適用於三進位電腦之直方圖可逆資訊隱藏技術之研究

全文

(1)國立高雄大學資訊工程研究所碩士論文. 適用於三進位電腦之直方圖可逆資訊隱藏技術之研究 The Study of Reversible Data Hiding Based on Histogram Modification over Ternary Computers. 研究生：鄭宇哲撰指導教授：陳建源博士. 中華民國一零一年七月.

(2) Acknowledgement 首先，我誠摯的感謝我的指導老師陳建源教授的提攜。在我這兩年的研究過程中，老師不斷教導我如何進行研究與創新、更引導我以多元的方向進行對問題的探討，讓我在研究過程中有豐碩的成就。此外，老師更不厭其煩的指導我在撰寫論文上的各種細節與邏輯，讓我更能以跨國際的角度審視自己與研究。對於老師的指導，除了感謝更是感謝。. 同時，也特別感謝在百忙之中撥空前來的口試委員，薛智誠教授與吳俊興教授。老師們在口試時提出的寶貴的意見，不僅讓我的論文得以更加完善，更讓我學習到如何深入簡出的方法，再次感謝老師們的指導。接著我要感謝在我碩士生涯中，陪我度過許多努力、困惑、進步、喜悅的實驗室學長姐與同學們。鈺峰學長、坪亨學長、筱林學姊、小猜、阿水、能寅，這兩年的碩士生活因為有你們讓我過得多采多姿，成為我一輩子最珍貴的記憶。. 最後要特別感謝我的家人，感謝您們支持我完成碩士學位，更感謝您們在我外出求學時的等待。還有淑真、以及所有曾經幫助過我的貴人們，因為有您們的幫助，讓我有今天的成果，謝謝。. i.

(3) 適用於三進位電腦之直方圖可逆資訊隱藏技術指導教授：陳建源博士（教授）國立高雄大學資訊工程研究所. 學生：鄭宇哲國立高雄大學訊工程研究所. 摘要本碩士論文將提出一種適用於三進位(Ternary)電腦之可逆資訊隱藏技術。我們的方法主要將三進位的機密資訊，利用改良後的棋盤式直方圖可逆資訊隱藏技術[21]藏入原始影像。若機密資訊為二進位 (Binary) 表示式，我們將機密資訊轉為三進位的 Non-adjacent Form (NAF)表示式，再將機密資訊藏入影像內。根據 NAF 表示式的特色，我們的方法在少量機密資訊下，可以減少藏入機密資訊時需要位移的像素數量，提升偽裝影像的品質。根據實驗結果與分析得知，我們的方法在最佳狀況下可以比楊教授等人的方法減少 33% 的像素值位移數量，並增加 1.76 的 PSNR 值。關鍵字：可逆式資訊隱藏、不相鄰資料表示法、棋盤式預測法、直方圖位移、三進位電腦。. ii.

(4) Reversible Data Hiding Based on Histogram Modification over Ternary Computers Advisor: Dr.(Professor) C. Y. Chen Institute of Computer Science and Information Engineering National University of Kaohsiung. Student: Y. Z. Jheng Institute of Computer Science and Information Engineering National University of Kaohsiung. ABSTRACT In this thesis, we propose a new data hiding method over ternary computers. We embed ternary secret data, NAF format, into cover image. According to the characteristics of NAF format, our method can reduce the number of shifted pixel values in embedded algorithm. This implies that more pixels are unchanged and keep higher PSNR of stego-image. According to experiment results and analysis, our method can reduce the number of shifted values by 33% and gain 1.76 higher PSNR in best case as compared with Yang et al.’s method. Keywords: Reversible Data Hiding, NAF, Chessboard Prediction, Histogram Modification, Ternary Computer.. iii.

(5) Contents Acknowledgement ...................................................................................................... i 摘要 ...................................................................................................................... ii Abstract .................................................................................................................. iii Content ................................................................................................................... iv Figures Content ....................................................................................................... v Tables Content ......................................................................................................... vi Chapter 1 Introduction ............................................................................................. 1 Chapter 2 Related Works .......................................................................................... 6 2.1 Ni et al.’s method ........................................................................................ 7 2.2 2.3 2.4. Lin et al.’s method ...................................................................................... 9 Yang et al.’s method .................................................................................. 12 Non-Adjacent Form(NAF) ......................................................................... 21. Chapter 3 Proposed Method.................................................................................................. 23 3.1 Embedded Algorithm .............................................................................................. 24 3.2 Extracted Algorithm ................................................................................................ 26 Chapter 4 Experiment Results .............................................................................................. 32 Chapter 5 Discussion ............................................................................................................ 35 Chapter 6 Conclusions .......................................................................................................... 37 References ................................................................................................................................ 38. iv.

(6) Figures Contents Figure 1. Figure 2. Figure 3. Figure 4. Figure 5. Figure 6. Figure 7. Figure 8. Figure 9.. An example for complex image. ......................................................................... 11 Two segments of Yang et al.’s method. ............................................................... 12 The cover image. ................................................................................................. 15 The predictive errors of white segment. .............................................................. 16 The histogram of predictive errors of white segment. ......................................... 16 Embedded histogram and shifted predictive errors of white segment. ............... 17 The stego-image after first algorithm. ................................................................. 17 The predictive errors of black segment. .............................................................. 17 The histogram of predictive errors in black segment. ......................................... 18. Figure 10. Embedded histogram and shifted predictive errors of black segment. .............. 18 Figure 11. The stego-image. ............................................................................................... 19 Figure 12. The shifted predictive errors of black segment. ................................................ 19 Figure 13. Figure 14. Figure 15. Figure 16.. The shifted predictive errors of white segment. ................................................ 20 The algorithm of NAF format. .......................................................................... 21 An example of NAF format. .............................................................................. 22 Embedded histogram and shifted predictive errors of white segment with our method. ................................................................................................ 28 Figure 17. The stego-image after first algorithm. ............................................................... 28 Figure 18. The predictive errors of black segment. ............................................................ 28 Figure 19. Figure 20. Figure 21. Figure 22. Figure 23. Figure 24. Figure 25.. The histogram of predictive errors in black segment. ....................................... 29 Embedded histogram and shifted predictive errors of black segment with our method. ................................................................................................ 29 The stegoimage. ................................................................................................. 30 The shifted predictive errors of black segment. ................................................ 30 The shifted predictive errors of white segment. ................................................ 31 Ten images in our experiments. ......................................................................... 32 The relationship between H 1 / H 2 and gained PSNR. .................................... 34. Figure 26.. The definition in this chapter. ............................................................................ 35. v.

(7) Tables Contents Table 1 The experiment results of Yang et al.’s and our method. ................................. 33 Table 2 The experiment results of other methods. ..................................................... 33. vi.

(8) Chapter 1. Introduction. To avoid that important information is taken by an illegal user during transmission, data hiding has paid more and more attention [2, 3, 4, 5, 6, 10, 12, 13, 15, 16, 17, 18, 20, 21]. The main idea of data hiding is to embed the secret data into cover image and generate quality stego-image with PSNR higher than 40dB. Since the stego-image is imperceptible when PSNR is higher than 40dB [13], the illegal user can not recognize whether the stego-image is embedded the secret data or not. Only the specific user can extract the secret data from the stego-image. In data hiding methods, there are two important issues: the capacity of secret data and the quality of stego-image. Because pixel values in the cover image are shifted to embed the secret data, the data hiding method with higher capacity usually generates stego-image with lower quality. How to embed more secret data without reducing the quality of stego-image has became an important object for researchers.[4, 9, 10, 13, 15, 16, 21]. An effective data hiding method with high quality was proposed in 2006 by Ni et al. [13]. Assume that the sender wants to send the secret data to the receiver. The sender at first embeds secret data into cover image using Ni et al.’s method and generates stego-image with PSNR higher than 48dB. The sender outputs all the numbers of pixel values in histogram. Then the sender finds the top two highest numbers as peak points and their corresponding zero points, the numbers are zero and closest to peak points. After shifting the pixel values between peak points and zero points, the spaces next to peak points are generated. The sender embeds the secret data “1” by shifting the pixel values from peak points to their nearby spaces and secret data “0” by remaining the pixel values. In the end, the sender transports the stego-image, peak points, zero points 1.

(9) and length of secret data to the receiver. Using acquired information, the receiver can extract secret data and discover the original cover image. Ni et al.’s method can keep high quality of stego-image since each difference between original and shifted pixel value is at most one. But this method has little capacity because it depends on top two highest numbers of pixel values. To gain more capacity, Lin et al. use the histogram of differences to embed the secret data in 2008[12]. In Lin et al.’s method, the sender computes all the absolute differences of pixel values which are adjacent to each other. Lin et al. call the differences as predictive errors. The sender outputs the numbers of predictive errors in histogram and finds a peak point and a zero point. Predictive errors between peak point and zero point are shifted to generate the space nearby peak point. The sender embeds the secret data “1” by shifting the predictive errors from peak point to its nearby space and secret data “0” by remaining the predictive errors. When all the predictive errors are scanned, the sender reverses predictive errors into pixel values and generates stego-image. Having similar values, the differences between nearing pixels are closed to zero so that the numbers of peak points are much more than Ni et al.’s. Since the capacity of data hiding depends on the sum of the number of peak points, Lin et al.’s method has more capacity than Ni et al.’s. Lin et al.’s method has higher capacity using histogram of predictive errors, but their method is not effective for complex images since some pixel values of complex images are salient. Yang et al. proposed a data hiding method to improve the algorithm of computing predictive errors in 2009[21]. In Yang et al.’s method, the sender divides the cover image into white and black segments, like a chessboard. The sender computes the predictive errors of white segment by subtracting the average of their circumferential pixel values in floor function. The statistics of predictive errors are compiled and 2.

(10) outputted in histogram. Through the histogram, the sender gets two peak points and their corresponding zero points. After shifting the pixel values between peak points and zero points, the spaces next to peak points are generated. The sender embeds the secret data “1” by shifting the pixel values from peak points to their nearby spaces and secret data “0” by remaining the pixel values. After all the predictive errors are scanned, the sender reverses predictive errors into pixel values and gets the stego-image. If there are remained secret data, the sender computes the predictive errors of black segment by subtracting the average of their circumferential pixel values in floor function. The sender compiles the embedded algorithm until all the secret data are embedded into cover image. After executing algorithm twice, the sender gets the stego-image. This method avoids the interference from salient pixel value. The capacity of Yang et al.’s method is higher than that of the presented data hiding methods. Although many studies are proposed to improve data hiding methods over binary computers, the studies seldom focus on data hiding methods over non-binary computers. Non-binary computers are used for special purpose nowadays like quantum computer [14]. The famous non-binary computer is the ternary computer [7]. Over the ternary computer, the balanced ternary numbering system [8] is described by Knuth as “the prettiest number system of all” [11] for its elegant arithmetic properties. The ternary concept will be found in optical computer [22] or quantum computer [14]. The main notation in the ternary computer is represented by {0, 1 , 1 }. Having the same concept, Non-Adjacent Form (NAF) is the ideal method to transform binary secret data into ternary format. The secret data in NAF format have three characteristics [8, 22]: the data are represented by {0, 1 , 1 }, 1 and 1 are never adjacent, and the probability of appearing 0 is about 66%. Based on the characteristics in NAF format, we propose a data hiding method over ternary computer with high quality of 3.

(11) stego-image. In our method, the sender embeds the secret data of NAF format into cover image. The sender divides the cover image into white and black segments. The sender computes the predictive errors of white segment by subtracting the average of their circumferential pixel values in floor function. The numbers of predictive errors are outputted in histogram. Through the histogram, the sender gets a peak point and its two corresponding zero points. Predictive errors between zero points are shifted except the peak point to generate the spaces nearby the peak point. The sender embed secret data “ 1 ” by shifting the predictive errors of peak point to their left space, secret data “1” by shifting the predictive errors of peak point to their right space and secret data “0” by remaining the pixel value. After all the predictive errors are scanned, the sender reverses predictive errors into pixel values and generates quality stego-image. If there are remained secret data, the sender computes the predictive errors of black segment by subtracting the average of their circumferential pixel values in floor function. The sender repeats the embedded algorithm until all the secret data are embedded into cover image. After executing algorithm twice, the sender gets the stego-image. In the end, the sender transports the stego-image, peak points, zero points and length of secret data to the receiver. Using acquired information, the receiver can extract secret data from the stego-image and discover cover image. From the past researches [2, 3, 5, 6, 9, 17], the average number of peak points should be shifted is 50% in binary format [1, 2, 3, 9, 17, 19] and 33% in NAF format [7, 8, 11]. We give some experiments by Yang et al.’s method and ours. According to experiment results, our method can keep higher PSNR when the number of first peak point is three times than the number of second peak point. Further more, we prove that our method can gain 1.76 PSNR more than Yang et al.’s method in best case. 4.

(12) Outline of this thesis is as follows. In Chapter 2, we introduce Ni et al.’s method, Lin et al.’s method, Yang et al.’s method and the format of NAF. Improved method is given in Chapter 3. The experiment results are shown in Chapter 4. In Chapter 5, we compare Yang et al.’s method and ours and compute the added PSNR in our method. Finally, we make a conclusion in Chapter 6.. 5.

(13) Chapter 2 Related Works In this chapter, we review the algorithm of Ni et al.’s method, Lin et al.’s method, Yang et al.’s method and the format of NAF. Before introducing these methods, we give the required definitions in this chapter. The cover image size is 512 * 512 . Every position of pixel (i, j ) in the cover image th th has a pixel value Pi , j {Pi , j | 1  i, j  512} . H mn is the m peak point of n. th th segment and Z mn is the m zero point of n segment. The length of secret data is. L and we assune that L is less than capacity in this thesis. Bk {Bk | k  L} is the k th th bit of secret data in binary format and N k {N k | k  L} is the k bit of secret data in. NAF format. Every Pi , j. has its own specific neighbor set S i , j . The predictive error of. (i, j ) is Di , j {Di , j | i  512, j  512} .. 6.

(14) 2.1 Ni et al.’s Method[13] Reversible data hiding based on histogram modification was proposed in 2006 by Ni et al. Assume that the sender wants to send the secret data B to the receiver. Before transporting secret data B to the receiver, the sender embeds secret data into cover image using Ni et al.'s method as follows. The sender outputs the numbers of pixel values in the cover image and gets histogram of pixel values. The sender gets { H 11 , H 21 ,. Z11 , Z 21 } where Z11  H11  H 21  Z 21 from the histogram. The pixel values in the range of ( H 11 , Z11 ) and ( H 21 , Z 21 ) are shifted by Formula (2-1) to make the spaces nearby peak points H 11 and H 21 ..  Pi , j  1, Pi , j   Pi , j  1,. if Z11  Pi , j  H11 if H 21  Pi , j  Z 21. (2-1). The sender scans pixel values from up to down and left to right until all the pixel values are scanned. When scanned pixel values are equal to peak points H 11 and H 21 , the sender embeds secret data by Formula (2-2)..  Pi , j  1, k  , if Pi , j  H11 and Bk  1  Pi , j   Pi , j  1, k  , if Pi , j  H 21 and Bk  1  P , k  , if Pi , j  H11 || H 21 and Bk  0  i, j. (2-2). The sender transports stego-image, L and ( H 11 , H 21 , Z11 , Z 21 ) to the receiver. Since shifted pixel values are not over lapping, the receiver can completely extract secret data and discover the original image. When receiving the information, the receiver scans the stego-image from up to down and left to right until all the pixel values are scanned. When scanned pixel values 7.

(15) are equal to peak points, the receiver extracts embedded data by Formula (2-3)..  Bk  1, k    Bk  0, k  . if Pi , j  H11  1 || H 21  1 if Pi , j  H11 || H 21. (2-3). To get the cover image, the user scans pixel values of stego-image again and shifts pixel values by Formula (2-4)..  Pi , j  Pi , j  1  Pi , j  Pi , j  1. if Z11  Pi , j  H11 if H 21  Pi , j  Z 21. (2-4). Since the redundant secret data are extracted when L  H11  H 21 , the user has to delete redundant Bk where k  L to get real secret data. In Ni et al.’s method, Stego-image remains very high quality because all the differences between embedded and original pixel values are at most one. Although their method has high quality of stego-image, Ni et al.’s method has little capacity because the capacity is depended on H11  H12 , the top two highest numbers of pixel values. To gain the capacity, Lin et al. propose an improving method in 2008.. 8.

(16) 2.2 Lin et al.’s Method[12] Lin et al. propose a method using the histogram of predictive errors to gain the capacity of data hiding methods. Before transporting secret data B to the receiver, the sender embeds secret data into cover image using Lin et al.'s method as follows. The sender computes all the predictive errors by Formula (2-5).. Di , j  Pi , j  Pi , j 1. (2-5). The sender outputs the numbers of predictive errors and gets histogram of predictive errors. The sender finds ( H 11 , Z11 ) where H 11  Z11 . The predictive errors in the range of ( H 11 , Z11 ) are shifted by Formula (2-6) to make the space nearby peak point. Di , j  Di , j  1,. if H11  Di , j  Z11. (2-6). The sender scans predictive errors from up to down and left to right until all the predictive errors are scanned. When scanned predictive errors are equal to peak point. H 11 , the sender shifts predictive errors and embeds secret data into cover image by Formula (2-7)..  Di , j  1, k  , Di , j    Di , j , k  ,. if Di , j  H11 and Bk  1 if Di , j  H11 and Bk  0. (2-7). Predictive errors are scanned again and reversed into pixel values by Formula (2-8).. 9.

(17)  if Pi , 0  Pi ,1  Pi , 0  Pi , 0     Pi ,1  Di ,0 else   Pi , 0  Di , 0 if Pi ,0  Pi ,1   Pi ,1   else  Pi ,1    Pi ,i1  Di , j 1 if Pi , j 1  Pi ,1  Pi , j    Pi ,i1  Di , j 1 else . (2-8). The sender transports stego-image, L , ( H 11 , Z11 ) to the receiver. When receiving the information, the receiver computes all the predictive errors by Formula (2-5). The receiver scans the predictive errors from up to down and left to right until all the predictive errors are scanned. When scanned predictive errors are equal to peak point H 11 , the receiver extracts embedded data by Formula (2-9)..  Bk  1, k    Bk  0, k  . if Di , j  H11  1 if Di , j  H11. (2-9). After extracting embedded data, the receiver shifts predictive errors by Formula (2-10).. Di , j  Di , j - 1 if H11  1  Di , j  Z11  else  Di , j  Di , j. (2-10). To get the cover image, the receiver scans predictive errors again and reverses predictive errors into pixel values by Formula (2-8). Since the redundant secret data are extracted when L  H11 , the user has to delete redundant Bk where k  L to get real secret data. Since differences between neighboring pixel values are usually closed to zero, Lin et al.’s method has higher H 11 . In other words, Lin et al.’s method has higher capacity than Ni et al.’s method. But Lin et al.’s method is not always effect. In complex image like Figure 1, neighboring pixel values are very different. To improve the accuracy of 10.

(18) predictive errors in complex image, Yang et al. propose an improving data hiding method in 2009.. Figure 1.. An example for complex image.. 11.

(19) 2.3. Yang et al.’s Method[21]. Yang et al. propose a data hiding method to improve the accuracy of predictive errors in complex image. Before transporting secret data B to the receiver, the sender embeds secret data into cover image using Yang et al.'s method as follows. The sender divides the cover image into white and black segments like a chessboard. Figure 2 is the two segments of Yang et al.’s method. Figure 2.. Two segments of Yang et al.’s method.. The sender compiles the embedded algorithm with one segment at once. In first part of embedded algorithms, the sender computes all the predictive errors in white segment by Formula (2-11)..  S i , j  {Px , y | x, y  [1,512], i  x  j  y  1}    Pi , j    Pi , j Si , j     Di , j  Pi , j   S  i, j     . (2-11).. The sender outputs the numbers of predictive errors of white segment and gets histogram of predictive errors. The sender find ( H 11 , H 21 , Z11 , Z 21 ) where. Z11  H11  H 21  Z 21 . The predictive errors in the range of ( H 11 , Z11 ) and ( H 21 , Z 21 ) are shifted by Formula (2-12) to make the spaces nearby peak points. 12.

(20)  Di , j  1, Di , j    Di , j  1,. if Z11  Di , j  H11 if H 21  Di , j  Z 21. (2-12). The sender scans predictive errors from up to down and left to right until all the predictive errors are scanned. When scanned predictive errors are equal to peak points ( H 11 , H 21 ), the sender shifts predictive errors and embeds secret data into cover image by Formula (2-13).. Di , j  1, k  ,  Di , j  Di , j  1, k  ,  Di , j , k  ,. if Di , j  H 21 and Bk  1 if Di , j  H11 and Bk  1. (2-13). if Di , j  H11 || H 21 and Bk  0. Predictive errors are scanned again and reversed into pixel values by Formula (2-14)..  S i , j  {Px , y | x, y  [1,512], i  x  j  y  1}    Pi , j    Pi , j Si , j   P  D  i , j i , j    Si, j      . (2-14). If there are remained secret data, the sender compiles second part of embedded algorithm for black segment as follows. In second part of embedded algorithms, the sender computes all the predictive errors in black segment by Formula (2-11). The sender outputs the numbers of predictive errors of black segment and gets histogram of predictive errors. The sender finds ( H 12 , H 22 , Z12 , Z 22 ) where Z12  H12  H 22  Z 22 . The predictive errors in the range of ( H 12 , Z12 ) and ( H 22 , Z 22 ) are shifted by Formula (2-15) to generate the spaces nearby peak points..  Di , j  1, Di , j    Di , j  1,. if Z12  Di , j  H12 if H 22  Di , j  Z 22. (2-15). The sender scans predictive errors from up to down and left to right until all the predictive errors are scanned. When scanned predictive errors are equal to peak points ( H 12 , H 22 ), the sender shifts predictive errors and embeds secret data into cover image 13.

(21) by Formula (2-16)..  Di , j  1, k  ,  Di , j   Di , j  1, k  ,   Di , j , k  ,. if Di , j  H 22 and Bk  1 if Di , j  H12 and Bk  1. (2-16). if Di , j  H12 || H 22 and Bk  0. Predictive errors are scanned again and reversed into pixel values by Formula (2-14). The sender transport stego-image, L , ( H 11 , H 21 , Z11 , Z 21 ) and ( H 12 , H 22 , Z12 , Z 22 ) to the receiver. When receiving the information, the receiver divides stego-image into white and black segments as Figure 2 and compiles the extracted algorithm with one segment at once. In first part of extracted algorithms, the receiver computes all the predictive errors in black segment by Formula (2-11). The receiver scans the predictive errors from up to down and left to right until all the predictive values are scanned. When the scanned predictive errors are equal to peak points ( H 12 , H 22 ), the receiver extracts embedded data by Formula (2-17)..  Bk  1, k    Bk  0, k  . if Di , j  H12  1 || H 22  1 | if Di , j  H12 || H 22. (2-17). After extracting embedded data, the receiver shifts predictive errors by Formula (2-18)..  Di , j  1, Di , j    Di , j  1,. if Z12  Di , j  H12 if H 22  Di , j  Z 22. (2-18). To get the cover image, the receiver scans predictive errors again and reverses predictive errors into pixel values by Formula (2-14). Then the receiver swaps black segment for white segment and computes second part of extracted algorithms as follows. In second part of extracted algorithms, the receiver computes all the predictive errors in white segment by Formula (2-11). The receiver scans the predictive errors from up to down and left to right until all the predictive values are scanned. When the scanned 14.

(22) predictive errors are equal to peak points ( H 11 , H 21 ), the receiver extracts embedded data by Formula (2-19)..  Bk  1, k    Bk  0, k  . if Di , j  H11  1 || H12  1 | if Di , j  H11 || H12. (2-19). After extracting embedded data, the receiver shifts predictive errors by Formula (2-20)..  Di , j  1, Di , j    Di , j  1,. if Z11  Di , j  H11 if H12  Di , j  Z12. (2-20). To get the cover image, the receiver scans predictive errors again and reverses predictive errors into pixel values by Formula (2-14). Since the redundant secret data are extracted when L  H11  H 21  H12  H 22 , the receiver has to combine the extracted secret data of two segments and delete redundant Bk where k  L to get real secret data. To understand Yang et al.’s method, we give the following example. Assume that the sender wants to embed the secret data B  {01101111010100) into the cover image in Figure 3.. Figure 3.. The cover image.. The sender divides the cover image into white and black segments and computes all the predictive errors of white segment by Formula (2-11). The results are shown in Figure 4. 15.

(23) Figure 4.. The predictive errors of white segment.. The sender outputs the numbers of predictive errors in white segment and gets histogram of predictive errors as Figure 5. 8. 2. 2. -2. Figure 5.. -1. 0. 1. 2. The histogram of predictive errors of white segment.. Obviously, the sender gets H11  0 , H 21  1 , Z11  1 and Z 21  2 . Then the sender shifts predictive errors in the range (-1,0) and (0,2) to generate the spaces nearby peak points ( H11, H 21 ) by Formula (2-12). The sender scans predictive errors of white segment from up to down and left to right. Predictive errors are shifted to embed secret data (0110111101) into the image in Figure 4 by Formula (2-13). After shifting predictive errors, the sender gets the embedded histogram and shifted predictive errors as Figure 6.. 16.

(24) 6. 2. 2. 1. 1 -2. Figure 6.. -1. 0. 1. 2. Embedded histogram and shifted predictive errors of white segment.. The sender scanned predictive errors again and reverses the predictive errors into pixel values by Formula (2-14). The results are shown in Figure 7.. Figure 7.. The stego-image after first algorithm.. To embed remained secret data (0100) , the sender swaps white segment for black segment and computes all predictive errors of black segment by Formula (2-11). The results are shown in Figure 8.. Figure 8.. The predictive errors of black segment.. 17.

(25) The sender outputs the numbers of predictive errors in black segment and gets histogram of predictive errors as Figure 9. 4. 3. 3. 2. -2. Figure 9.. 1. -1. 0. 1. 2. 3. 4. 5. The histogram of predictive errors in black segment.. Obviously, the sender gets H12  0 , H 22  1 , Z12  2 and Z 22  3 . Then the sender shifts predictive errors in the range (-2,0) and (0,3) to generate the spaces nearby peak points H 11 and H 12 by Formula (2-15). The sender scans predictive errors of black segment from up to down and left to right. Predictive errors are shifted to embed secret data (0100) into the image in Figure 8 by Formula (2-16). After shifting predictive errors, the sender gets the embedded histogram and shifted predictive errors as Figure 10.. 4 3 2. -2. Figure 10.. -1. 0. 2. 1. 1. 2. 1 3. 4. 5. Embedded histogram and shifted predictive errors of black segment.. The sender scans predictive errors again and reverses the predictive errors into pixel values by Formula (2-14). The results are shown as Figure 11.. 18.

(26) Figure 11.. The stego-image.. The secret data are completely embedded into cover image so far. The sender transports stego-image, L , H 11 , H 21 , Z11 , Z 21 , H 12 , H 22 , Z12 and Z 22 to the receiver. When receiving the information, the receiver divides the stego-image into white and black segments as Figure 11. In the first part of extracted algorithms, the receiver computes all the predictive errors of black segment by Formula (2-11). The results are shown in Figure 12.. Figure 12.. The shifted predictive errors of black segment.. The receiver scans predictive errors from up to down and left to right. The embedded data are extracted by Formula (2-17). After extracting embedded data, predictive errors are shifted by Formula (2-18). The receiver scans predictive errors again and reverses the predictive errors into pixel values by Formula (2-14). The results are shown in Figure 7. After compiling first part of extracted algorithms, the receiver gets the 19.

(27) embedded data (0100000) . In second part of extracted algorithms, the receiver swaps black segment for white segment and computes all the predictive errors of white segment by Formula (2-11). The results are shown in Figure 13.. Figure 13.. The shifted predictive errors of white segment.. The receiver scans predictive errors from up to down and left to right. The embedded data are extracted by Formula (2-19). After extracting embedded data, predictive errors are shifted by Formula (2-20). The receiver scans predictive errors again and reverses the predictive errors into pixel values by Formula (2-14). The results are as Figure 3. After compiling second part of extracted algorithms, the receiver gets the embedded data. (0110111101) . Since the redundant secret data are extracted when. L  H11  H 21  H12  H 22 , the receiver has to combine the extracted data of two. segments and delete redundant Bk where k  L to get real secret data. Finally, the receiver gets the cover image and secret data B  (01101111010100) .. 20.

(28) 2.4 Non-Adjacent Form(NAF) format[1, 7, 8] NAF format transforms the binary string into the ternary string, presented by { 1 , 0, 1}. Through NAF format, the string has three important characteristics: 1. the data are represented by {0, 1 , 1 } 2. The probability of appearing 0 is about 66%. 3. 1 and 1 are never adjacent. We use these two characteristics to hide the secret data with high quality of stegoimage. Before we introduce the NAF algorithm, we define that B  ( B1 B2 B3 ......Bk ) is the binary format of secret data and N  ( N1 N 2 N 3 ......N k ) is the NAF format of secret data. Let R be the carrier in NAF algorithm. The NAF algorithm is listed as follows. let N L1  0 , R  0 for k  1 : L, k is odd switch. Bk Bk 1 R. case ‘00’: N k N k 1 =00, R  0 ;break; case ‘10’: N k N k 1 =10, R  0 ;break; case ‘01’: if Bk  2 =1 then N k N k 1 =0 1 , R  1 ; else N k N k 1 =01, R  0 ;break; case ‘11’: N k N k 1 = 1 0, R  1 ;break; case ‘001’: N k N k 1 =00, R  1 ;break; end_switch end_for Figure 14.. The algorithm of NAF format. 21.

(29) To understand the NAF algorithm, we use the secret data in Yang et al.’s method as an example. We have secret data B  ( B1 B2 B3 B4 B5 B6 B7 B8 B9 B10 B11B12 B13B14 ) =(01101111010100).We set R  0 and show the steps of NAF algorithm as Figure 15. Step 1.. B1 B2  R  01, N1 N 2  0 1, R  1. Step 2.. B3 B4  R  01, N 3 N 4  0 1, R  1. Step 3.. B5 B6  R  001, N 5 N 6  00, R  1. Step 4.. B7 B8  R  001, N 7 N 8  00, R  1. Step 5.. B9 B10  R  11, N 9 N10  1 0, R  1. Step 6.. B11B12  R  11, N11N12  1 0, R  1. Step 7.. B13 B14  R  10, N11N12  10, R  0. Figure 15.. An example of NAF format. According to Figure 15, we get the NAF format of secret data N =(0 1 0 1 0000 1 0 1 010).. 22.

(30) Chapter 3 Proposed Method In this chapter, we present our method and give an example to understand our method easily. Assume that the sender embeds the secret data N into the cover image using our method and sends the stego-image to the receiver. As mentioned in Chapter 2, the cover image size is 512 * 512 . Every position of pixel (i, j ) in the cover image th th has a pixel value Pi , j {Pi , j | 1  i, j  512} . H mn is the m peak point of n segment. th th and Z mn is the m zero point of n segment. By the way, it is worth to notice that. we only get one peak point in each segment. The length of secret data is L and we let. L be less than capacity in this chapter. Bk {Bk | k  L} is the k th bit of secret data in th binary format and N k {N k | k  L} is the k bit of secret data in NAF format. Every. Pi , j. has. its. own. specific neighbor set. S i , j . The predictive error is. Di , j {Di , j | i  512, j  512} .. Now the sender wants to embed secret data N , transformed by NAF algorithm if the secret data are represented in binary format, into the cover image. The sender performs the following algorithm.. 23.

(31) 3.1 Embedded Algorithm Input: cover image, N . Output: stego-image , L , H 11 , Z11 , Z 21 , H12 , Z12 and Z 22 . Step 1.. Read and divide the cover image into white and black segments as Figure 2.. Step 2.. Compute all predictive errors in white segment by Formula (3-1)..  S i , j  {Px , y | x, y  [1,512], i  x  j  y  1}    Pi , j    Pi , j Si , j   D  P  i , j i , j    S i , j      . (3-1).. Step 3.. Output the numbers of Di , j in white segment in histogram.. Step 4.. Find H 11 , Z11 and Z 21 where Z11  H11  Z 21 .. Step 5.. Shift the predictive errors by Formula (3-2) to generate spaces, H11  1 and H11  1 , nearby the peak point H 11 ..  Di , j  1, Di , j   Di , j  1, Step 6.. if Z11  Di , j  H11 if H11  Di , j  Z 21. (3-2). Scan predictive errors in white segment from up to down and left to right until all the predictive errors are scanned. Shift the predictive errors to embed the secret data by Formula (3-3)..  Di , j  1, k  , if Di , j  H11 and N k  1  Di , j   Di , j  1, k  , if Di , j  H11 and N k  1  if Di , j  H11 and N k  0  Di , j , k  , Step 7.. (3-3). Scan white segment again and reverse predictive errors into pixel values by Formula (3-4). 24.

(32)  S i , j  {Px , y | x, y  [1,512], i  x  j  y  1}    Pi , j    Pi , j Si , j     Pi , j  Di , j   S  i, j     . (3-4). Step 8.. Compute all predictive errors in black segment by Formula (3-1).. Step 9.. Out put the numbers of Di , j in black segment in histogram.. Step 10.. Find H12 , Z12 and Z 22 where Z12  H12  Z 22 .. Step 11.. Shift the predictive errors by Formula (3-5) to generate spaces, H12  1 and H12  1 , nearby the peak point H12 ..  Di , j  1, Di , j   Di , j  1, Step 12.. if Z12  Di , j  H12 if H12  Di , j  Z 22. (3-5). Scan predictive errors in black segment from up to down and left to right until all the predictive errors are scanned. Shift the predictive errors to embed the secret data by Formula (3-6)..  Di , j  1, k  , if Di , j  H12 and N k  1  Di , j   Di , j  1, k  , if Di , j  H12 and N k  1  if Di , j  H12 and N k  0  Di , j , k  , Step 13.. ( 3-6). Scan black segment again and reverse predictive errors into pixel values by Formula (3-4). Step 14.. Transport stego-image, L , H 11 , Z11 , Z12 , H 12 , Z12 and Z 22 to the receiver.. 25.

(33) 3.2 Extracted Algorithm After receiving the information from the sender, the receiver performs the following algorithm to get the secret data and the original cover image.. Input: stego-image, L , H 11 , Z11 , Z12 , H 12 , Z12 and Z 22 . Output: secret data, cover image. Step 1.. Read and divide stego-image into white and black segments as Figure 2.. Step 2.. Compute all predictive errors in black segment by Formula (3-1).. Step 3.. Scan pixel values in black segment from up to down and left to right until all the secret data in black segment are scanned. Extract secret data by Formula (3-7).. Bk  1, k     Bk  0, k   Step 4.. if Di , j  H12  1 || H12  1 if Di , j  H12. (3-7). After extracting embedded data, the receiver shifts predictive errors by Formula (3-8)..  Di , j  1, Di , j    Di , j  1, Step 5.. if Z12  Di , j  H12 if H12  Di , j  Z 22. (3-8). Scan black segment again and return predictive errors to pixels by Formula (3-4).. Step 6.. Compute all predictive errors in white segment by Formula (3-1).. Step 7.. Scan pixel values in white segment from up to down and left to right until all the secret data in white segment are scanned. Extract secret data by Formula (3-9).. 26.

(34) Bk  1, k     Bk  0, k   Step 8.. if Di , j  H11  1 || H11  1 if Di , j  H11. (3-9). After extracting embedded data, the receiver shifts predictive errors by Formula (3-10)..  Di , j  1, Di , j    Di , j  1, Step 9.. if Z11  Di , j  H11 if H11  Di , j  Z 21. (3-10). Scan white segment again and return predictive errors to pixels by Formula (3-4).. Step 10.. Combine all extracted data and delete redundant bits N k {N k | k  L} .. Step 11.. Output secret data N and the cover image.. We implement the same example in Chapter 2 by using our method as follows. Now the sender wants to embed the secret data N  (0 1 0 1 0000 1 0 1 010) into the cover image in Figure 3. The sender divides the cover image into white and black segments and computes all the predictive errors of white segment by Formula (3-1) as Figure 4. Then the sender outputs the numbers of predictive errors in white segment and gets histogram of predictive errors as Figure 5. Different from Yang et al.’s method, our method only gets H11  0 , Z11  3 and Z 21  1 . Then the sender shifts predictive errors in the range (  3 , 0 ) and ( 0 , 1 ) to generate the spaces nearby peak point H11 by Formula (3-2). The sender scans predictive errors from up to down and left to right. Predictive errors are shifted to embed secret data (0 1 00 1 000) by Formula (3-3). After shifting predictive errors, the sender gets the embedded histogram and shifted predictive errors as Figure 16.. 27.

(35) 6. 2. 2 2. -3. Figure 16.. -2. -1. 0. 1. 2. Embedded histogram and shifted predictive errors of white segment with our method.. The sender reverses the predictive errors into pixel values by Formula (3-4). The results are shown in Figure 17.. Figure 17.. The stego-image after first algorithm.. To embed remained secret data (1 0 1 010) , the sender swaps white segment for black segment and computes all the predictive errors of black segment by Formula (3-1) as Figure 18.. Figure 18.. The predictive errors of black segment.. 28.

(36) Given the numbers of predictive errors in black segment, the sender gets histogram of predictive errors as Figure 19. 6 3. 3 1. -2. Figure 19.. -1. 0. 1. 2. 3. 4. 5. The histogram of predictive errors in black segment.. Obviously, the sender gets H12  0 , Z12  2 and Z 22  1 . Then the sender shifts predictive errors in the range (  2 , 0 ) and ( 0 , 1 ) to generate the spaces nearby peak point by Formula (3-5). The sender scans predictive errors of black segment from up to down and left to right. Predictive errors are shifted to embedded secret data. N  (1 0 1 010) into image by Formula (3-6). After shifting predictive errors, the sender can get the embedded histogram and shifted predictive errors as Figure 20.. 3. 3. 3. 2. 1. 1 -2. Figure 20.. -1. 0. 1. 2. 3. 4. 5. Embedded histogram and shifted predictive errors of black segment with our method.. The sender reverses the predictive errors into pixel values by Formula (3-4). The results are shown in Figure 21.. 29.

(37) Figure 21.. The stego-image.. Now the secret data are completely embedded into the cover image. The sender transports stego-image, L , H 11 , Z11 , Z12 , H 12 , Z12 and Z 22 to the receiver. After receiving the information, the receiver divides the stego-image into white and black segments as Figure 19. Then the receiver computes all the predictive errors of black segment by Formula (3-1). The results are shown in Figure 22.. Figure 22.. The shifted predictive errors of black segment.. Then the receiver scans predictive errors from up to down and left to right. The receiver extracts secret data ( 100 1 01) by Formula (3-7). After extracting embedded data, the receiver shifts predictive errors by Formula (3-8) as Figure 23. The receiver scans black segment again and reverses the predictive errors into pixel values by Formula (3-4). The results are shown in Figure 15. To extract the secret data in white segment, the receiver computes the predictive errors of white segment by Formula (3-1). The results are 30.

(38) shown in Figure 23.. Figure 23.. The shifted predictive errors of white segment.. Then the receiver scans predictive errors from up to down and left to right for the segment. The embedded data (0 1 00 1 000) are extracted by Formula (3-9). After extracting embedded data, the receiver shifts the predictive errors by Formula (3-10) as Figure 4. The receiver scans white segment again and reverses the predictive errors into pixel values by Formula (3-10) and gets the cover image as Figure 3. Since the redundant secret data are extracted when L  H11  H12 , the receiver has to combine the extracted secret data of two segment and delete redundant N k where k  L to get real secret data N  (0 1 0 1 0000 1 0 1 010) . Comparing Figure 11 and Figure 21, more three pixel values are not shifted in our method. We compile some experiments and show the results in the next chapter.. 31.

(39) Chapter 4 Experiment Results In this chapter, we show the experiment results of Yang et al.’s and our methods. We let M 1 be the number of highest peak point and M 2 be the number of second highest. peak. point.. Using. the. example. in. Chapter. 3,. we. have. ( H11 , H 21 , H12 , H 22 )  (2, 6, 3, 6) and get (M 1 , M 2 )  (6  6, 2  3)  (12, 5) .. We construct the secret data randomly, and embed the secret data in ten cover images, as Figure 24, using Yang et al.’s and our methods. In Table 1., we know the higher PSNR is kept with our improved method if M 1 / M 2  3 .. airplane. peppers. boat. lena. holidayhouse. swimmingpool. snowtree. dubai. room. cloudsea. Figure 24.. Ten images in our experiment.. 32.

(40) Table 1. Image Name. The experiment results of Yang et al.’s and our method.. M2. M1. M1 / M 2. PSNR Gained PSNR Yang et al.'s mrthod. Our method. airplane. 11976. 7099. 1.687. 49.935. 49.915. -0.020. peppers. 45572. 23765. 1.918. 50.212. 50.149. -0.063. boat. 57120. 28196. 2.026. 50.288. 50.226. -0.062. lena. 61481. 29854. 2.059. 50.350. 50.290. -0.060. holidayhouse. 114699. 38156. 3.006. 50.671. 50.674. 0.003. swimmingpool. 100613. 19511. 5.157. 50.587. 50.692. 0.105. snowtree. 90318. 15678. 5.761. 50.466. 50.572. 0.106. dubai. 112162. 17352. 6.464. 50.674. 50.824. 0.150. room. 176220. 21630. 8.147. 51.183. 51.488. 0.306. cloudsea. 195970. 21070. 9.301. 51.328. 51.733. 0.405. In addition, we also make some experiments with Lin et al.’s method[9], and other well-known data hiding methods like Gradient Adjacent Predictive Method[5, 6, 14] and Block Predictive Method[2, 13]. The experiment results are shown in Table 2.. Table 2. Lin et al.'s method[9]. The experiment results of other methods.. Yang et al.'s method[17]. GAP[5, 6, 14]. Block[2, 13]. M 1 / M 2 Gained PSNR M 1 / M 2 Gained PSNR M 1 / M 2 Gained PSNR M 1 / M 2 Gained PSNR airplane. 2.05. -0.09. 1.69. -0.02. 2.51. -0.06. 1.36. -0.14. peppers. 1.63. -0.07. 1.92. -0.06. 1.94. -0.07. 1.07. -0.11. boat. 1.68. -0.09. 2.03. -0.06. 1.99. -0.08. 1.07. -0.14. lena. 1.67. -0.09. 2.06. -0.06. 2.01. -0.08. 1.12. -0.05. holidyhouse. 1.90. -0.13. 3.01. 0.00. 2.85. -0.01. 1.33. -0.16. swimmingpool. 4.70. 0.10. 5.16. 0.11. 4.41. 0.10. 1.96. -0.08. snowtree. 4.50. 0.07. 5.76. 0.11. 4.86. 0.11. 2.03. -0.07. dubai. 5.50. 0.14. 6.46. 0.15. 5.09. 0.15. 2.18. -0.06. room. 6.12. 0.26. 8.15. 0.31. 5.13. 0.26. 1.51. -0.26. cloudsea. 6.27. 0.30. 9.30. 0.41. 4.29. 0.20. 1.46. -0.27. 33.

(41) To find the relationship between M 1 / M 2 and gained PSNR, we show all the results and get Figure 25. M1 / M 2 0.50 0.40 0.30 0.20 0.10 0.00 -0.10. 0.00. 1.00. 2.00. 3.00. 4.00. 5.00. 6.00. 7.00. 8.00. 9.00. 10.00 Gained PSNR. -0.20 -0.30 -0.40 Figure 25.. The relationship between H 1 / H 2 and gained PSNR. From all the experiment results, higher PSNR is kept in our method if. M 1 / M 2  3 . And the higher M 1 / M 2 the higher PSNR is kept. To prove our experiment results, we make discussion and analysis in next chapter.. 34.

(42) Chapter 5 Discussion In this chapter we compare Yang et al.’s method with ours. Given an example of histogram in Figure 26., we can find the first two peak points, H 1 and H 2 , and their corresponding zero points, Z1 and Z 2 . We let M 1 be the number of. H1 , M 2 be. the number of H 2 , O1 bethe sum of the numbers in the range of ( H1 , Z1 ), O2 be the sum of the numbers in the range of ( H1 , H 2 ) and O3 be the sum of the numbers of predictive errors in the range of. Figure 26.. ( H 2 , Z 2 ).. The distribution of histogram.. Assuming the secret data is presented by {0, 1} randomly, the probability of appearing 0 is 1/2. In Yang et al.’s method, the number of pixel values which should be. 1 2. shifted is: O1  O3  ( M 1  M 2 ) . In NAF format, the sum of probability appearing 1 and 1 is 1/3. Therefore, the number of pixel values which we should shift is:. 1 O1  O2  O3  M 2  M1 . As compared with Yang et al.’s method, the percentage of 3 our method reduces the number of shifted pixel values is about 33%. 35.

(43) 1 1 ( M 1  M 2 ))  (O1  O2  O3  M 1  M 2 ) 2 3 1 O1  O3  ( M 1  M 2 ) 2 1 1 ( M 1  M 2 )  O2  M 2  M 1 3 2 1 O1  O3  ( M 1  M 2 ) 2 3M 1  3M 2  6O2  6 M 2  2 M 1  6O1  6O3  3M 1  3M 2 (O1  O3 . . (5.1). M 1  3M 2  6O2 3M 1  3M 2  6O1  6O3. 1 In general, we have that O2 is zero. Thus, if M 2  M 1 , the number of shifted pixel 3 values in our method is less than that in Yang et al.’s method. According to Formula (5.1), in best case, our method reduce 33% of shifted values of Yang et al.’s methods. Now. we. compute. MSE . 1 m1  mn i 0. n 1.  j 0. the. added. I (i, j )  K (i, j ). 2. PSNR. of. most.. Given. the. formula. of.  MAX I2   , and the formula of PSNR  10  log 10   MSE . we can get the added PSNR is about 1.76 by Formula (5.2) ..    256 2   256 2     10  log 10  10  log 10   N   2N       3     256 2 N   10  log 10   2  2 N 256     3  3  10  log 10   2  1.7609 36. (5.2).

(44) Chapter 6 Conclusion In this thesis, we propose a reversible data hiding method which can be used in ternary computers. We embed ternary secret data, or transformed from binary format by NAF algorithm, into cover image. Since the probability of appearing 0 is about 66% in NAF format, our method can reduce the number of shifted values by 33%. In addition, the added PSNR of our method is 1.76 more than Yang et al.’s data hiding method in best case. From the studies [7, 8], ternary numbering system has higher balance than binary ones. We believe ternary computer will become more and more available. Having the same peril during sending information, data hiding over ternary computer should be noticed. Our method not only is proper to ternary computes, but also improves the communication between binary and ternary computers. We believe this study is worth to continue.. 37.

(45) References [1] S. Arno, “Signed Digit Representations of Minimal Hamming Weight,” IEEE Transaction on Computers, Vol. 42, pp. 1007-1010, 1993. [2] X. Bo, Y. Lizhi, and H.Yongfeng, “Reversible Data Hiding Using Histogram Shifting in Small Blocks,” IEEE International Conference on Communications, pp. 1-6, 2010. [3] M. M. Carli, G. Boato, and K. Egiazarian, “Reversible Watermarking via Histogram Shifting and Least Square Optimization,” Proceedings of 12th ACM Workshop on Multimedia and Security, pp. 147-152, 2010. [4] M. U. Celik, G. Sharma, A. M. Tekalp, and E. Saber, “Reversible Data Hiding,” IEEE International Conference on Image Process, Vol. 2, no. 2-3, pp. 157-160, 2002. [5] M. Chen, Z. Chen, X. Zeng, and Z. Xiong, “Reversible Data Hiding Using Additive Prediction-Error Expansion,” Proceedings of the 11th ACM workshop on Multimedia and security, pp. 19-24, 2009. [6] D. Coltue, “Improved Embedding for Prediction Based Reversible Watermarking,” IEEE Transaction on Information Forensics and Security, Vol. 6, pp. 873-882, 2011. [7] M. Glusker, D. M. Hogan, and P. Vass, “The Ternary Calculating Machine of Thomas Fowler,” IEEE Annals of the History of Computing, Vol. 27, No. 3, pp. 4-22, 2005. [8] B. Hayes, “Third Base,” American Scientist, Vol. 89, No. 6, pp. 490–494, 2001. [9] A. Kingstin, and F. Autrusseau, “Lossless Image Compression via Predictive Coding of Discrete Radon Projections,” Signal Processing: Image Communication, Vol. 23, pp. 313-324, 2008. [10] H. J. Kim, V. Sachnev, Y. Q. Shi, J. Nam, and H. G. Choo, “A Novel Difference Expansion Transform for Reversible Data Embedding,” IEEE Transactions on Information Forensics and Security, Vol. 3, pp. 456-465, 2008. [11] D.E. Knuth, “The Art of Computer Programming,” Seminumerical Algorithms, Vol. 2, third ed. Addison-Wesley, 1998. [12] C. C. Lin, W. L. Tai, and C. C. Chang, “Multilevel Reversible Data Hiding Based 38.

(46) on Histogram Modification of Difference Images,” Pattern Recognit, Vol. 41, pp. 3582-3591, 2008. [13] Z. Ni, Y. Q. Shi, N. Ansari, and W. Su, “Reversible Data Hiding,” IEEE Transaction on Circuits and Systems for Video Technology, Vol. 16, pp. 354-362, 2006. [14] P. Pečar, A.Ramšak, N. Zimic, M. Mraz, and I. L. Bajec, “Adiabatic Pipelining : a Key to Ternary Computing with Quantum Dots,” Nanotechnology, Vol. 19, No. 49, pp. 1-12, 2008. [15] W. L. Tai, C. M. Yeh, and C. C. Chang, “Reversible Data Hiding Based on Histogram Modification of Pixel Differences,” IEEE Transaction on Circuits and Systems for Video Technology, Vol. 19, pp. 906-910, 2009. [16] P. Tsai, “Histogram-Based Reversible Data Hiding for Vector Quantization Compressed Images,” IET Image Processing, Vol. 3, pp. 100-114, 2009. [17] P. Tsai, Y. C. Hu, and H. L. Yeh, “Reversible Image Hiding Scheme Using Predictive Coding and Histogram Shifting,” Signal Processing, Vol. 89, pp. 1129-1143, 2009. [18] C. M. Wang, N. I. Wu, C. S. Tsai, and M. S. Hwang, “A High quality Steganographic Method with Pixel-Value Differencing and Modulus Function,” Journal of Systems and Software, Vol. 81, pp. 150-158, 2008. [19] X. Wu, and N. Memon, “Context-Based, Adaptive, Lossless Image Coding,” IEEE Transaction on Communications, Vol. 45, pp. 437-444, 1997. [20] H. W. Yang, K. F. Hwang, and S. S. Chou, “Interleaving Max-Min Difference Histogram Shifting Data Hiding Method,” Journal of Software, Vol. 5, pp. 615-621, 2010. [21] C. H. Yang, and M. H. Tsai, “Improving Histogram-Based Reversible Data Hiding by Interleaving Predictions,” IET Image Processing, Vol. 4, pp. 223-234, 2010. [22] J. Yi, H. Huacan, and L. Yangtian, “Ternary Optical Computer Architecture,” Physica Scripta, Vol. T118, pp. 98-101, 2005.. 39.

(47)