Information hiding based on binary encoding ... - Semantic Scholar
Information hiding based on binary encoding methods and pixel scrambling techniques Kuang Tsan Lin Department of Mechanical and Computer Aided Engineering St. John’s University 499, Section 4, Tam King Road, Tamsui, Taipei County 25135, Taiwan ([email protected]) Received 23 September 2009; revised 1 December 2009; accepted 4 December 2009; posted 4 December 2009 (Doc. ID 117621); published 7 January 2010
Novel information hiding for digital images based on binary encoding methods and pixel scrambling techniques is presented. First, a pixel scrambling technique is used to rearrange the pixels of a covert image to form a scrambled matrix by using a specified scrambling rule. Then, the gray values of all the pixels in the scrambled matrix are sequentially transformed into many sets of eight-digit binary codes. Subsequently, the eight-digit binary codes are encoded into a host image to form an overt image by using a specific encoding rule. Besides the eight-digit binary codes (information codes), the overt image contains five other groups of binary codes (specification codes), i.e., identification codes, gray-level codes, dimension codes, scrambling number codes, and scrambling time codes, to denote the parameters used for scrambling and encoding. According to the test results, the proposed method performs well. Moreover, the overt image and the host image look almost the same, and the decoded covert image is exactly the same as the original covert image. © 2010 Optical Society of America OCIS codes: 100.0100, 100.2000, 100.4998.
1. Introduction
Image scrambling techniques are important image encoding methods that have been applied to digital image processing, information hiding, and digital watermarking to enhance information security. Image scrambling techniques can be classified into two types according to image pixel arrangements. For the first type, matrix transformations, e.g., the Arnold transformation [1–3] and the p-Fibonacci transformation [4,5], are used. Because successful inverse transformations need the correct transformation parameters, the scrambling techniques can be protected by the secrete parameters. Although the methods perform well, the reconstruction of the encoded images requires external keys, and the transformations is complex. For the second type, coordinate movements, e.g., the cellular automata [6,7], the torus automorphism [8], and chaotic sequences [9], are used. Because successful image reconstruction needs the correct scrambling path 0003-6935/10/020220-09$15.00/0 © 2010 Optical Society of America 220
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parameters, the scrambling techniques can be protected by the secrete parameters. Although the methods perform well, external keys are needed for the image reconstruction, and some distortion may be found in reconstructed images. However, some studies have explored assembling image scrambling techniques and other image encrypting methods to increase image security. Ye and Li [10] combined an image scrambling technique and a watermarking technique. Hennelly and Sheridan [11] and Zhao et al. [12,13] combined image scrambling techniques and fractional Fourier transforms to hide covert images. Meng et al. [14] used an image scrambling technique and an iterative Fresnel transform technique to hide covert images. Wang et al. [15] combined an image scrambling technique and a binary Fourier transform technique by using computer-generated hologram techniques to hide covert images. Liu and Zhang [16] combined a point-set projection technique with a watermarking technique to hide covert images. Lu et al. [17] integrated an image scrambling technique with a double random-phase encoding technique to hide covert images. All the combination methods had very good
robustness, but they all needed external keys during covert-image reconstruction and reconstructed covert images may contain some distortion. This paper will propose a method that combines a binary encoding method [18] and a pixel scrambling technique to hide covert images in host images to form overt images. The proposed method can be applied to equilateral or nonequilateral images, and overt images look almost the same as their corresponding host images. External keys are not needed during covert-image reconstruction, but conventions to derive encoding parameters are needed for decoding covert images. In addition, the proposed method is simple and secure. Most of all, and there is no distortion for reconstructed covert images. The proposed encoding method can be applied to both digital and optical types of overt image. Although the encoding mechanisms of the two types of overt image are different, their encoding techniques are similar. Therefore, this paper will focus on just the digital type of overt image. For the optical type of overt image, one can refer to [18,19]. 2. Theory A. Pixel Scrambling and Descrambling Techniques for Encoding Information
Let C be an r × c covert image to be encoded and let D be an r × c matrix formed from the pixel scrambling of C. The processes for deriving D from C are presented below. First, map the pixels of C to form a pixel string A with r × c elements according to a specified order (from the first row to the last row and from the first column to the last column for the same row). Then, take every nth pixel in A (from the first pixel to the last pixel and recycled) to form the elements of D in sequence (from the first row to the last row and from the first column to the last column for the same row). After a pixel in A has been taken once, it is removed from A. The pixel-taking operations are repeated until all of the pixels in A are removed and relocated in D. An example for the scrambling processes from a 3 × 3 covert image C to a 3 × 3 matrix D is shown in Fig. 1. The descrambling processes from an r × c matrix D to an r × c covert image C are presented below. Because the three parameters r, c, and n are all known in advance for this case, we can simulate the scrambling processes introduced by the last paragraph to derive the mapping between the elements of C and D. The mapping from D to C is the inverse mapping from C to D for element positions, so we can easily map the elements from D to C by referencing C and D. An example of the descrambling from a 3 × 3 matrix D to a 3 × 3 covert image C is shown in Fig. 2. For the case with n ¼ 2, because r ¼ 3 and c ¼ 3 are known, the pixel mapping between the covert image C in Fig. 1(a) and the matrix D in Fig. 1(c) can be derived by a scrambling operation as shown in Fig. 1. Then, the pixel mapping from C to D is known; e.g., the element 11 of D corresponds to the element
Fig. 1. Example for pixel scrambling from C to D. (a) C, (b) A, (c) D for n ¼ 2, (d) D for n ¼ 3.
12 of C, and the element 22 of D corresponds to the element 11 of C. Therefore, the pixel mapping from D to C can be easily derived [Fig. 2(b)]; e.g., the element 11 of C corresponds to the element 22 of D and the element 22 of C corresponds to the element 23 of D . On the other hand, the case with n ¼ 3 for the pixel mapping between D and C can be derived according to the pixel mapping between the covert image C in Fig. 1(a) and the matrix D in Fig. 1(d); so C [Fig. 2(c)] can be easily derived. B. Encoding and Decoding Methods for a PixelScrambled Matrix
Let H be an M × N host image used to encode a pixelscrambled matrix D to form an M × N overt image H , where the matrix D is pixel scrambled from a covert image C. All of the pixels in H are classified into six categories to respectively encode identification codes, gray-level codes, dimension codes, scrambling number codes, scrambling time codes, and information codes. The identification codes are used to check whether the codes in H are encoded with the proposed encoding method; the gray-level codes are used 10 January 2010 / Vol. 49, No. 2 / APPLIED OPTICS
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lar to Eq. (1), but every symbol k should be replaced by the symbol g, and g0 ¼ 2. The dimension codes are two sets of eight-digit binary codes. The first set of binary codes is used to denote the row dimension r ðr ≥ 1Þ of C, and it includes r1 , r2 , r3 , r4 , r5 , r6 , r7 , and r8 . The relationship between r and r1 –r8 is similar to Eq. (1), but every k should be replaced by r, and r0 ¼ 1. The second set of binary codes is used to denote the column dimension c ðc ≥ 1Þ of C, and it includes c1, c2 , c3 , c4 , c5 , c6 , c7 , and c8 . The relationship between c and c1 –c8 is similar to Eq. (1), but every k should be replaced by c, and c0 ¼ 1. The scrambling number codes are eight-digit binary codes n1 , n2 , n3 , n4 , n5 , n6 , n7 , and n8 , and they are used to denote the n ðn ≥ 2Þ value for forming D from C. The relationship between n and n1 –n8 is similar to Eq. (1), but every k should be replaced by n, and n0 ¼ 2. The scrambling time codes are eight-digit binary codes t1, t2 , t3 , t4 , t5 , t6 , t7 , and t8 , and they are used to show t ðt ≥ 1Þ times that pixel scrambling is repeated. The relationship between t and t1 –t8 is similar to Eq. (1), but every k should be replaced by t, and t0 ¼ 1. The information codes are used in the encoding process to encrypt D into the M × N host image H to form an M × N overt image H as shown below. Fig. 2. Example for pixel descrambling from D to C . (a) D , (b) C for n ¼ 2, (c) C for n ¼ 3.
to denote the gray-level value of C; the dimension codes are used to denote the two dimensions (the parameters r and c) of C; the scrambling number codes are used to denote the pixel number jumping (the parameter n); the scrambling time codes are used to denote the times of repeated scrambling. The information codes are used to encrypt D, and they are encoded in the second to the last row of H . On the other hand, the other five categories of codes are encoded in the first row of H with the arrangements specified by the designer. For reducing the number of the equations used in this paper, we define a common equation, i.e., Eq. (1). Let an integer k be composed of eight binary codes k1, k2 , k3 , k4 , k5 , k6 , k7 , k8 , and another integer be denoted k0 according to k¼
8 X i¼1
ki × 2i−1 þ k0 :
ð1Þ
For the identification codes, the number of codes must be large enough to avoid incorrect judgment, and the codes are binary, e.g., 110010101001011 001010011. The gray-level codes are used to denote the graylevel value g ðg ≥ 2Þ of the covert image C, and they are eight-digit binary codes g1 , g2 , g3 , g4 , g5 , g6 , g7 , and g8 . The relationship between g and g1 –g8 is simi222
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1. Set a binary array R with N elements. Some of the elements of R are determined according to identification codes, gray-level codes, dimension codes, scrambling number codes, and scrambling time codes. The other elements of R are not used, and they are all set to be 0. 2. Map the elements of D to form the elements of a scrambled-data string E according to Eððr0 − 1Þ × c þ c0 Þ ¼ Dðr0 ; c0 Þ;
ð2Þ
where 1 ≤ r0 ≤ r and 1 ≤ c0 ≤ c. Ph−1 3. After 2h−1 < g ≤ 2h and EðkÞ ¼ i¼0 ai ðkÞ2i−1 are known, transform the scrambled-data string E into a binary-data string G according to Gði þ h × ðk − 1ÞÞ ¼ ai ðkÞ:
ð3Þ
4. Use the elements of G (from the first element to the last element) to form the elements of a ðM − 1Þ × N data matrix S (from the first row to the last row and from the left side to the right side for the same row). Since the data number L of G may be smaller than ðM − 1Þ × N, there are ðM − 1Þ × N − L dummy elements in S not formed from the elements of G. As a result, the values of dummy elements are all set to be 0. 5. Combine the row array R with N elements and the ðM − 1Þ × N matrix S to form a M × N binary matrix T. The first row of T is copied from R, while other rows of T are copied from S in sequence. 6. Modulate the element Hðu; vÞ of the host image H to form the element H0 ðu; vÞ of a modulated matrix
H0 according to H0 ðu; vÞ ¼ 2 floorðHðu; vÞ=2Þ;where the function floorðxÞ modulates the value of x to the nearest integer xn ð≤ xÞ, and every H0 ðu; vÞ is an even integer. 7. An overt image H is formed by adding the corresponding elements of matrices T and H 0 , i.e., H ðu; vÞ ¼ Tðu; vÞ þ H 0 ðu; vÞ:
ð5Þ
Figure 3 shows an assumed host matrix H, an assumed binary matrix T, the resulting modulated matrix H0 , and the resulting overt matrix H . The decoding processes to reconstruct a scrambled matrix D from the overt matrix H are presented below. 1. Use the overt matrix H to determine a M × N binary matrix T according to Tðm0 ; n0 Þ ¼ H ðm0 ; n0 Þ − 2 floorðH ðm0 ; n0 Þ=2Þ;
ð6Þ
where 1 ≤ m0 ≤ M, 1 ≤ n0 ≤ N, and Tðm0 ; n0 Þ is 0 or 1. 2. Separate the M × N T into a 1 × N row array R (the first row) and a ðM − 1Þ × N matrix S (the second to the last row). 3. According to the identification codes in R, we can check whether this overt image contains the codes proposed by this paper. 4. Use the elements of S (from the first row to the last row and from the left side to the right side for the same row) to form the elements of a binary-data string G (from the first element to the last element). 5. Derive an r × c (r and c can be determined from the two sets of dimension codes in R) data string E from G according to EðkÞ ¼
m X
0
Gðmðk − 1Þ þ m0 Þ2m−m ;
ð7Þ
m0 ¼1
where 1 ≤ k ≤ r × c and m is determined from 2m−1 < g ≤ 2m . (g can be determined from the gray-level codes in R.) 6. Transform E into the r × c scrambled matrix D . The element D ðr0 ; c0 Þ (1 ≤ r0 ≤ r and 1 ≤ c0 ≤ c) of D is determined according to D ðr0 ; c0 Þ ¼ Eððr0 − 1Þ × c þ c0 Þ:
δ¼
i¼1
Pc
j¼1 ½K −1;0 ði; jÞ
where K m;n ¼ j½Dði þ m; j þ nÞ − Dði; jÞ2
ð8Þ
For an r × c image D scrambled from an r × c image C, the image scrambling degree δ is defined as [20]
Pr
Fig. 3. (a) Assumed host matrix H, (b) assumed binary matrix T, (c) resulting modulated matrix H 0 , (d) resulting overt image H .
− ½Cði þ m; j þ nÞ − Cði; jÞ2 j:
ð9bÞ
A greater δ value indicates that C and D are more different.
þ K 1;0 ði; jÞ þ K 0;−1 ði; jÞ þ K 0;1 ði; jÞ 2552 × r × c
;
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ð9aÞ
223
For checking image quality, this paper will resort to peak signal-to-noise ratio (PSNR) values of the two images H and H . The definition of PSNR is [21] PSNR ¼ 10 × log
M×N ; MSE
ð10aÞ
where MSE ¼
M X N 1 X ½Hði; jÞ − H ði; jÞ2 : M × N i¼1 j¼1
ð10bÞ
If the PSNR value is higher than 30, it will be difficult for the naked eye to identify the difference between H and H ; i.e. the image H looks almost the same as H [22]. 3. Experiment
Figure 4 shows two covert images for tests. The first covert image is a 64 × 120 binary image [Fig. 4(a)] and the other is a 64 × 120 256-gray-level image [Fig. 4(b)]. Figure 5 shows a 256 × 256 256-gray-level picture used as the host image H. The test of the covert image C in Fig. 4(a) is introduced below. Since the dimension of H is 256 × 256, the dimension of the binary row array R is 1 × 256. The 1st to 24th elements of R are used as the identification codes, and they are designated as 11001010 1001011001010011. The 25th to 32nd elements of R are used as the gray-level codes. Since the gray-level value is 2, the gray-level codes are designated as P 00000000 (i.e., 2 ¼ 8i¼1 gi × 2i−1 þ 2). The 33rd to 48th elements of R are used as the dimension codes. Since the size of C is 64 × 120, the two sets of dimension P8 codesi−1for r and c are 00111111 (i.e., P8 64 ¼ r × 2 þ 1) and 01110111 (i.e., 120 ¼ i¼1 i i¼1 ci × 2i−1 þ 1), respectively. The 49th to 56th elements of R
Fig. 4. Covert images for tests. (a) 64 × 120120 binary image, (b) 64 × 120 256-gray-level image. 224
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Fig. 5. Host image for encoding to form different overt images.
are used as the scrambling number codes. Since the scrambling number value n isP set to be 17 here, the codes are 00001111 (i.e., 17 ¼ 8i¼1 ni × 2i−1 þ 2). The 57th to 64th elements of R are used as the scrambling time codes. Since the scrambling times value t is set P to be 5 here, the codes are 00000100 (i.e., 5 ¼ 8i¼1 ti × 2i−1 þ 1). The 65th to 256th elements of R (dummy elements) are not used, and all of them are set to be 0. First we scramble the 64 × 120 covert image C into a 64 × 120 matrix D by using the proposed pixel scrambling technique with n ¼ 17 and t ¼ 5. The pixel-scrambled matrix D is shown in Fig. 6(a). Then we transform D into a scrambled-data string E with 7680 (¼64 × 120) elements and transform E into a binary-data string G with 61440 (¼64 × 120 × 8) elements. Subsequently, we copy the elements of G to form the elements of a 256 × 256 data matrix S. The 61441st to 65280th elements of G are all set to be 0. Furthermore, we transform G into a 255 × 256 matrix S, and we combine the 1 × 256 binary row array R and the 255 × 256 matrix S to form a 256 × 256 binary matrix T. The matrix T is shown in Fig. 6(b), where a white pixel denotes “1” and a black pixel denotes “0”. In addition, the host image H is modulated to form a modulated image H0 . Then, the corresponding elements of the matrices T and H0 are added to form an overt image H . The overt image H is shown in Fig. 6(c), and it looks almost the same as the host image H in Fig. 5. The above processes are used to encrypt the covert image C in Fig. 4(a). (1) C is used to form D and R; (2) D and R are used to form T; (3) H is modulated to form H 0 ; (4) combine T and H are combined to form H . The decrypting processes are (1) H is modulated to form H0 ; (2) H 0 is subtracted from H form T; (3) T is separated into R and D; (4) D and R are used to form C. Because the matrices reconstructed in the decrypting processes are all the same as the matrices created in the encrypting processes, we do not show them again.
Fig. 7. (a) Matrix D with n ¼ 14 and t ¼ 2, (b) matrix T with n ¼ 14 and t ¼ 2.
tween the original covert image C and the reconstructed covert image C is infinity; i.e., there is no distortion during the covert image reconstruction. 4.
Fig. 6. (a) Matrix D with n ¼ 17 and t ¼ 5, (b) matrix T with n ¼ 17 and t ¼ 5, (c) overt matrix H for encoding the covert image in Fig. 4(a).
The image encrypting test for the covert image in Fig. 4(b) is similar to that for the covert image in Fig. 4(a), so we do not introduce the former in detail. For the covert image in Fig. 4(b), the parameters n ¼ 14 and t ¼ 2 are used. The pixel-scrambled matrix D is shown in Fig. 7(a), and the binary matrix T is shown in Fig. 7(b). The PSNR values between H and H are all greater than 46 for the two covert images. Therefore, the two images H and H look almost identical for the two cases. Moreover, for each case the PSNR value be-
Discussion
Only the binary encoding method and the pixel scrambling technique are demonstrated for clear and brief explanations in this paper, but changing the definitions of parameters or changing specified pixels for encoding parameters can easily increase the security of the proposed method. Therefore, ease of use and high security are the main advantages of the proposed method. On the other hand, needing bigger host-image sizes (for binary encoding) and needing longer running times (for pixel scrambling) are the main disadvantages of the proposed method. We calculate the image scrambling degree percentages (100% × scrambling degree=maximum scrambling degree) of the scrambled images for different n and t values. For t ¼ 1, the image scrambling degree percentages for n ¼ 2 to n ¼ 128 are shown in Fig. 8. In Fig. 8(a), corresponding to the covert image in Fig. 4(a), the image scrambling degree percentages are always greater than 80% for n ≥ 6. (The case with n ¼ 20 has the maximum image scrambling degree, 1.85.) In Fig. 8(b), corresponding to the covert image in Fig. 4(b), the image scrambling degree percentages are always greater than 80% for n ≥ 4. (The case with n ¼ 16 has the maximum image scrambling degree 0.27.) For specified n values, the image scrambling degree percentages for t ¼ 1 to t ¼ 300 are shown in 10 January 2010 / Vol. 49, No. 2 / APPLIED OPTICS
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Fig. 9. In Fig. 9(a), corresponding to the covert image in Fig. 4(a), the condition n ¼ 17 is used, and the -
image scrambling degree percentages are always greater than 80% for t ≥ 1. (The case with t ¼ 249 has the maximum image scrambling degree, 1.94.) In Fig. 9(b), corresponding to the covert image in Fig. 4(b), the condition n ¼ 14 is used, and the image scrambling degree percentages are always greater than 80% for t ≥ 1. (The case with t ¼ 53 has the maximum image scrambling degree 0.27.) The image scrambling degree percentages for different n and t values are shown in Fig. 10. In Fig. 10(a), corresponding to the covert image in Fig. 4(a), when n ≥ 6 and t ¼ 1, the image scrambling degree percentages are always greater than 80%. In Fig. 10(b), corresponding to the covert image in Fig. 4(b), when n ≥ 4 and t ¼ 1, the image scrambling degree percentages are always greater than 80%. According to the data in Figs. 8–10, we can find that the scrambling number n ≥ 6 can always make the image scrambling degree percentages greater than 80% for every case with t ¼ 1. Because a case with t ¼ 1 needs encoding only one time and a case with t ≥ 2 needs encoding plural times, encoding cases with t ≥ 6 and t ¼ 1 are preferred if saving time is important for a user. Figure 11 shows the three Lena images [from Fig. 4(b)] scrambled by using the p-Fibonacci transformation with p ¼ 1 [Fig. 11(a)], the Arnold transformation [Fig. 11(b)], and the proposed method [Fig. 11(c)]. The image scrambling degree percentages for different t values for the p-Fibonacci transformation and the Arnold transformation are shown in Figs. 12(a) and 12(b), respectively. In Fig. 12(a),
Fig. 9. (Color online) Image scrambling degrees with different t values and (a) n ¼ 7 for Fig. 4(a), (b) n ¼ 20 for Fig. 4(b).
Fig. 10. (Color online) Conditions for image scrambling degree percentages greater than 80% for (a) Fig. 4(a) and (b) Fig. 4(b).
Fig. 8. (Color online) Image scrambling degrees with different n values and t ¼ 1 for (a) Fig. 4(a) and (b) Fig. 4(b).
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Fig. 12. (Color online) Image scrambling degree percentages for t ¼ 1 for (a) the p-Fibonacci transformation and (b) the Arnold transformation.
Fig. 11. Different scrambled Lena images for t ¼ 1 by using (a) the p-Fibonacci transformation, (b) the Arnold transformation, (c) the proposed method.
the image scrambling degree percentages are not stable, and they are greater than 80% for some discrete t values. (The case with t ¼ 141 has the maximum image scrambling degree, 0.25.) In Fig. 12(b), the image scrambling degree percentages are stable and they are always greater than 80% for t ≥ 4. (The case with t ¼ 5 has the maximum image scrambling degree 0.28.) In contrast, the image scrambling degree percentages based on the proposed method are very stable, and they are always greater than 80% for t ≥ 1 and n ≥ 6 [Fig. 8(b)]. For t ¼ 1, the image scrambling degree is 0.20 for the p-Fibonacci transformation, the image scrambling degree is only 0.07 for the Arnold transformation, and the image scrambling degree is 0.27 for the proposed method (n ≥ 4). Therefore, the image scrambling degree of the proposed method is higher than the p-Fibonacci or Arnold transformation for t ¼ 1. Therefore, the proposed method performs better than the p-Fibonacci
transformation and the Arnold transformation for the smallest t. 5.
Conclusion
The proposed method can encrypt a covert image by combining the binary encoding method and the pixel scrambling technique for a host image H to derive a overt image H . The two images H and H look almost the same, and the reconstructed covert image is exactly the same as the original covert image. Furthermore, the extraction of a hidden covert image from an overt image without authorization can be very difficult, whereas the extraction of the hidden covert image can be easy for authorized users. Moreover, the host image H is not needed to decode the encoded image from the overt image H . References 1. Y. L. Yang, N. Cai, and G. Q. Ni, “Digital image scrambling technology based on the symmetry of Arnold transform,” J. Beijing Inst. Technol. 15, 216–220 (2006). 2. B. Li and J. W. Xu, “Period of Arnold transformation and its application in image scrambling,” J. Cent. South Univ. Technol. 12, 278–282 (2005). 3. T. Kong and D. Zhang, “A new anti-Arnold transformation algorithm,” J. Software 15, 1558–1564 (2004). 10 January 2010 / Vol. 49, No. 2 / APPLIED OPTICS
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4. D. Qi, J. Zou, and X. Han, “A new class of scrambling transformation and its application in the image information covering,” Sci. China Ser. E Technol. Sci. 43, 304–312 (2000). 5. Y. Zhou, S. Agaian, V. M. Joyner, and K. Panetta, “Two Fibonacci p-code based image scrambling algorithms,” Proc. SPIE 6812, 681215 (2008). 6. O. Lafe, “Data compression and encryption using cellular automata transforms,” Eng. Applic. Artif. Intell. 10, 581–591 (1997). 7. G. Z. Hernández and H. J. Herrmann, “Cellular automata for elementary image enhancement,” Graphical Models Image Process. 58, 82–89 (1996). 8. C. C. Chang and J. C. Chuang, “An image intellectual property protection scheme for gray-level images using visual secret sharing strategy,” Pattern Recognit. Lett. 23, 931–941 (2002). 9. H. Zhang, J. Huang, and Z. Li, “New method of digital image scrambling based on binary tree generated by chaotic sequences,” Proc. SPIE 6790, 67905D (2007). 10. R. Ye and H. Li, “A novel image scrambling and watermarking scheme based on cellular automata,” in 2008 International Symposium on Electronic Commerce and Security (2008), pp. 938–941. 11. B. Hennelly and J. T. Sheridan, “Optical image encryption by random shifting in fraction Fourier domains,” Opt. Lett. 28, 269–271 (2003). 12. J. Zhao, H. Lu, X. Song, J. Li, and Y. Ma, “Optical image encryption based on multistage fractional Fourier transforms and pixel scrambling technique,” Opt. Commun. 249, 493–499 (2005). 13. J. Zhao, H. Lu, and Q. Fan, “Color image encryption based on fractional Fourier transforms and pixel scrambling technique,” Proc. SPIE 6279, 62793B (2007).
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14. X. F. Meng, L. Z. Cai, X. L. Yang, X. X. Shen, and G. Y. Dong, “Information security system by iterative multiple-phase retrieval and pixel random permutation,” Appl. Opt. 45, 3289 (2006). 15. Y. Y. Wang, Y. R. Wang, Y. Wang, H. J. Li, and W. J. Sun, “Optical image encryption based on binary Fourier transform computer-generated hologram and pixel scrambling technology,” Opt. Lasers Eng. 45, 761–765 (2007). 16. Q. Liu and X. Zhang, “Secure 3D watermarking algorithm based on point set projection,” Proc. SPIE-Int. Soc. Opt. Eng. 6790, 67904R (2007). 17. H. Lu, J. Zhao, Q. Fan, Y. Xu, and X. Wan, “Iterative double random phase encryption based on pixel scrambling technology,” Acta Photonica Sin. 34, 1069–1073 (2005). 18. K. T. Lin, “Digital information encrypted in an image using binary encoding,” Opt. Commun. 281, 3447–3453 (2008). 19. S. L. Yeh, “Dot-matrix hologram with an encrypted figure,” Opt. Eng. 45, 095801 (2006). 20. X. Y. Yu, H. E. Ren, S. Li, and X. D. Zhang, “A new measurement method of image encryption,” J. Phys. Conf. Ser. 48, 408–411 (2006). 21. M. Kutter and F. A. P. Petitcolas, “A fair benchmark for image watermarking systems,” Proc. SPIE 3657, 226–239 (1999). 22. T. K. Shih, L. C. Lu, and R. C. Chang, “An automatic image in paint tool,” in Proceedings of the Eleventh ACM International Conference on Multimedia, (Association for Computing Machinery, 2003), pp. 102–103.
Novel information hiding for digital images based on binary encoding methods and pixel scrambling techniques is presented. First, a pixel scrambling technique is used to rearrange the pixels of a covert image to form a scrambled matrix by using a specified scrambling rule. Then, the gray values of all the pixels in the scrambled matrix are sequentially transformed into many sets of eight-digit binary codes. Subsequently, the eight-digit binary codes are encoded into a host image to form an overt image by using a specific encoding rule. Besides the eight-digit binary codes (information codes), the overt image contains five other groups of binary codes (specification codes), i.e., identification codes, gray-level codes, dimension codes, scrambling number codes, and scrambling time codes, to denote the parameters used for scrambling and encoding. According to the test results, the proposed method performs well. Moreover, the overt image and the host image look almost the same, and the decoded covert image is exactly the same as the original covert image. © 2010 Optical Society of America OCIS codes: 100.0100, 100.2000, 100.4998.
1. Introduction
Image scrambling techniques are important image encoding methods that have been applied to digital image processing, information hiding, and digital watermarking to enhance information security. Image scrambling techniques can be classified into two types according to image pixel arrangements. For the first type, matrix transformations, e.g., the Arnold transformation [1–3] and the p-Fibonacci transformation [4,5], are used. Because successful inverse transformations need the correct transformation parameters, the scrambling techniques can be protected by the secrete parameters. Although the methods perform well, the reconstruction of the encoded images requires external keys, and the transformations is complex. For the second type, coordinate movements, e.g., the cellular automata [6,7], the torus automorphism [8], and chaotic sequences [9], are used. Because successful image reconstruction needs the correct scrambling path 0003-6935/10/020220-09$15.00/0 © 2010 Optical Society of America 220
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parameters, the scrambling techniques can be protected by the secrete parameters. Although the methods perform well, external keys are needed for the image reconstruction, and some distortion may be found in reconstructed images. However, some studies have explored assembling image scrambling techniques and other image encrypting methods to increase image security. Ye and Li [10] combined an image scrambling technique and a watermarking technique. Hennelly and Sheridan [11] and Zhao et al. [12,13] combined image scrambling techniques and fractional Fourier transforms to hide covert images. Meng et al. [14] used an image scrambling technique and an iterative Fresnel transform technique to hide covert images. Wang et al. [15] combined an image scrambling technique and a binary Fourier transform technique by using computer-generated hologram techniques to hide covert images. Liu and Zhang [16] combined a point-set projection technique with a watermarking technique to hide covert images. Lu et al. [17] integrated an image scrambling technique with a double random-phase encoding technique to hide covert images. All the combination methods had very good
robustness, but they all needed external keys during covert-image reconstruction and reconstructed covert images may contain some distortion. This paper will propose a method that combines a binary encoding method [18] and a pixel scrambling technique to hide covert images in host images to form overt images. The proposed method can be applied to equilateral or nonequilateral images, and overt images look almost the same as their corresponding host images. External keys are not needed during covert-image reconstruction, but conventions to derive encoding parameters are needed for decoding covert images. In addition, the proposed method is simple and secure. Most of all, and there is no distortion for reconstructed covert images. The proposed encoding method can be applied to both digital and optical types of overt image. Although the encoding mechanisms of the two types of overt image are different, their encoding techniques are similar. Therefore, this paper will focus on just the digital type of overt image. For the optical type of overt image, one can refer to [18,19]. 2. Theory A. Pixel Scrambling and Descrambling Techniques for Encoding Information
Let C be an r × c covert image to be encoded and let D be an r × c matrix formed from the pixel scrambling of C. The processes for deriving D from C are presented below. First, map the pixels of C to form a pixel string A with r × c elements according to a specified order (from the first row to the last row and from the first column to the last column for the same row). Then, take every nth pixel in A (from the first pixel to the last pixel and recycled) to form the elements of D in sequence (from the first row to the last row and from the first column to the last column for the same row). After a pixel in A has been taken once, it is removed from A. The pixel-taking operations are repeated until all of the pixels in A are removed and relocated in D. An example for the scrambling processes from a 3 × 3 covert image C to a 3 × 3 matrix D is shown in Fig. 1. The descrambling processes from an r × c matrix D to an r × c covert image C are presented below. Because the three parameters r, c, and n are all known in advance for this case, we can simulate the scrambling processes introduced by the last paragraph to derive the mapping between the elements of C and D. The mapping from D to C is the inverse mapping from C to D for element positions, so we can easily map the elements from D to C by referencing C and D. An example of the descrambling from a 3 × 3 matrix D to a 3 × 3 covert image C is shown in Fig. 2. For the case with n ¼ 2, because r ¼ 3 and c ¼ 3 are known, the pixel mapping between the covert image C in Fig. 1(a) and the matrix D in Fig. 1(c) can be derived by a scrambling operation as shown in Fig. 1. Then, the pixel mapping from C to D is known; e.g., the element 11 of D corresponds to the element
Fig. 1. Example for pixel scrambling from C to D. (a) C, (b) A, (c) D for n ¼ 2, (d) D for n ¼ 3.
12 of C, and the element 22 of D corresponds to the element 11 of C. Therefore, the pixel mapping from D to C can be easily derived [Fig. 2(b)]; e.g., the element 11 of C corresponds to the element 22 of D and the element 22 of C corresponds to the element 23 of D . On the other hand, the case with n ¼ 3 for the pixel mapping between D and C can be derived according to the pixel mapping between the covert image C in Fig. 1(a) and the matrix D in Fig. 1(d); so C [Fig. 2(c)] can be easily derived. B. Encoding and Decoding Methods for a PixelScrambled Matrix
Let H be an M × N host image used to encode a pixelscrambled matrix D to form an M × N overt image H , where the matrix D is pixel scrambled from a covert image C. All of the pixels in H are classified into six categories to respectively encode identification codes, gray-level codes, dimension codes, scrambling number codes, scrambling time codes, and information codes. The identification codes are used to check whether the codes in H are encoded with the proposed encoding method; the gray-level codes are used 10 January 2010 / Vol. 49, No. 2 / APPLIED OPTICS
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lar to Eq. (1), but every symbol k should be replaced by the symbol g, and g0 ¼ 2. The dimension codes are two sets of eight-digit binary codes. The first set of binary codes is used to denote the row dimension r ðr ≥ 1Þ of C, and it includes r1 , r2 , r3 , r4 , r5 , r6 , r7 , and r8 . The relationship between r and r1 –r8 is similar to Eq. (1), but every k should be replaced by r, and r0 ¼ 1. The second set of binary codes is used to denote the column dimension c ðc ≥ 1Þ of C, and it includes c1, c2 , c3 , c4 , c5 , c6 , c7 , and c8 . The relationship between c and c1 –c8 is similar to Eq. (1), but every k should be replaced by c, and c0 ¼ 1. The scrambling number codes are eight-digit binary codes n1 , n2 , n3 , n4 , n5 , n6 , n7 , and n8 , and they are used to denote the n ðn ≥ 2Þ value for forming D from C. The relationship between n and n1 –n8 is similar to Eq. (1), but every k should be replaced by n, and n0 ¼ 2. The scrambling time codes are eight-digit binary codes t1, t2 , t3 , t4 , t5 , t6 , t7 , and t8 , and they are used to show t ðt ≥ 1Þ times that pixel scrambling is repeated. The relationship between t and t1 –t8 is similar to Eq. (1), but every k should be replaced by t, and t0 ¼ 1. The information codes are used in the encoding process to encrypt D into the M × N host image H to form an M × N overt image H as shown below. Fig. 2. Example for pixel descrambling from D to C . (a) D , (b) C for n ¼ 2, (c) C for n ¼ 3.
to denote the gray-level value of C; the dimension codes are used to denote the two dimensions (the parameters r and c) of C; the scrambling number codes are used to denote the pixel number jumping (the parameter n); the scrambling time codes are used to denote the times of repeated scrambling. The information codes are used to encrypt D, and they are encoded in the second to the last row of H . On the other hand, the other five categories of codes are encoded in the first row of H with the arrangements specified by the designer. For reducing the number of the equations used in this paper, we define a common equation, i.e., Eq. (1). Let an integer k be composed of eight binary codes k1, k2 , k3 , k4 , k5 , k6 , k7 , k8 , and another integer be denoted k0 according to k¼
8 X i¼1
ki × 2i−1 þ k0 :
ð1Þ
For the identification codes, the number of codes must be large enough to avoid incorrect judgment, and the codes are binary, e.g., 110010101001011 001010011. The gray-level codes are used to denote the graylevel value g ðg ≥ 2Þ of the covert image C, and they are eight-digit binary codes g1 , g2 , g3 , g4 , g5 , g6 , g7 , and g8 . The relationship between g and g1 –g8 is simi222
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1. Set a binary array R with N elements. Some of the elements of R are determined according to identification codes, gray-level codes, dimension codes, scrambling number codes, and scrambling time codes. The other elements of R are not used, and they are all set to be 0. 2. Map the elements of D to form the elements of a scrambled-data string E according to Eððr0 − 1Þ × c þ c0 Þ ¼ Dðr0 ; c0 Þ;
ð2Þ
where 1 ≤ r0 ≤ r and 1 ≤ c0 ≤ c. Ph−1 3. After 2h−1 < g ≤ 2h and EðkÞ ¼ i¼0 ai ðkÞ2i−1 are known, transform the scrambled-data string E into a binary-data string G according to Gði þ h × ðk − 1ÞÞ ¼ ai ðkÞ:
ð3Þ
4. Use the elements of G (from the first element to the last element) to form the elements of a ðM − 1Þ × N data matrix S (from the first row to the last row and from the left side to the right side for the same row). Since the data number L of G may be smaller than ðM − 1Þ × N, there are ðM − 1Þ × N − L dummy elements in S not formed from the elements of G. As a result, the values of dummy elements are all set to be 0. 5. Combine the row array R with N elements and the ðM − 1Þ × N matrix S to form a M × N binary matrix T. The first row of T is copied from R, while other rows of T are copied from S in sequence. 6. Modulate the element Hðu; vÞ of the host image H to form the element H0 ðu; vÞ of a modulated matrix
H0 according to H0 ðu; vÞ ¼ 2 floorðHðu; vÞ=2Þ;where the function floorðxÞ modulates the value of x to the nearest integer xn ð≤ xÞ, and every H0 ðu; vÞ is an even integer. 7. An overt image H is formed by adding the corresponding elements of matrices T and H 0 , i.e., H ðu; vÞ ¼ Tðu; vÞ þ H 0 ðu; vÞ:
ð5Þ
Figure 3 shows an assumed host matrix H, an assumed binary matrix T, the resulting modulated matrix H0 , and the resulting overt matrix H . The decoding processes to reconstruct a scrambled matrix D from the overt matrix H are presented below. 1. Use the overt matrix H to determine a M × N binary matrix T according to Tðm0 ; n0 Þ ¼ H ðm0 ; n0 Þ − 2 floorðH ðm0 ; n0 Þ=2Þ;
ð6Þ
where 1 ≤ m0 ≤ M, 1 ≤ n0 ≤ N, and Tðm0 ; n0 Þ is 0 or 1. 2. Separate the M × N T into a 1 × N row array R (the first row) and a ðM − 1Þ × N matrix S (the second to the last row). 3. According to the identification codes in R, we can check whether this overt image contains the codes proposed by this paper. 4. Use the elements of S (from the first row to the last row and from the left side to the right side for the same row) to form the elements of a binary-data string G (from the first element to the last element). 5. Derive an r × c (r and c can be determined from the two sets of dimension codes in R) data string E from G according to EðkÞ ¼
m X
0
Gðmðk − 1Þ þ m0 Þ2m−m ;
ð7Þ
m0 ¼1
where 1 ≤ k ≤ r × c and m is determined from 2m−1 < g ≤ 2m . (g can be determined from the gray-level codes in R.) 6. Transform E into the r × c scrambled matrix D . The element D ðr0 ; c0 Þ (1 ≤ r0 ≤ r and 1 ≤ c0 ≤ c) of D is determined according to D ðr0 ; c0 Þ ¼ Eððr0 − 1Þ × c þ c0 Þ:
δ¼
i¼1
Pc
j¼1 ½K −1;0 ði; jÞ
where K m;n ¼ j½Dði þ m; j þ nÞ − Dði; jÞ2
ð8Þ
For an r × c image D scrambled from an r × c image C, the image scrambling degree δ is defined as [20]
Pr
Fig. 3. (a) Assumed host matrix H, (b) assumed binary matrix T, (c) resulting modulated matrix H 0 , (d) resulting overt image H .
− ½Cði þ m; j þ nÞ − Cði; jÞ2 j:
ð9bÞ
A greater δ value indicates that C and D are more different.
þ K 1;0 ði; jÞ þ K 0;−1 ði; jÞ þ K 0;1 ði; jÞ 2552 × r × c
;
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ð9aÞ
223
For checking image quality, this paper will resort to peak signal-to-noise ratio (PSNR) values of the two images H and H . The definition of PSNR is [21] PSNR ¼ 10 × log
M×N ; MSE
ð10aÞ
where MSE ¼
M X N 1 X ½Hði; jÞ − H ði; jÞ2 : M × N i¼1 j¼1
ð10bÞ
If the PSNR value is higher than 30, it will be difficult for the naked eye to identify the difference between H and H ; i.e. the image H looks almost the same as H [22]. 3. Experiment
Figure 4 shows two covert images for tests. The first covert image is a 64 × 120 binary image [Fig. 4(a)] and the other is a 64 × 120 256-gray-level image [Fig. 4(b)]. Figure 5 shows a 256 × 256 256-gray-level picture used as the host image H. The test of the covert image C in Fig. 4(a) is introduced below. Since the dimension of H is 256 × 256, the dimension of the binary row array R is 1 × 256. The 1st to 24th elements of R are used as the identification codes, and they are designated as 11001010 1001011001010011. The 25th to 32nd elements of R are used as the gray-level codes. Since the gray-level value is 2, the gray-level codes are designated as P 00000000 (i.e., 2 ¼ 8i¼1 gi × 2i−1 þ 2). The 33rd to 48th elements of R are used as the dimension codes. Since the size of C is 64 × 120, the two sets of dimension P8 codesi−1for r and c are 00111111 (i.e., P8 64 ¼ r × 2 þ 1) and 01110111 (i.e., 120 ¼ i¼1 i i¼1 ci × 2i−1 þ 1), respectively. The 49th to 56th elements of R
Fig. 4. Covert images for tests. (a) 64 × 120120 binary image, (b) 64 × 120 256-gray-level image. 224
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Fig. 5. Host image for encoding to form different overt images.
are used as the scrambling number codes. Since the scrambling number value n isP set to be 17 here, the codes are 00001111 (i.e., 17 ¼ 8i¼1 ni × 2i−1 þ 2). The 57th to 64th elements of R are used as the scrambling time codes. Since the scrambling times value t is set P to be 5 here, the codes are 00000100 (i.e., 5 ¼ 8i¼1 ti × 2i−1 þ 1). The 65th to 256th elements of R (dummy elements) are not used, and all of them are set to be 0. First we scramble the 64 × 120 covert image C into a 64 × 120 matrix D by using the proposed pixel scrambling technique with n ¼ 17 and t ¼ 5. The pixel-scrambled matrix D is shown in Fig. 6(a). Then we transform D into a scrambled-data string E with 7680 (¼64 × 120) elements and transform E into a binary-data string G with 61440 (¼64 × 120 × 8) elements. Subsequently, we copy the elements of G to form the elements of a 256 × 256 data matrix S. The 61441st to 65280th elements of G are all set to be 0. Furthermore, we transform G into a 255 × 256 matrix S, and we combine the 1 × 256 binary row array R and the 255 × 256 matrix S to form a 256 × 256 binary matrix T. The matrix T is shown in Fig. 6(b), where a white pixel denotes “1” and a black pixel denotes “0”. In addition, the host image H is modulated to form a modulated image H0 . Then, the corresponding elements of the matrices T and H0 are added to form an overt image H . The overt image H is shown in Fig. 6(c), and it looks almost the same as the host image H in Fig. 5. The above processes are used to encrypt the covert image C in Fig. 4(a). (1) C is used to form D and R; (2) D and R are used to form T; (3) H is modulated to form H 0 ; (4) combine T and H are combined to form H . The decrypting processes are (1) H is modulated to form H0 ; (2) H 0 is subtracted from H form T; (3) T is separated into R and D; (4) D and R are used to form C. Because the matrices reconstructed in the decrypting processes are all the same as the matrices created in the encrypting processes, we do not show them again.
Fig. 7. (a) Matrix D with n ¼ 14 and t ¼ 2, (b) matrix T with n ¼ 14 and t ¼ 2.
tween the original covert image C and the reconstructed covert image C is infinity; i.e., there is no distortion during the covert image reconstruction. 4.
Fig. 6. (a) Matrix D with n ¼ 17 and t ¼ 5, (b) matrix T with n ¼ 17 and t ¼ 5, (c) overt matrix H for encoding the covert image in Fig. 4(a).
The image encrypting test for the covert image in Fig. 4(b) is similar to that for the covert image in Fig. 4(a), so we do not introduce the former in detail. For the covert image in Fig. 4(b), the parameters n ¼ 14 and t ¼ 2 are used. The pixel-scrambled matrix D is shown in Fig. 7(a), and the binary matrix T is shown in Fig. 7(b). The PSNR values between H and H are all greater than 46 for the two covert images. Therefore, the two images H and H look almost identical for the two cases. Moreover, for each case the PSNR value be-
Discussion
Only the binary encoding method and the pixel scrambling technique are demonstrated for clear and brief explanations in this paper, but changing the definitions of parameters or changing specified pixels for encoding parameters can easily increase the security of the proposed method. Therefore, ease of use and high security are the main advantages of the proposed method. On the other hand, needing bigger host-image sizes (for binary encoding) and needing longer running times (for pixel scrambling) are the main disadvantages of the proposed method. We calculate the image scrambling degree percentages (100% × scrambling degree=maximum scrambling degree) of the scrambled images for different n and t values. For t ¼ 1, the image scrambling degree percentages for n ¼ 2 to n ¼ 128 are shown in Fig. 8. In Fig. 8(a), corresponding to the covert image in Fig. 4(a), the image scrambling degree percentages are always greater than 80% for n ≥ 6. (The case with n ¼ 20 has the maximum image scrambling degree, 1.85.) In Fig. 8(b), corresponding to the covert image in Fig. 4(b), the image scrambling degree percentages are always greater than 80% for n ≥ 4. (The case with n ¼ 16 has the maximum image scrambling degree 0.27.) For specified n values, the image scrambling degree percentages for t ¼ 1 to t ¼ 300 are shown in 10 January 2010 / Vol. 49, No. 2 / APPLIED OPTICS
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Fig. 9. In Fig. 9(a), corresponding to the covert image in Fig. 4(a), the condition n ¼ 17 is used, and the -
image scrambling degree percentages are always greater than 80% for t ≥ 1. (The case with t ¼ 249 has the maximum image scrambling degree, 1.94.) In Fig. 9(b), corresponding to the covert image in Fig. 4(b), the condition n ¼ 14 is used, and the image scrambling degree percentages are always greater than 80% for t ≥ 1. (The case with t ¼ 53 has the maximum image scrambling degree 0.27.) The image scrambling degree percentages for different n and t values are shown in Fig. 10. In Fig. 10(a), corresponding to the covert image in Fig. 4(a), when n ≥ 6 and t ¼ 1, the image scrambling degree percentages are always greater than 80%. In Fig. 10(b), corresponding to the covert image in Fig. 4(b), when n ≥ 4 and t ¼ 1, the image scrambling degree percentages are always greater than 80%. According to the data in Figs. 8–10, we can find that the scrambling number n ≥ 6 can always make the image scrambling degree percentages greater than 80% for every case with t ¼ 1. Because a case with t ¼ 1 needs encoding only one time and a case with t ≥ 2 needs encoding plural times, encoding cases with t ≥ 6 and t ¼ 1 are preferred if saving time is important for a user. Figure 11 shows the three Lena images [from Fig. 4(b)] scrambled by using the p-Fibonacci transformation with p ¼ 1 [Fig. 11(a)], the Arnold transformation [Fig. 11(b)], and the proposed method [Fig. 11(c)]. The image scrambling degree percentages for different t values for the p-Fibonacci transformation and the Arnold transformation are shown in Figs. 12(a) and 12(b), respectively. In Fig. 12(a),
Fig. 9. (Color online) Image scrambling degrees with different t values and (a) n ¼ 7 for Fig. 4(a), (b) n ¼ 20 for Fig. 4(b).
Fig. 10. (Color online) Conditions for image scrambling degree percentages greater than 80% for (a) Fig. 4(a) and (b) Fig. 4(b).
Fig. 8. (Color online) Image scrambling degrees with different n values and t ¼ 1 for (a) Fig. 4(a) and (b) Fig. 4(b).
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Fig. 12. (Color online) Image scrambling degree percentages for t ¼ 1 for (a) the p-Fibonacci transformation and (b) the Arnold transformation.
Fig. 11. Different scrambled Lena images for t ¼ 1 by using (a) the p-Fibonacci transformation, (b) the Arnold transformation, (c) the proposed method.
the image scrambling degree percentages are not stable, and they are greater than 80% for some discrete t values. (The case with t ¼ 141 has the maximum image scrambling degree, 0.25.) In Fig. 12(b), the image scrambling degree percentages are stable and they are always greater than 80% for t ≥ 4. (The case with t ¼ 5 has the maximum image scrambling degree 0.28.) In contrast, the image scrambling degree percentages based on the proposed method are very stable, and they are always greater than 80% for t ≥ 1 and n ≥ 6 [Fig. 8(b)]. For t ¼ 1, the image scrambling degree is 0.20 for the p-Fibonacci transformation, the image scrambling degree is only 0.07 for the Arnold transformation, and the image scrambling degree is 0.27 for the proposed method (n ≥ 4). Therefore, the image scrambling degree of the proposed method is higher than the p-Fibonacci or Arnold transformation for t ¼ 1. Therefore, the proposed method performs better than the p-Fibonacci
transformation and the Arnold transformation for the smallest t. 5.
Conclusion
The proposed method can encrypt a covert image by combining the binary encoding method and the pixel scrambling technique for a host image H to derive a overt image H . The two images H and H look almost the same, and the reconstructed covert image is exactly the same as the original covert image. Furthermore, the extraction of a hidden covert image from an overt image without authorization can be very difficult, whereas the extraction of the hidden covert image can be easy for authorized users. Moreover, the host image H is not needed to decode the encoded image from the overt image H . References 1. Y. L. Yang, N. Cai, and G. Q. Ni, “Digital image scrambling technology based on the symmetry of Arnold transform,” J. Beijing Inst. Technol. 15, 216–220 (2006). 2. B. Li and J. W. Xu, “Period of Arnold transformation and its application in image scrambling,” J. Cent. South Univ. Technol. 12, 278–282 (2005). 3. T. Kong and D. Zhang, “A new anti-Arnold transformation algorithm,” J. Software 15, 1558–1564 (2004). 10 January 2010 / Vol. 49, No. 2 / APPLIED OPTICS
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