number of instances. Here, we have to proceed the learning process with a small data base and a large dimension. Thus, the overfitting and the curse of dimensionality problem may occurs.

Fortunately, when the SVM uses the cross validation procedure, the cost parameter that controls the over/under-fitting phenomenon, is set to a value that allows a better handling of the curse of dimensionality problem and then, can prevent the overfitting problem. A cross validation procedure splits the original training data into one or more training subsets. More precisely, the v-fold cross validation divides the training set into v subsets of equal size, v-1 subsets are used for training and the rest subset is left for testing.

Experimental results

Data Acquisition

The Dresden Image Database is designed to fill the needs for digital image forensics applications by providing a useful resource for investigating camera-based image forensic methods [START_REF] Gloe | The dresden image database for benchmarking digital image forensics[END_REF]. It provides 16,000 authentic digital full-resolution natural images in the JPEG format, and of 1,500 uncompressed raw images. It covers different camera settings, environments and specific scenes, facilitate rigorous analyses of manufacturer, model or device dependent characteristics and their relation to other influencing factors. In our experiments 14 different camera models were used, as outlined in table 1. A set of 100 images for the training and an another one of 100 images for the test are used from Dresden database for each camera model. As a result 1400 images for training and 1400 images for testing were used from 14 camera model which are randomly selected.

Experimental Protocol

Since each color channel is denoised separately, an image is decomposed into its three color channels (R, G, B). It is recommended that when image blocks are used in forensic investigation, they should be taken from the image center before POL-PRNU extraction stage. This will reduce false positive rate [START_REF] Li | Empirical investigation into the correlation between vignetting effect and the quality of sensor pattern noise[END_REF]. The images from the training and testing sets are cropped to obtain the 1024x1024 central images. The essential step is to extract POL-PRNU from all images by applying wavelet denoising filter on the original image. This step then followed by subtracting the denoised image from original as explained in section 3.1.

Two sets of features are extracted from linear pattern of POL-PRNU of each image. The first set is the co-occurrences matrix which consists of 10764 co-occurrences, and the additional interesting features of CFA interpolation characteristics.

Camera identification method Result(%) CFA 86.93 Co-occurrences 96.91 Compared method in [START_REF] Filler | Using sensor pattern noise for camera model identification[END_REF] 88.23 Proposed method 97.81 Table 3. Overall average identification rates for all the tested algorithms.

Conclusion

This paper proposes an algorithm for identifying camera sources combining techniques based on sensor pattern noise and machine learning. The algorithm is mainly composed of extracting two sets of features from the noise residual POL-PRNU. The first set is the co-occurrences matrix. The second set is the color dependencies from normalized cross correlation of the three color channels and their shifted versions . These sets of features served as input to the SVM which is used as a classifier. The effectiveness of the method for source camera model identification, was tested on a set of images from the Dresden data-base.

The results illustrate the efficiency of the proposed method since it provides an identification rate of 97.81%. Compared to Filler's method [START_REF] Filler | Using sensor pattern noise for camera model identification[END_REF], we increase the identification rate by 9.58% since it only achieved 88.23% of identification on the same data set.

One problem related to the PRNU correlation based methods is their weak detection rate if geometrical transformations such as, cropping or scaling, have been performed. The direct detection will not succeed because of the desynchronization introduced by additional distortion [START_REF] Goljan | Camera identification from cropped and scaled images[END_REF].

Our future work include enhancing the proposed method to a better classification accuracy by improving the former feature set, and considering the problem of the geometrical transformations. Adding an unknown class will also be one of the perspectives, as an class, to handle models which are not in the training set.

Fig. 2 .

 2 Fig. 2. The proposed system framework

For the feature normalization step, we used the method of min-max scaling for both training and testing sets. In this approach, the features will be rescaled, to a specific range [0,1]. This will avoid attributes in the greater numeric ranges dominating those in the smaller ranges. For the classification, LIBSVM package was used [START_REF] Chang | Libsvm: A library for support vector machines[END_REF] with the Radial Basis Function (RBF) and v-fold cross validation scheme. Although SVM is a binary classification model, LIBSVM package performs multi-classification by using one-versus-rest (OVR) approach. We used the kernel parameter γ = 2 -7 and cost parameter C = 4096 for the SVM after examining a grid search over a range of values. For γ ∈ {2 3 , 2 2 , 2 1 , ..., 2 -15 } and C ∈ {2 15 , 2 14 , 2 13 , ..., 2 -5 } as is recommended in [START_REF] Hsu | A practical guide to support vector classification[END_REF]. The training and testing sets consisted of 100 images each for each camera model. The method was implemented under corei7 processor with memory of 16 gega bytes. For the computation cost, the feature extraction process took few seconds for each image, while the training process took 30 minutes.

Filler et al [START_REF] Filler | Using sensor pattern noise for camera model identification[END_REF] proposed a method for camera model identification which aims to classify camera models using some features. We have implemented this method for comparison purposes on the same set of images from Dresden database. The later method [START_REF] Filler | Using sensor pattern noise for camera model identification[END_REF] proposed features are concerning statistical moments, cross correlation between color channels, block covariance, and cross correlation of the linear pattern. The images are cropped to 1024x1024. We did not perform the step of reducing feature space.

Table 2. Confusion matrix of the proposed method for the fourteen camera models, the symbol '-' refers to the values less than 1%.

Results and Discussion

A comparison is performed between our method and three other experiments.

From the proposed method, we take the first set of features (the co-occurrences matrix) and perform it alone. The experiment resulted in 96.91% as an identification accuracy. This proves the potential role of the statistical features represented by co-occurrences matrix.

The second experiment is performed by taking the set of CFA interpolation features alone from our proposed method. It gave a result of 86.93% of accuracy. This is considered acceptable but not enough, and still less than the result of the first experiment of the co-occurrences computed on the POL-PRNU.

The method presented in [START_REF] Filler | Using sensor pattern noise for camera model identification[END_REF] is tested under a similar conditions as discussed in section 4.2. This method only achieved 88.23% as an average identification accuracy. This low result may be explained by the use of probabilities of first order.

Back to our method, we gathered the two sets of proposed features. By combining the co-occurrences computed on the linear pattern from equation 4 and the CFA features, we gain almost 1% accuracy. The computation of the cooccurrences on the linear pattern is the most important features and the addition of CFA features allows to improve the efficiency. Our method achieved an average accuracy of 97.81%. We can see from the table 2 that some of the models are identified with a very high accuracy. For example, Samsung-L74wide, and Canon-Ixus55 achieve 99.91%, and 99.57% respectively. Also, most of the other camera models do so, except for the Sony-DSC-H50, and Sony-DSC-W170. They achieve the lowest rates of 93%, and 93.94% respectively, and this is, maybe, PRNU structure is very close between the two cameras.

Table 3 shows all the mentioned comparisons with their accuracy rates. Finally, we conclude that our method always performs better than the compared method. This is due to the strength of the descriptive features of the