SELF-AUTHENTICATION OF NATURAL COLOR IMAGES IN PASCAL TRANSFORM DOMAIN. E. E. Varsaki, V. Fotopoulos and A. N. Skodras

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1 SELF-AUTHENTICATION OF NATURAL COLOR IMAGES IN PASCAL TRANSFORM DOMAIN E. E. Varsaki, V. Fotopoulos and A. N. Skodras Digital Systems & Media Computing Laboratory School of Science and Technology, Hellenic Open University Tsamadou st., GR-6, Patras, Greece {e.varsaki, vfotop1, dsmc.eap.gr ABSTRACT This paper proposes a self-authentication scheme for color images, using a fragile data hiding method based on the Discrete Pascal Transform (DPT). The main contribution of this work is in the development of a scheme that is able to detect whether a color image has been tampered or not, to locate the exact tampered position, and to attempt to restore the original image content. This is achieved by embedding a low bit-depth scaled version of the image itself, resulting in very good stego image quality and satisfactory restoration. Index Terms self-authentication, data hiding, discrete Pascal transform. 1. INTRODUCTION The digital nature of multimedia documents facilitates modification, transformation and duplication. A central issue in multimedia manipulation is to reinforce their security and integrity. Most research to date intents to authenticate still images by hiding side information. In other words the problem of image authentication is solved by using fragile or semi-fragile data hiding schemes. Authentication concerns the image content. It is important to distinguish between malicious manipulations, which involve the altering of the content of the original image, and manipulations related to the use of an image, such as format conversions, compression, filtering and so on. In the first case the image content is altered as opposed to the second case. Image authentication can be used in applications where the content is important like cultural heritance, legal disputes, journalism, etc. Self-authentication methods attempt to embed the image into itself. In this way it is possible to recover the original image, even if the stego image has been altered by one or more attacks, malicious or not. First Fridrich et al. proposed a fragile method in which a compressed version of the original image is embedded into the LSB of its own pixels [1]. Several authentication methods have been proposed after that. The information is hidden in the LSB of the image [], or in the transform (DCT [3], wavelet [4]) coefficients. The recently presented discrete Pascal transform (DPT) by Aburdene et al. in [5] has become an important aspect of study. Kostopoulos et al. [6] proposed an interest watermarking scheme based on Pascal Triangle. However, the study on the applications of DPT is still of interest. This paper aims to evaluate the use of DPT in a selfauthentication scheme. The paper is organized as follows. In section the Discrete Pascal Transform is discussed. The embedding and extracting algorithms are presented in section 3. In section 4 the self-authentication scheme is analysed. Experimental results are shown in section 5. Finally conclusions are drawn in section 6.. DISCRETE PASCAL TRANSFORM AND DATA HIDING The discrete Pascal transform (DPT) X(k) of a 1D data vector x(n), for n=0,1,,,n-1 is defined in [7] as Xk ( ) = Pxn ( ), (1) where k=0,1,,,n-1 and P is the NxN Pascal transform matrix the elements of which are equal to p i, j N j ( 1) aij, for i j =, () 0, otherwise with i,j=0,1,,,n-1. The Pascal s transform matrix for N=, N=3 and N=4 are given in (3). For clarity reasons, the zero elements are denoted by a dot. The Pascal transform matrix is equal to its inverse, thus the inverse DPT is implemented by multiplying again with the Pascal transform matrix as in (4) /09/$ IEEE DSP 009

2 P =, P 3 = , P4 = (3) x() n = P X() k (4) N The Pascal transform operation is highly sensitive to noise and this is reflected on the DPT coefficients [8]. By extending to two dimensions, the DPT X(u,v) for the D data x(m,n), where M and N are the dimensions of the D signal, is defined as T X ( uv, ) = P xmnp (, ). (5) where m=0,1,,m-1 and n=0,1,,n-1. Coefficients in Pascal transform represent weighted differences in neighboring data values. Aburdene in [5] has proved that any sharp change in the pixel values of an image will produce large transform coefficients. When a variation occurs the DPT coefficients change up to the variation location. In the following we study how the DPT coefficient are affected when the identity or the anti-diagonal identity matrix is added to the D data. For simplicity let us assume a x block of the D data as follows: a11 a1 A = a1 a. (6) The DPT of A is calculated according to (5) as: X = a11 a1 a11 a1 a1 + a M a a a Let I be the x identity matrix and A be the x result of the addition of A and I. The DPT of A is calculated in (9). a a 1 0 a + 1 a A = + = a1 a 0 1 a1 a + 1 a ( a11 + 1) a1 X = ( a11 + 1) a1 ( a11 + 1) a1 a1 + ( a + 1) a11 a11 a1 1 1 X = + a11 a1 a11 a1 a1 + a X = X + 1, (9) From eq. (9) we see that the addition of the identity matrix increases the DPT coefficients by one, except for coefficient X (,) which increases by. In a similar way, it is seen that the X 4 DPT of a 4x4 D data block, to which the I 4 matrix is added, equals to N (7) (8) X 4 = X4 +, (10) where X 4 is the DPT of the 4x4 D data block prior to the addition. In general, the DPT coefficients of a D data block A V, for V even or odd, increases by the VxV Pascal matrix, when the VxV identity matrix is added to the A V block. Likewise, let A be the x result block of the addition of the x A data and I, where I is the x anti-diagonal identity matrix. X is the DPT of A and equals to (11). 0 1 X = X +, 1 (11) It is seen that the X (1,1) coefficient remains unchanged. The rest of the coefficients are decreased by one except for X (,), which is increased by. The X 4 DPT of a 4x4 data block in which the 4x4 antidiagonal identity matrix I 4 is added, equals to X 4 = X4 +. (1) In general, the DPT coefficients of an VxV sized A V data block (where V is even), monotonically decreases from the anti-diagonal identity diagonal of the block towards the X V (V,V) coefficient, when the VxV anti-diagonal identity matrix is added to the A V block. This does not hold for the odd sized matrices. In this case the the DPT coefficients are increased, as can be seen in eq. (13-14) below for the 3x3 matrices A 3 and X 3. a11 a1 a A 3 = a1 a 1 a + 3 (13) a a3 a X 3 = X , (14) In eq. (14) X 3 is the DPT of A 3 3x3 data block and X 3 is the DPT of the 3x3 D data block prior the addition. We have observed that the X V (V,V) coefficient of natural photos has a zero mean Gaussian distribution. The coefficients are positive or negative. It is possible to change the coefficient sign, by just adding the identity or the antidiagonal identity matrix VxV matrix to the VxV pixel block. This property of the DPT can be used in order to invisibly embed data into a block of pixels. Based on the above

3 discussion the embedding algorithm is presented in the next section. 3. DATA EMBEDDING AND EXTRACTING ALGORITHMS 3.1. Embedding Algorithm Let x(m,n) be a color component of the original image of MxN pixels, with m=1,,,m and n=1,,,n. The embedding algorithm is applied into RGB color space, but it is expandable to other color spaces as YUV. Let b i, with i=1,,,p be the to hide size P. The and the original image are the inputs of the algorithm. The output is the stego-image s, which contains the secret and is not visible for a human observer. The embedding algorithm is as follows: Step1: Divide each R, G and B color component of the original image into 4x4 non-overlapping blocks. The block size selection is discussed later. Step: For each bit b i check whether it is equal to 0 or not. If this is not the case go to step 7. Step3: Compute the DPT of the corresponding x i 4x4 block, X i. Step4: Check whether -0-T<X i (4,4) -T. If this true continue with the next bit (step ). Step5: If X i (4,4)<-T-0 add the identity 4x4 matrix I to the x i corresponding 4x4 pixel block and compute again the DPT of the modified block. Repeat this step until the inequality in step 4 is satisfied and then continue with the next bit. Step6: While X i (4,4)>-T subtract the identity 4x4 matrix I from the x i corresponding 4x4 pixel block and compute again the DPT of the modified x i block. Repeat this step until inequality in step 4 is satisfied and then continue with the next bit (step ). Step7: Compute the DPT of the corresponding x i 4x4 block, X i. Step8: Check whether X i (4,4) DPT coefficient satisfies the equation: T<X i (4,4) T+0. If this is true continue with step. Step9: If X i (4,4)<T add the 4x4 identity matrix I to the corresponding x i block. Compute again the DPT of the altered x i block and repeat until inequality in step 8 is satisfied. Step10: While X i (4,4)>T+0 subtract the 4x4 identity matrix I from the corresponding x i block and compute again the DPT. Repeat until the inequality in step 8 is satisfied and then continue with step. Step11: Form the stego image from the changed and unchanged x i blocks of the original image. xi + I Message bi xi -I Xi(4,4)<T i = i + 1 Original image x P = (MxN)/16 Divide x into P 4x4 blocks i = 1 bi = 0 Compute Xi the DPT of i block T<Xi(4,4)<T+0 i = P Stego image s Compute Xi the DPT of i block -T-0<Xi(4,4)<-T Fig. 1. Block diagram of the embedding algorithm. xi -I xi + I Xi(4,4)<-T-0 In some cases steps 5,6,9 and 10 may result in overflow. To avoid such errors, additional control is applied. When overflow occurs and no condition in steps 4 and 8 is satisfied, then the above steps may be modified as follows: Modified step5 or 9: Subtract the anti-diagonal identity matrix from the corresponding 4x4 x i block and compute again the DPT. Repeat this step until inequality in step 4 or 8 is satisfied and then continue with the next bit. If this is not the case go to next step. Modified step6 or 10: Add the anti-diagonal identity matrix to the corresponding 4x4 x i block and compute again the DPT. Repeat this step until inequality in step 4 or 8 is satisfied. Continue with next bit. Fig. 1 depicts the block diagram of the embedding algorithm. T is an upper threshold defined for authentication purposes. A range of 0 is selected because the X 4 (4,4) coefficient is changed by 0, when identity or anti-diagonal identity 4x4 matrix is added to the 4x4 block as presented in section. 3.. Extracting Algorithm The hidden may be retrieved by collecting the embedding bit of each 4x4 block. The extracting algorithm is shown in Fig.. The procedure simply reads the S(4,4) DPT coefficient of the 4x4 stego blocks in order to reply 0 or 1 for the hidden. The extracting algorithm is as follows:

4 Step1: Divide each R, G and B component of the stegoimage s into 4x4 non overlapping blocks. Step: For each s i block compute the DPT coefficients, S i. The bits are extracted in the same order as been embedded. Step3: If -0-T<S i (4,4)<T the bit extracted is equal to zero (b i =0). Step4: If T<S i (4,4)<T+0 then b i =1. Step5: Form the hidden from the extracted b i bits. i = i + 1 Extract bi = 0 Stego image s P = (MxN)/16 Divide s into P 4x4 blocks i = 1 Compute Si the DPT of i block -T-0<Si(4,4)>-T i = P Form the hidden Extract bi = 1 T<Si(4,4)>T+0 Tampered Fig.. Block diagram of the extracting algorithm Fig. shows the extracting algorithm, where the decision about extracting 1 or 0 is based on whether coefficient S i (4,4) belongs to the positive [T, T+0] range or the negative [-T-0, -T] range, respectively. If for a block the S i (4,4) coefficient does not satisfy one of the above conditions, then the image is considered as tampered. 4. IMAGE SELF-AUTHENTICATION The general idea is to embed secret data into an original color image so that any attempt to alter the content of the image will also alter the. In order to locate the tampered regions of the image, the hidden is extracted. Then the is compared with the image content and the tampered blocks are located. The extracted restores the content of the tampered image. For that reason the embedded is a scaled version of the image itself. Since the embedding algorithm is designed to be easily destroyed but also able to recover tampered regions, the is reversed. This is made in order to extract the original image from an un-tampered place and by this restore it. The authentication scheme is described latter (Fig. 3) Message preparation The original image is 1:4 scaled to each dimension. This color scaled version or the image is inverted to gray level. That version is denoted as a(h,l), where h=1,,,m/4 and l=1,, N/4, with MxN the original image size. Only the three most significant bits (MSBs) of its pixel values are used to compose the. In this way each pixel of a(h,l) image is embedded into three 4x4 blocks of the original color image. The is embedded in reverse order. For example for the upper left corner block the 3MSB of pixel a(m/4, N/4) is embedded. In this way, the tampered region can be restored by the that is embedded in the non-tampered regions of the image. 4.. Message embedding Embedding procedure takes place after the preparation. The first MSBs of the gray level and scaled version of the original image are embedded into the 4x4 blocks of the original image s R component. The next MSBs are embedded into the 4x4 blocks of the G component and the last MSBs are embedded into the 4x4 blocks of the B component. The embedding procedure is as described in section 3. DPT X(4,4) coefficient is enforced to be in the [T, T+0] or the [-T-0, -T] range, depending on the bit. Threshold T selection is discussed in experimental results Tamper detection At the receiving part the image y(m,n) needs to be searched to check whether it is authentic or has been altered in some way. The tamper detection is performed during the extracting procedure in which the exact tampered position/s is/are located. At the next phase the image is restored. The extracting procedure is performed as described in section 3. Each image s color component is divided into 4x4 non-overlapping blocks. Then DPT is computed and Y(4,4) coefficient is checked. If Y(4,4) [T, T+0], then the extracting bit equals to 1. If Y(4,4) [-T-0, -T], then the extracting bit equals to 0. If neither of the above is true, then the corresponding block is altered and the extracted bit equals to null. The is extracted, reversed and forms the a (h,l). Changes between a (h,l) and a(h,l) identify the tampered locations.

5 RGB to Gray level Threshold in 3MSB Extracting Original Image R G Scale 1:4 B 3 rd MSB nd MSB 1 st MSB Embeding 1 st MSB into R vector nd MSB into G vector 3 rd MSB into B vector Reverse Tamper Tampered image Extracted RGB to Gray level Tamper Detection Threshold in 3MSB Restored image Divide each vector in 4x4 blocks Scale 1:4 Fig. 3. Block diagram of the authentication scheme 4.4. Image restoration In the next step the received image is scaled 1:4 and a gray level version c(h,l) of the received image is produced. Then the 3 MSB of its pixels are compared with a (h,l) extracted image and the changes between the two images define the tampered locations. It is clear that for each tampered region, the extracted image locates two different places that differ from the received. For example let a (i,j) be a destroyed pixel with i [1,M/4] and j [1,N/4]. Then a (i,j) c(i,j), but also a (M/4-i,N/4-j) c(m/4-i,n/4-j), because a (h,l) is reversely embedded. To determine which region is the tampered one, the DPT of the corresponding a (h,l) differences block of the received image y(m,n) is calculated. If Y(4,4) [T, T+0] or Y(4,4) [-T-0, -T], then the block is tampered. For example let a (i,j) differ from the c(i,j) and c(i,j) corresponds to a tampered block. a (i,j) is not embedded into the tampered block but a (M/4-i,N/4-j) is. So the tampered block can be replaced by a (i,j) which represents the original content of the block. In this way the tampered image can be restored. 5. EXPERIMENTAL RESULTS The experiments test the self-authentication method s performance for various block sizes and upper and lower threshold values T. It is important to select the appropriate block size for performance tuning. As mentioned in section, odd block sizes are not preferred because they lead to overflows. Even-sized blocks are preferred because they can obviate overflows. In embedding, when overflows happen, the anti-diagonal identity matrix is subtracted instead of adding the identity matrix. In the case that the image is divided in 4x4 blocks, the maximum possible change to the X(4,4) DPT coefficient is 18*0=560. Testing over ten color images, X(4,4) DPT coefficient has an average minimum value of -700 and average maximum value of 700. Consequently, when overflow occurs, the subtraction of anti-diagonal identity matrix ensures that no underflow occurs. In x blocks the X(,) coefficient is even smaller. In contrary, when odd sized blocks are used, there is no possibility to correct the overflow. The capacity of the proposed scheme is defined as follows P = ( M N)/ t (15) where t is the block size. The algorithm is tested for different block sizes. There is a trade-off between image quality and payload. Experimental results are shown in Fig. 4. It is observed that a block size of 4x4 is an good trade-off between image quality and capacity. Fig. 4. Trade-off between Capacity and PSNR for the Lenna image (56x56) Tests between tampered and non tampered images were performed. For the case of tampered images the extracted image a (h,l) differs from the received image c(h,l). In order to locate the exact tampered locations the selection of threshold T is of paramount importance. If the embedded scheme uses a low T value (e.g. T=0), then the DPT coefficient X i (4,4) of a test image s block i, is near 0. It is very probable for the corresponding tampered block the DPT coefficient Y i (4,4) to be in this range. Consequently, it is not possible to locate the tampered region. Tests for different T values were conducted. The results are shown in Table I and in Fig. 5.

6 TABLE I IMAGE QUALITY OF FRUITS IMAGE FOR DIFFERENT THRESHOLDS FOR 4X4 BLOCK SIZE T PSNR [db] Fig. 5. Experimental results of a tampered image using different thresholds. Fig. 6. Image restoration for different distortions using threshold equal to 100 and 4x4 block size. The tampered images on the upper row and the restored images in the lower row. From Fig.5 we see that when X(4,4) [-80,- 0] [0,80] the method fails to recognize a number of tampered blocks. Less errors are present when X(4,4) [- 100,-80] [80,100]. Of course higher thresholds have better detection results, but image quality is degraded. The proposed self-authentication scheme can restore a wide range of alterations (Fig.6). However, the image content is important in the restoration results. In natural images like that of Fig. 6 where texture is present, the restoration is successful. In other cases, as for example in compound documents, this may result in unexpected failure. For instance, if letters are important to be recovered from a small sized image, then a x embedding version has to be applied, and the 1: scaled image is embedded into itself. 6. CONLUSIONS components of the image. The embedding algorithm takes into account the DPT coefficients of every 4x4 block. The DPT coefficients are forced to be relatively high. In this way the tampered region is located and restored. The drawback of the method is its inability to recover alterations made in the center of the image or when more than one diametrical regions are tampered. This is an issue of future work and improvement. ACKNOWLEDGEMENTS This work was supported by the project PENED (Contract no 03E 83) co-funded by the European Union - European Social Fund (75%), the Greek Government - Ministry of Development - General Secretariat of Research and Technology (5%) and the Private Sector in the frames of the European Competitiveness Programme (Third Community Support Framework). 7. REFERENCES [1] J. Fridrich, M. Goljan, Protection of Digital Images Using Self Embedding, Symposium on Content Security Data Hiding in Digital Media, New Jersey Institute of Technology, Newark, NJ, USA, pp. 1-10, [] M. H. Hassan, S.A.M. Gilani A Fragile Watermarking Scheme for Color Image Authentication, Proc. of World Academy of Science, Engineering and Technology, Vol. 13, pp , 006. [3] C-Y. Lin, S-F. Chang, Semi-fragile Watermarking for Authenticating JPEG Visual Content, Proc. SPIE international Conf. on Security and Watermarking of Multimedia Contents II, vol.3971, San Jose, Calif. USA, 000. [4] J. Wang, S. Lian, G. Liu, Y. Dai, Z. Liu, Z. Ren, Secure Multimedia Watermarking Authentication in Wavelet Domain, Journal of Electronic Imaging, 17(3), July 008. [5] M. F. Aburdene and T. J. Goodman, The Discrete Pascal Transform and its Applications, IEEE Signal Processing Letters, Vol.1,.7, pp , July 005. [6] H. Kostopoulos, S. Kandiliotis, I. Kostopoulos, M. Xenos A Digital Image Watermarking Technique Using Modulated Pascal s Triangles, IASTED International Conf. Signal Processing, Pattern Recognition & Applications, pp8-86, Rhodes, Greece, 003. [7] A.N. Skodras, Efficient Computation of the Discrete Pascal Transform, Proceedings of the 14 th European Signal Processing Conference (EUSIPCO 006), Florence, Italy, Sept [8] T.J. Goodman, M.F. Aburdene, On the Pascal Transform and Threshold Selection, Proc. 41 st Annual Conf. on Information Sciences and Systems (CiSS 07), pp , Baltimore, MD, March 007. A self-authentication scheme is proposed that is able to distinguish tampered regions in a modified image and restore the original one. This is implemented by embedding the 3 MSB of the scaled gray level version of the original image. Each MSB is embedded into one of the color