Digital Imaging Sensor Identification (Further Study)

Transcription

1 Digital Imaging Sensor Identification (Further Study) o Chen, Jessica Fridrich *, and iroslav Goljan Department of Electrical and Computer Engineering SUY Binghamton, Binghamton, Y ABSTRACT In this paper, we revisit the prolem of digital camera sensor identification using photo-response non-uniformity noise (PRU). Considering the identification tas as a joint estimation and detection prolem, we use a simplified model for the sensor output and then derive a aximum Lielihood estimator of the PRU. The model is also used to design optimal test statistics for detection of PRU in a specific image. To estimate unnown shaping factors and determine the distriution of the test statistics for the image-camera match, we construct a predictor of the test statistics on small image locs. This enales us to otain conservative estimates of false rejection rates for each image under eyman- Pearson testing. We also point out a few pitfalls in camera identification using PRU and ways to overcome them y preprocessing the estimated PRU efore identification. 1. ITRODUCTIO The prolem of estalishing the origin of digital media otained through an imaging sensor is important whenever digital content is presented as silent witness in the court. For example, in a child pornography case, estalishing solid evidence that a given image was otained using a suspect s camera is oviously very important. The identification technology could also e used to lin a camcorder confiscated inside a movie theater to other, previously pirated content. In these applications, it is necessary to lin an image or a video-clip to a specific piece of hardware and not just a camera rand or model [1 3]. To solve this prolem, an equivalent of iometrics for cameras or fingerprint is needed. One of the first fingerprints for digital imaging sensors were defective pixels (hot and dead pixels) [4]. Luas et al. [5] proposed the PRU and its detection inside a given image using correlation in a manner similar to digital watermaring methods. In this paper, we also use the PRU ut construct the estimation and detection methods from a model of the sensor output using the apparatus of statistical signal estimation and detection. This enales us to estimate the detection errors more accurately and mae etter use of availale data. In particular, the numer of images needed to estimate the PRU can e significantly smaller than what was reported in [5]. oreover, y estimating the error proailities for each image separately, rather than for all images from the camera, more reliale results can e otained. In Section, we descrie a simplified sensor output model that will e used in this paper. Estimation of PRU and its detection is detailed in Section 3. The predictor of the test statistics is descried in Section 4. The eyman-pearson testing methodology employed in this paper appears in Section 5. Experimental results for 5 cameras are discussed in Section 6. In Section 7, we point out the importance of preprocessing the PRU to decrease the false alarm rate (false camera identification). The paper is summarized in Section 8, where we also outline future research directions. Everywhere in this paper, oldface font will denote vectors of length specified in the text, e.g., X and Y are vectors of length n and X[i] denotes the i-th component of X. Unless mentioned otherwise, all operations among vectors, such as product, ratio, raising to a power, etc., are element-wise. The norm of X is denoted as X = X X with n X Y [] i [] i eing the dot product of two vectors. Denoting the mean values with a ar, the normalized i X Y 1 correlation is ( XX) ( YY) corr( XY, ). XX YY * fridrich@inghamton.edu; phone ; fax

2 . IAGIG SESOR OUTPUT ODEL The chain of processing that occurs in digital cameras is quite complex and varies greatly for different camera types and manufacturers. It includes signal quantization, white alance, demosaicing (color interpolation), color correction, gamma correction, filtering, and, optionally, JPEG compression. Because the processing details are not always easily availale (e.g., hard-wired or proprietary), we decided to use a simplified model [6] that captures the most essential elements of typical in-camera processing. This enales us to develop a low-complexity camera ID procedure applicale to a wide spectrum of cameras. We acnowledge that a more accurate model tailored to a specific camera would liely produce more reliale camera ID results at the cost of increased complexity. Let I[i] e the signal in one color channel at pixel i, i = 1,, n, generated y the sensor efore demosaicing is applied and let Y[i] e the incident light intensity at pixel i. Here, we assume that the pixels are indexed, for example, in a row-wise manner. Dropping the pixel indices for etter readaility, we use the following model of the sensor output I g ( 1K) Y q, (1) where g is the color channel gain, is the gamma correction factor (typically, 1/.), K is a zero-mean multiplicative factor responsile for PRU (the sensor fingerprint), and, s, r, q stand for the following noise sources: dar current, shot noise, read-out noise, and quantization noise. The gain factor g adjusts the pixel intensity level according to the sensitivity of the pixel in the red, green, and lue spectral ands to otain the correct white alance. We remind that all operations in (1) are element-wise. Because the dominant term in the square racet in (1) is the light intensity Y, we can factor it out. Using the first two terms in Taylor expansion of (1 + x) = 1 + x + O(x ), from (1) we otain s r (0) (0) I I I K, () (0) where I (gy) is the sensor output in the asence of noise; is a complex of independent random noise components. 3. CAERA IDETIFICATIO ETHODOLOGY The camera identification methodology is a joint estimation and detection prolem. First, we estimate the PRU K from a set of images taen y the camera. Then, using hypothesis testing we detect the presence of the term I (0) K in a specific image I whose origin is in question. In this section, we descrie the details of oth tass. The first step is host signal rejection to improve the SR etween the signal of interest and oserved data. We suppress the influence of the noiseless image I (0) y sutracting from oth sides of () an estimate Iˆ (0) F() I of I (0) otained using a denoising filter F (0) (0) (0) (0) W IIˆ IKI I ˆ + ( I I) K, or W IK. (3) The term is a comination of with the additional distortion introduced y the denoising filter. Woring with the noise residual W significantly improves the SR for our signal of interest IK and thus improves the reliaility of the camera identification process. However, the denoising filter also shapes the signal we are trying to estimate or detect and it also maes the noise highly non-stationary. For example, in textured areas is larger and the signal IK is attenuated y a multiplicative factor that depends on local texture. We use a wavelet ased denoising filter F that removes from images Gaussian noise with variance F (in this paper, we used F = 3.) ore details aout this filter can e found in our previous wor [5] or in the original pulication [7].

3 3.1 L estimation of PRU We now explain the methodology for estimating the PRU K from images I 1,, I otained y the same camera. We will assume for simplicity that the images are relatively smooth and non-saturated, such as lue sy images. Because we can mae aritrary test images with the camera, this assumption is reasonale. For such images, the model (3) is approximately accurate. From (3), we have for each = 1,, W I K W I Iˆ Iˆ I (0) (0),, F( I). (4) Assuming that for each pixel i the sequence is WG (white Gaussian noise) with variance 1[ i], [ i],, [ i],, the log-lielihood of otaining the measured data W / I given K is L ( /( ) ) ( W / I K) log( /( I) ) 1 1 /( I ) W I K. (5) Taing partial derivatives of (5) with respect to individual elements of K and simplifying, we finally otain the L estimate for the PRU L( W / I K ) W /( I ) K 0, K 1 /( I ) Kˆ 1 1 WI ( I ). (6) We also compute the second partial derivative and otain the Cramer-Rao Lower Bound (CRLB) on the variance of ˆK, 1 L( W / I K) 1 ˆ L( K) ( ) var( ) I K E K 1 K ( I ) 1. (7) Due to the linearity of the signal model, our L estimator is actually the minimum variance uniased (VU) estimator and the CRLB is its variance. Expression (7) also gives us clues on what images should e used for the est estimation of PRU. First, we want the luminance I to e as high as possile ut not saturated ecause saturated pixels (I [i] 55 for an 8-it grayscale image) carry no information aout PRU. Second, var( Kˆ ) is proportional to, which is the comination of various noise sources and the term I0 I ˆ0 introduced y denoising (3). Thus, the test images should e as smooth as possile. In summary, the est candidates for PRU estimation are images of right (ut not saturated) uniformly white scenes. In practice, one might tae out-of-focus images of cloudy sy after zooming in. Finally, we note that var( Kˆ ) is approximately inversely proportional to the numer of images used for estimation and thus decreases as 1/. 3. PRU detection The model (3) with the assumption that is a WG is an approximately valid representation of reality as long as the image is spatially homogenous with no saturated areas, such as sy images. As explained in the previous section, such images are the est for PRU estimation. In most real-life images, however, the PRU noise IK ˆ is modulated. It is attenuated in areas of the image that were flattened y processing, such as JPEG compression. Also, in overexposed regions of the image saturated at the maximum value of the dynamic range, the effect of the PRU (and any other noise) is not present at all. Additional attenuation is produced y the denoising filter ecause we may e sutracting a portion of the PRU noise with the denoised image. As a result, we accept the following model for the camera output

4 W TIKˆ, (8) where now T[i] is a multiplicative attenuation factor and [] i is a sequence of independent Gaussian variales with unequal variances [] i (colored Gaussian noise). We formulate the prolem of detection of PRU in the noise residual W Iˆ I0 of a given image I as a inary hypothesis testing H 0 : W H 1 : W TX, (9) where X IKˆ is the non-attenuated PRU signal. Here, more accurately we should have written H 0 : W TIKˆ ', where K ˆ ' corresponds to a PRU from some other camera. However, ecause the comined noise term dominates the contriution from the PRU, we consider the PRU as a wea signal and include it in the noise term. To estimate the shaping factor T and the variance, we can either accept a parametric model for them and estimate the parameters, or we can estimate T and locally from the image. Finding a good model, however, is not an easy tas ecause oth factors depend on the complex interplay etween the denoising filter and the local texture and content of the image. Liewise, due to insufficient data, it is not possile to accurately estimate these two nonstationary factors at every pixel. Instead, we opted for the following approach. We divide the image into smaller disjoint locs and assume that within each loc {1,, } T and are constant equal to T and, respectively. Both T and will e estimated from a predictor constructed in the next section. Allowing these estimates to e accurate up to an unnown multiplicative factor common to all locs (see the discussion at the end of Section 4), we arrive at the following hypothesis testing prolem H 0 : W H 1 : W atx. (10) where now [] i is a sequence of zero-mean independent Gaussian variales with nown variance equal to ˆ on the -th loc, and T[i] is piece-wise constant equal to Tˆ, which is also nown. Both a and are unnown multiplicative factors. The optimal detector for (10) is the normalized Generalized atched Filter (see Chapter 4.4 in [8]) Tˆ ( ) X W 1 ˆ ˆ T 1 X 1 ˆ W 1 ˆ. (11) We now explain the estimation of the shaping factor T and variance. Under hypothesis H 1, the noise residual W comes from the tested camera and the normalized correlation calculated for pixels in the -th loc is corr( T X, W ) corr( X, T X ) T X X X T X TX. (1) Because is zero mean and uncorrelated with X[i], the mixed term X is small compared to the other terms in (1). Thus, T X 1 1, (13) T X E, 1 1 T X E TX, The locs will e descried y their index sets B {1,, n}. Signals constrained to the -th loc will e denoted with suscript, e.g., X, W, etc.

5 where ETX, TX T X, E, B denote the energy of the signal of interest TX and the noise in the -th loc, respectively ( B is the cardinality of B ). Finally, we otain from (13) an estimate for the SR etween the signal of interest and noise E E, TX, 1 1. (14) Expression (14) assumes nowledge of the correlation, which we only now under hypothesis H 1 ut not under H 0. We address this prolem y constructing a predictor of the normalized correlation on small locs ased on our nowledge of ˆK, Î 0, and some features extracted from the loc that we expect to most influence (see Section 4). In other words, we will calculate for each loc an estimate (prediction) ˆ of what the correlation should e if I was otained y the camera with PRU ˆK (hypothesis H 1 ). From (8), W EW, ETX, E, and thus using (14) we otain estimates for oth T and : ˆ (1 ˆ ) E, ˆ ˆ W, T EW,. (15) X In the expression for ˆ in (15), we sipped the multiplicative factor 1/ B ecause it is the same for all locs if all locs are of the same size and thus does not influence the value of (11). We note that once the shaping factors T and the noise variance are estimated, the test statistics for the generalized matched filter (10) is calculated y evaluating (11) after sustituting for Tˆ and ˆ their corresponding estimates (15) where, (16) 1 Tˆ X W ˆ ˆ Ti 1 X i i i 1 ˆ W i 1 ˆ i i. (17) 4. CORRELATIO PREDICTOR In this section, we construct a predictor of the correlation corr( X, W ) on small locs for images coming from the same camera as the PRU (the matched case corresponding to hypothesis H 1 ). From experiments, we determined that the most influential factors are 1) Image intensity; ) Texture; 3) Signal flattening. The predictor will e a mapping from some feature vector to a real numer in the interval [0,1] the predicted value of. The loc size can not e too small to avoid the lac of statistically significant data ut not too large ecause then the assumption of stationarity of the shaping factor T and the variance is less liely to hold. For typical sizes of digital camera images with 1 million pixels or more, we recommend square locs with B = 1818 pixels. Image intensity. Because the PRU noise is multiplicative, the correlation is higher in areas of high intensity. However, due to the finite dynamic range, the PRU noise is not present in saturated regions ( I[] i = 55) and is

6 attenuated for Icrit I[] i 55, where the critical value of intensity I crit depends on the camera. Thus, we define the first feature as the average image intensity attenuated close to the maximum dynamic range where att(x) is the attenuation function f 1 I ([]) B att I i, (18) i B ([] I i I crit ) / e, I[ i] I, att([]) I i crit I[]/ i Icrit, I[] i Icrit, (19) where is a constant. For example, for our tested Canon G camera, we determined from experiments I crit = 50, = 6. Texture. The correlation tends to e smaller in textured areas for two reasons the variance is larger (and thus (13) smaller) and the shaping factor T < 1 (the denoising filter removes part of the signal of interest). Since the denoising filter that we use is constructed in the wavelet domain (8-tap Dauechis), we conveniently use this transformed signal and calculate the texture feature f T as the average reciprocal power of wavelet coefficients w in the LH, HL, and HH suands in the first three levels of the wavelet decomposition of the loc B : f T 1 1, (0) S 1 var( w[ i]) is where S is the union of indices of all wavelet coefficients from all 9 suands and var(w[i]) = min {var 3 (w[i]), var 5 (w[i]), var 7 (w[i]), var 9 (w[i])} is the variance of wavelet coefficients in the neighorhood of the coefficient w[i] estimated from local 33, 55, 77, and 99 square neighorhoods (all these quantities are calculated during denoising). The purpose of the reciprocal function is to normalize the feature to the interval [0, 1]. Signal flattening. Image processing that is of low-pass filtering nature, such as JPEG compression, further attenuates the PRU noise and thus decreases the correlation. In a relatively flat and high intensity (ut not saturated) area, the predictor would thus incorrectly predict a high correlation. These flattened areas will typically have a low value of the local variance. Thus, we added the third feature f S defined as the ratio of pixels in the loc with average local variance elow a certain threshold 1 fs { ib I [ i ] ci [ i]}, (1) B where c is an appropriately chosen constant that depends on the variance of the PRU K (e.g., c = 0.03 for Canon G) and [] i is the local variance of image intensity I[i] at pixel i estimated from a local 55 neighorhood. I From experiments, the correlation coefficients strongly depend on the collective influence of texture and intensity. Sometimes, highly textured regions are also high-intensity regions. Thus, we included the following comined texture-intensity feature calculated in the wavelet domain from the same wavelet coefficients as f T f TI 1 att( I[ i]) S 1 w [ i], () is where in () for each i, att(i[i]) is the attenuated pixel intensity after susampling the image to an appropriate size that matches the corresponding suand. Having specified the features, machine learning can now e used to learn the relationship etween the features and the correlation. In this paper, we opted for a simple polynomial multivariate least square fitting. Let e a column vector of K normalized correlations (1) calculated for K image locs and f I, f T, f S, and f TI e the corresponding K- dimensional feature vectors. We model as a linear comination of the features and their second-order terms

7 [ ] f [ ] f [ ] f [ ] f [ ] f [ ] f [ ] f [ ] f [ ]... [ ], (3) 0 1 I T 3 S 4 TI 4 I I 5 I T where [] is the modeling noise and is the vector of coefficients to e determined. In (3), there are total of =15 terms. Rewriting (3) in a matrix form, we have H, (4) where H is a K15 matrix of features and their multiplications and = ( 1,, 15 ) is the unnown vector parameter. Applying the least square estimator (LSE) to estimate, we otain 1 ˆ T T and the estimated correlation ˆ [ f I, f T, f S, f TI, f I f I, f I f T,...]ˆ. HH H (5) (6) True correlation Predicted Statistics Estimated correlation True Statistics (a) () Figure 1. a) Scatter plot of vs. ˆ for K=30, locs from 300 images for Canon G; ) True test statistics (16) vs. ˆ ˆ for 85 Canon G images not used for calculating the PRU or the predictor. 1 The images used to uild the predictor should e as diverse as possile so that the features extracted from the locs cover as large portion of the feature space as possile. This is in contrast to the requirements for calculating the PRU where we desire to have images mostly without texture. By overlapping the locs, one can extract several hundreds of locs from one image, depending on the image size. In practice, we have otained very good predictors even from as few as 10 images. We note that if the image under investigation I is a JPEG image, the predictor should also e trained on JPEG images of approximately the same quality factor ecause JPEG compression influences the correlation. If the image has undergone an unnown processing that influences the values of correlations, the predictions may e off y a multiplicative factor ecause the PRU will e attenuated in such images. This is why we allow in our model (10) unnown multiplicative factors a and. We remar that the features can e defined in other ways and evaluated in different domains. We tested predictors ased on features calculated all from the spatial domain and otained a very similar performance. Liewise, other machine learning tools that we tested, e.g. neural networs, provided very similar results. It is very liely that a more detailed study of the influence of image properties on the test statistics comined with etter machine-learning tools will further improve the predictor and lead to a more accurate camera ID process.

8 5. EYA-PEARSO TESTIG We now descrie the eymann-pearson hypothesis testing approach to decide whether or not a given image I was taen with a specific camera whose PRU K has een estimated in Section 3. We would lie to estimate the proaility of maing an incorrect decision. Recalling (9), false acceptance occurs when hypothesis H 0 is true ut we decide H 1, while false rejection occurs when we accept H 0 when H 1 is true. To estimate the false acceptance ratio (FAR), we tested 500 images from coming from over 1000 different cameras against the PRU K. The test statistics (11) was modeled as a Generalized Gaussian GG( 0, 0, 0 ) where 0 is the mean, 0 the shape parameter, and 0 the width parameter. To estimate the false rejection rate (FRR), we need to otain a pdf for the test statistics for images from the same camera. However, ecause the statistics heavily depend on the image content, modeling the distriution for very diverse images is nearly impossile as it strongly depends on the availale images. oreover, this would lead to overly conservative estimates for good images and too optimistic estimates for highly textured images. We should e evaluating the FRR against images of approximately the same content (close in the feature space). The predictor constructed in the previous section will enale us to achieve this goal. First, note that the experimental predictor data displayed graphically in Fig. 1 can e used to estimate the pdf of the prediction error for each estimated loc correlation ˆ ˆ. (7) From Figure 1a we see that the variance of does not depend on the predicted correlation value. Thus, we model the prediction error as identically distriuted (ut not necessarily independent) Generalized Gaussian variales with pdf f (x) = f GG (x; 0,, ). The errors from different locs, however, are not independent ecause it is liely that neighoring locs will have similar texture and thus the prediction errors will also e similar. To otain a conservative estimate of the FRR, we made the assumption that the prediction errors are completely correlated, in ˆ which case the pdf of the test statistics ecomes f ((x 1 )/c)/c, where c 1 and ˆ 1 1. This is the pdf of for images of approximately the same content (measured in the feature space). We set the decision threshold Th to otain FAR f GG ( x; 0, 0, 0) dx = 10 5, and calculate the FRR for images of Th approximately the same content as Th FRR f (( x 1) / c ) / c dx. (8) We note that we decide that I was taen y the camera when > Th, given y (16). 6. EXPERIETS We selected five cameras for our tests: Canon G with a 4 megapixel (P) CCD, Olympus C765-1 and Olympus C765- with a 4P CCD, Sigma SD-9 with a 3P COS Foveon sensor, and Olympus C3030 with a 3P CCD. We note that the two Olympus C765 cameras are of exactly the same rand. The PRU was calculated from 30 lue sy images or uniformly lit test images otained using a light ox. The predictors were trained on more than 10,000 locs from 0 images. We first calculated the test statistics for the H 0 hypothesis y testing the PRU from each camera against 500 images from 1000 different cameras (including the images from the other four tested cameras). The distriution of the test statistics was used to calculate the threshold Th giving FAR = Then, we tested images from the correct camera and evaluated the FRR for each image using (8) and (9). f ( x;,, ) ( / (1/ )) e GG x

9 Tale 1. True and predicted values of test statistics and FRR for eight test images from each of five cameras. Canon RAW JPEG 90 JPEG 75 G True/Pred. FRR True/Pred. FRR True/Pred. FRR / e 0.01/ e 0.014/ e 0.066/ e / e 0.00/ e / e / e / e / e / e / e / e / e / e / e / e / e / e / e / e / / e / e 4 Olympus RAW JPEG 90 JPEG 75 C True/Pred. FRR True/Pred. FRR True/Pred. FRR / e 0.015/ e / e / e 0.0/ e 0.016/ e / e / e 0.01/ e / e / e 0.019/ e / e / e / e / e / e / e / e / e / e / e / e / e 7 Olympus RAW JPEG 90 JPEG 75 C 765- True/Pred. FRR True/Pred. FRR True/Pred. FRR / e / e / e / e / e 0.03/ e / e / e 0.011/ e /0.058.e / e / e / e / e / e / e / e / e / e / e / e / e / e / e 5 Sigma RAW JPEG 90 JPEG 75 SD-9 True/Pred. FRR True/Pred. FRR True/Pred. FRR / e 0.04/ e / e / e / e 0.018/ e / e / e 0.019/ e / e / e 0.0/ e / e / e 0.031/ e / e / e / e / e / e / e / e / e / e

10 Olympus RAW JPEG 90 JPEG 75 C3030 True/Pred. FRR True/Pred. FRR True/Pred. FRR / e 0.044/ e 0.01/ e 0.081/ e / e / e / e / e / e / e / e / e / e / e / e / e / e / e / e / e / e / e / e / e 5 In the tales (Tale 1) we display the true value of the test statistics (16), its predicted value ˆ ˆ 1, and the FRR (at FAR=10 5 ) for eight selected images. We always selected the worse image in the set (o. 1), the est image (o. 8), and 6 other in-etween representative images. Fig. shows the eight images from Cannon G. We note that the worst images were always heavily textured images, often comined with very right (saturated) and dar areas. The PRU in such images is largely attenuated and difficult to detect Figure. Eight Canon G images used in experiments. 7. PRU PRE-PROCESSIG Besides the PRU, the signal ˆK estimated in Section 3.1 will contain all components that are systematically present in every image, such as artifacts due to color interpolation, on-sensor signal transfer [9], and JPEG compression (lociness). These artifacts are not unique to the sensor and are shared among cameras of the same rand or cameras sharing the same imaging sensor design **. This may cause the estimated PRUs etween two different cameras to e slightly correlated, which increases the false identification rate and decreases the reliaility of the camera ID process. Thus, to further improve the accuracy of the camera ID algorithm, we suppress these artifacts y pre-processing the estimated PRU efore it is used for identification. We measure the success of removing the unwanted artifacts y ** This oservation was already made in [5] ut was not further investigated.

11 calculating the correlation etween PRUs from different cameras (of the same rand or sharing the same sensor design), aiming for the correlation to e as close to zero as possile. We identified three main causes responsile for the artifacts. 1. Color interpolation. Cameras equipped with a CFA (color filter array) only capture one color at each pixel, while the remaining colors must e interpolated from neighoring pixels. Although there exists a large numer of interpolation algorithms, all involve some common elements. In particular, after quantization the sensor raw output is first adjusted for gain (ecause of varying sensitivity of silicone to light of different wavelengths). The resulting colors are then interpolated. The asymmetry of most color interpolation algorithms comined with slightly offset gains might introduce small ut measurale iases into the interpolated colors. Because CFAs contain a periodic structure, these iases will show up as a periodic ias in column and row averages of the estimated PRU. Thus, we can remove such artifacts y zeroing out the means of rows and columns of the estimated PRU.. Row-wise and column-wise operation of sensors and processing circuits. The row-wise and column-wise character of operations of digital imaging sensors and/or image processing circuits [9] also introduces a ias into each column and row. This linear pattern is also removed y zeroing out columns and rows. 3. JPEG lociness artifacts. Strong JPEG compression, especially in digital camcorders, causes lociness artifacts that can propagate into the estimated PRU [10]. These artifacts manifest themselves as peas and ridges in the magnitude of the PRU in the Fourier domain. We note that the first two artifacts are specific to the sensor design, the color filter array (CFA), and color interpolation. Thus they might e potentially useful for identification of the camera rand or model. We again leave this idea to our future wor and now focus on methods for removing the artifacts. To suppress the artifacts, we pre-process the estimated PRU y zeroing out the means of rows and columns and y further filtering/masing of the PRU in the Fourier domain. In column and row zeroing, we first sutract column averages from each pixel in each column (for each color channel separately) and then sutract row averages from every pixel in the row. This maes the column and row averages of the estimated PRU equal to zero. We denote this matrix operation as Z ( Kˆ ). The linear pattern, a potentially a useful forensic entity y itself, is the difference LP( Kˆ ) Kˆ Z ( Kˆ ). The linear pattern is wea compared to ˆK with SR elow 10 for compact or SLR cameras and it can e stronger for cheap cameras, e.g., cell-phone cameras. Figure 3 shows the enhanced linear pattern estimated from Canon G camera. This camera has the Bayer CFA with periodicity along the rows and columns, which clearly shows up in the liner pattern. Figure 3. Linear pattern of Canon G (detail). Figure 4 Fourier transform of the PRU after zero-meaning. To remove the JPEG lociness and remaining periodic patterns from the zero-meaned PRU, we apply the Wiener filter in the Fourier domain and only eep the noise component: ˆ 1 WF( Z ( K)) =F { F( Z ( Kˆ )) W( F( Z ( ˆK )))}, Gamal et al. [9] model the readout process as a first order isotropic autoregressive process.

12 where W is the Wiener filter with variance determined from the magnitude of the Fourier transform F ( Z ( Kˆ )) and window size 33. The resulting PRU has a much flatter frequency spectrum than Z ( Kˆ ) (see Figure 4). Tale. Correlations etween ˆK for Canon G and ˆK for Canon S40 efore and after pre-processing. G S40 Red Green Blue ˆK G vs. ˆK S Z ( K ˆ G) vs. Z ( Kˆ S40) WF( Z ( K ˆ )) vs. WF( Z ( Kˆ )) G S40 In Tale we show the correlations etween differently processed PRUs estimated from two Canon cameras, G and S40. Both cameras share the same CCD sensor. Both PRU estimates were generated from 0 images (high quality JPEGs in case of G and uncompressed images for S40). Another example shown in Tale 3 involves two 1.3 p cell phone cameras LG VX8100 and Samsung A900. Only 10 images of sy and gray wall were the source for ˆK LG and 15 similar images for ˆK SA. The zero-meaning efficiently reduced the correlation etween the PRU estimates. Further filtering in the Fourier domain removed the lociness caused y JPEG compression. Tale 3. Correlations etween ˆK for LG VX8100 and ˆK Samsung A900 efore and after pre-processing. LG SA Red Green Blue ˆK LG vs. ˆK SA Z ( K ˆ LG ) vs. Z ( K ˆ SA ) WF( Z ( K ˆ )) vs. WF( Z ( Kˆ )) LG SA 8. COCLUSIOS In this paper, we present an improved method for camera ID ased on joint estimation and detection of the camera photo-response non-uniformity (PRU) in images. The method can e used whenever there is a need to answer the question whether or not a given image was taen with a specific camera that is either in our possession or images provaly taen y that camera are availale. The method uses the same principles as the approach proposed y Luas et al. [5] ut uses more advanced signal estimation and detection methods. First, the availale images are denoised and the PRU is estimated from the denoising residuals. In contrast to the intuitive approach reported in [5] in which the residuals were averaged, we start with a simplified linearized model of camera output and formulate the prolem as parameter estimation in noisy oservations. Under the assumption that the corrupting noise sources are Gaussian, the maximum lielihood estimator is VU. The detection of presence of the PRU in a given image then amounts to detection of a nown signal attenuated y local image properties in independent Gaussian noise with unequal variances. The attenuation factor and the noise variance are estimated from a specially constructed predictor of normalized correlations on small image locs. The optimal detector for this detection prolem is the normalized correlation of pre-whitened signals (generalized matched filter). Using the eyman-pearson hypothesis testing, we estimate the proaility of false rejection (falsely deciding that the camera did not tae the image when it did) when setting the proaility of false identification to The correlation predictor enales us to estimate the proaility of false rejection for images of similar content (images producing

13 similar values of the test statistics). This leads to much more accurate error estimates ecause the reliaility of the camera identification method is nown to strongly depend on image content. Finally, we report some new results regarding the PRU estimation. Besides the PRU, the estimated signal will contain all the components that are systematic (present in every image), artifacts due to JPEG compression (lociness), column (or row) artifacts due to on-sensor signal transfer, and artifacts due to color interpolation. These artifacts slightly increase the value of the test statistics among cameras of the same rand or cameras sharing the same imaging sensor design. This increases the false identification rate and decreases the reliaility of the camera ID process. To suppress the effect of these artifacts, we proposed to pre-process the estimated PRU y zeroing out the means of rows and columns and further filtering the PRU in the Fourier domain. We note that some of these artifacts are specific to the sensor design, the CFA, and color interpolation, and thus might e potentially useful for identification of the camera rand or model. Our future research will e focused to applying the developed methodology to digital forgery detection. ACKOWLEGEET The wor on this paper was supported y the AFOSR grant numer FA The U.S. Government is authorized to reproduce and distriute reprints for Governmental purposes notwithstanding any copyright notation there on. The views and conclusions contained herein are those of the authors and should not e interpreted as necessarily representing the official policies, either expressed or implied, of Air Force Research Laoratory, or the U.S. Government. REFERECES 1.. Kharrazi, H. T. Sencar, and. emon, Blind Source Camera Identification, Proc. ICIP 04, Singapore, Octoer 4 7, A.C. Propescu and H. Farid, Statistical Tools for Digital Forensic, in J. Fridrich (ed.): 6 th International Worshop on Information Hiding, LCS vol. 300, Springer-Verlag, Berlin-Heidelerg, ew Yor, pp , A. Swaminathan,. Wu, and K.J.R. Liu, on-intrusive Forensic Analysis of Visual Sensors Using Output Images, IEEE Int. Conf. on Acoustics, Speech, and Signal Processing (ICASSP'06), ay K. Kurosawa, K. Kuroi, and. Saitoh, CCD Fingerprint ethod Identification of a Video Camera from Videotaped Images, Proc of ICIP 99, Koe, Japan, pp , Octoer J. Luáš, J. Fridrich, and. Goljan, Digital Camera Identification from Sensor Pattern oise, IEEE Transactions on Information Security and Forensics, vol. 1(), pp , June G. Healey and R. Kondepudy, Radiometric CCD Camera Caliration and oise Estimation, IEEE Transactions on Pattern Analysis and achine Intelligence, vol. 16(3), pp , arch, K. ihca, I. Kozintsev, and K. Ramchandran, Spatially Adaptive Statistical odeling of Wavelet Image Coefficients and its Application to Denoising, Proc. IEEE Int. Conf. Acoustics, Speech, and Signal Processing, Phoenix, Arizona, vol. 6, pp , arch S.. Kay, Fundamentals of Statistical Signal Processing, Volume II, Detection theory, Prentice Hall, A. El Gamal, B. Fowler, H. in, and X. Liu, odeling and Estimation of FP Components in COS Image Sensors. Proceedings of the SPIE, Solid State Sensor Arrays: Development and Applications II, vol , San Jose, CA, pp , January Chen, J. Fridrich, and. Goljan, Source Digital Camcorder Identification Using CCD Photo Response onuniformity, Proc. SPIE Electronic Imaging, Security, Steganography, and Watermaring of ultimedia Contents IX, vol. 6505, San Jose, California, January 8 Feruary 1, 007.