Abstract

A novel information security system based on multiple-phase retrieval by an iterative Fresnel-transform algorithm and pixel random permutation (PRP) technique is proposed. In this method a series of phase masks cascaded in free space are employed and the phase distributions of all the masks are adjusted simultaneously in each iteration. It can achieve faster convergence and better quality of the recovered image compared with double-phase encoding and a similar approach in the spatial-frequency domain with the same number of phase masks and can provide a higher degree of freedom in key space with more geometric parameters as supplementary keys. Furthermore, the security level of this method is greatly improved by the introduction of the PRP technique. The feasibility of this method and its robustness against occlusion and additional noise attacks are verified by computer simulations. The performance of this technique for different numbers of phase masks and quantized phase levels is investigated systematically with the correlation coefficient and mean square error as convergence criterions.

© 2006 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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2005 (6)

G. Situ and J. Zhang, "Image hiding with computer-generated phase codes for optical authentication," Opt. Commun. 245, 55-65 (2005).
[CrossRef]

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).
[CrossRef]

G. S. Spagnolo, C. Simonetti, and L. Cozzella, "Content fragile watermarking based on a computer generated hologram coding technique," J. Opt. 7, 333-342 (2005).
[CrossRef]

X. F. Meng, L. Z. Cai, M. Z. He, G. Y. Dong, and X. X. Shen, "Cross-talk-free double-image encryption and watermarking with amplitude-phase separate modulations," J. Opt. 7, 624-631 (2005).
[CrossRef]

M. Z. He, L. Z. Cai, Q. Liu, W. C. Wang, and X. F. Meng, "Multiple image encryption and watermarking by random phase matching," Opt. Commun. 247, 29-37 (2005).
[CrossRef]

M. Z. He, L. Z. Cai, Q. Liu, and X. L. Yang, "Phase-only encryption and watermarking based on phase-shifting interferometry," Appl. Opt. 44, 2600-2606 (2005).
[CrossRef] [PubMed]

2004 (5)

L. Z. Cai, M. Z. He, Q. Liu, and X. L. Yang, "Digital image encryption and watermarking by phase-shifting interferometry," Appl. Opt. 43, 3078-3084 (2004).
[CrossRef] [PubMed]

G. Situ and J. Zhang, "A lensless optical security system based on computer-generated phase only masks," Opt. Commun. 232, 115-122 (2004).
[CrossRef]

H. T. Chang and C. T. Chen, "Asymmetric-image verification for security optical system based on joint transform correlator architecture," Opt. Commun. 239, 43-54 (2004).
[CrossRef]

T. Nomura, K. Uota, and Y. Morimoto, "Hybrid optical encryption of a 3-D object using a digital holographic technique," Opt. Eng. 43, 2228-2232 (2004).
[CrossRef]

H. Kim, D. H. Kim, and Y. H. Lee, "Encryption of digital hologram of 3-D object by virtual optics," Opt. Express 12, 4912-4921 (2004).
[CrossRef] [PubMed]

2003 (3)

2002 (5)

2001 (4)

X. Tan, O. Matoba, Y. Okada-Shudo, M. Ide, T. Shimura, and K. Kuroda, "Secure optical memory system with polarization encryption," Appl. Opt. 40, 2310-2315 (2001).
[CrossRef]

H. T. Chang, "Image encryption using separable amplitude-based virtual image and iteratively retrieved phase information," Opt. Eng. 40, 2165-2171 (2001).
[CrossRef]

J. Rosen and B. Javidi, "Hidden images in halftone pictures," Appl. Opt. 40, 3346-3353 (2001).
[CrossRef]

D. Abookasis, O. Arazi, J. Rosen, and B. Javidi, "Security optical systems based on a joint transform correlator with significant output images," Opt. Eng. 40, 1584-1589 (2001).
[CrossRef]

2000 (5)

1999 (1)

1996 (1)

R. K. Wang, I. A. Watson, and C. Chatwin, "Random phase encoding for optical security," Opt. Eng. 35, 2464-2468 (1996).
[CrossRef]

1995 (2)

1993 (1)

1982 (1)

Abookasis, D.

D. Abookasis, O. Arazi, J. Rosen, and B. Javidi, "Security optical systems based on a joint transform correlator with significant output images," Opt. Eng. 40, 1584-1589 (2001).
[CrossRef]

Arazi, O.

D. Abookasis, O. Arazi, J. Rosen, and B. Javidi, "Security optical systems based on a joint transform correlator with significant output images," Opt. Eng. 40, 1584-1589 (2001).
[CrossRef]

Cai, L.

L. Yu, X. Peng, and L. Cai, "Parameterized multi-dimensional data encryption by digital optics," Opt. Commun. 203, 67-77 (2002).
[CrossRef]

Cai, L. Z.

X. F. Meng, L. Z. Cai, M. Z. He, G. Y. Dong, and X. X. Shen, "Cross-talk-free double-image encryption and watermarking with amplitude-phase separate modulations," J. Opt. 7, 624-631 (2005).
[CrossRef]

M. Z. He, L. Z. Cai, Q. Liu, W. C. Wang, and X. F. Meng, "Multiple image encryption and watermarking by random phase matching," Opt. Commun. 247, 29-37 (2005).
[CrossRef]

M. Z. He, L. Z. Cai, Q. Liu, and X. L. Yang, "Phase-only encryption and watermarking based on phase-shifting interferometry," Appl. Opt. 44, 2600-2606 (2005).
[CrossRef] [PubMed]

L. Z. Cai, M. Z. He, Q. Liu, and X. L. Yang, "Digital image encryption and watermarking by phase-shifting interferometry," Appl. Opt. 43, 3078-3084 (2004).
[CrossRef] [PubMed]

Chang, H. T.

H. T. Chang and C. T. Chen, "Asymmetric-image verification for security optical system based on joint transform correlator architecture," Opt. Commun. 239, 43-54 (2004).
[CrossRef]

C. H. Yeh, H. T. Chang, H. C. Chien, and C. J. Kuo, "Design of cascaded phase keys for a hierarchical security system," Appl. Opt. 41, 6128-6134 (2002).
[CrossRef] [PubMed]

H. T. Chang, W. C. Lu, and C. J. Kuo, "Multiple-phase retrieval for optical security systems by use of random-phase encoding," Appl. Opt. 41, 4825-4834 (2002).
[CrossRef] [PubMed]

H. T. Chang, "Image encryption using separable amplitude-based virtual image and iteratively retrieved phase information," Opt. Eng. 40, 2165-2171 (2001).
[CrossRef]

Chatwin, C.

R. K. Wang, I. A. Watson, and C. Chatwin, "Random phase encoding for optical security," Opt. Eng. 35, 2464-2468 (1996).
[CrossRef]

Chen, C. T.

H. T. Chang and C. T. Chen, "Asymmetric-image verification for security optical system based on joint transform correlator architecture," Opt. Commun. 239, 43-54 (2004).
[CrossRef]

Chien, H. C.

Cozzella, L.

G. S. Spagnolo, C. Simonetti, and L. Cozzella, "Content fragile watermarking based on a computer generated hologram coding technique," J. Opt. 7, 333-342 (2005).
[CrossRef]

Dong, G. Y.

X. F. Meng, L. Z. Cai, M. Z. He, G. Y. Dong, and X. X. Shen, "Cross-talk-free double-image encryption and watermarking with amplitude-phase separate modulations," J. Opt. 7, 624-631 (2005).
[CrossRef]

Flenup, J. R.

He, M. Z.

M. Z. He, L. Z. Cai, Q. Liu, W. C. Wang, and X. F. Meng, "Multiple image encryption and watermarking by random phase matching," Opt. Commun. 247, 29-37 (2005).
[CrossRef]

X. F. Meng, L. Z. Cai, M. Z. He, G. Y. Dong, and X. X. Shen, "Cross-talk-free double-image encryption and watermarking with amplitude-phase separate modulations," J. Opt. 7, 624-631 (2005).
[CrossRef]

M. Z. He, L. Z. Cai, Q. Liu, and X. L. Yang, "Phase-only encryption and watermarking based on phase-shifting interferometry," Appl. Opt. 44, 2600-2606 (2005).
[CrossRef] [PubMed]

L. Z. Cai, M. Z. He, Q. Liu, and X. L. Yang, "Digital image encryption and watermarking by phase-shifting interferometry," Appl. Opt. 43, 3078-3084 (2004).
[CrossRef] [PubMed]

Hennelly, B.

Ide, M.

Javidi, B.

Joseph, J.

Kim, D. H.

Kim, H.

Kishk, S.

Kreske, K.

Kuo, C. J.

Kuroda, K.

Lee, Y. H.

Li, J.

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).
[CrossRef]

Li, Y.

Liu, Q.

Lu, H.

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).
[CrossRef]

Lu, W. C.

Ma, Y.

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).
[CrossRef]

Mait, J. N.

Matoba, O.

Meng, X. F.

X. F. Meng, L. Z. Cai, M. Z. He, G. Y. Dong, and X. X. Shen, "Cross-talk-free double-image encryption and watermarking with amplitude-phase separate modulations," J. Opt. 7, 624-631 (2005).
[CrossRef]

M. Z. He, L. Z. Cai, Q. Liu, W. C. Wang, and X. F. Meng, "Multiple image encryption and watermarking by random phase matching," Opt. Commun. 247, 29-37 (2005).
[CrossRef]

Mifune, Y.

Mikan, S.

Morimoto, Y.

Nomura, T.

T. Nomura, K. Uota, and Y. Morimoto, "Hybrid optical encryption of a 3-D object using a digital holographic technique," Opt. Eng. 43, 2228-2232 (2004).
[CrossRef]

T. Nomura, S. Mikan, Y. Morimoto, and B. Javidi, "Secure optical data storage with random phase key codes by use of a configuration of a joint transform correlator," Appl. Opt. 42, 1508-1514 (2003).
[CrossRef] [PubMed]

B. Javidi and T. Nomura, "Securing information by use of digital holography," Opt. Lett. 25, 28-30 (2000).
[CrossRef]

B. Javidi and T. Nomura, "Polarization encoding for optical security systems," Opt. Eng. 39, 2439-2443 (2000).
[CrossRef]

Okada-Shudo, Y.

Peng, X.

L. Yu, X. Peng, and L. Cai, "Parameterized multi-dimensional data encryption by digital optics," Opt. Commun. 203, 67-77 (2002).
[CrossRef]

Refrégier, P.

Rosen, J.

Shen, X. X.

X. F. Meng, L. Z. Cai, M. Z. He, G. Y. Dong, and X. X. Shen, "Cross-talk-free double-image encryption and watermarking with amplitude-phase separate modulations," J. Opt. 7, 624-631 (2005).
[CrossRef]

Sheridan, J. T.

Shimura, T.

Simonetti, C.

G. S. Spagnolo, C. Simonetti, and L. Cozzella, "Content fragile watermarking based on a computer generated hologram coding technique," J. Opt. 7, 333-342 (2005).
[CrossRef]

Singh, K.

Situ, G.

G. Situ and J. Zhang, "Image hiding with computer-generated phase codes for optical authentication," Opt. Commun. 245, 55-65 (2005).
[CrossRef]

G. Situ and J. Zhang, "A lensless optical security system based on computer-generated phase only masks," Opt. Commun. 232, 115-122 (2004).
[CrossRef]

G. Situ and J. Zhang, "A cascaded iterative Fourier transform algorithm for optical security applications," Optik (Stuttgart) 114, 473-477 (2003).
[CrossRef]

Song, X.

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).
[CrossRef]

Spagnolo, G. S.

G. S. Spagnolo, C. Simonetti, and L. Cozzella, "Content fragile watermarking based on a computer generated hologram coding technique," J. Opt. 7, 333-342 (2005).
[CrossRef]

Tajahuerce, E.

Takai, N.

Tan, X.

Unnikrishnan, G.

Uota, K.

T. Nomura, K. Uota, and Y. Morimoto, "Hybrid optical encryption of a 3-D object using a digital holographic technique," Opt. Eng. 43, 2228-2232 (2004).
[CrossRef]

Wang, R. K.

R. K. Wang, I. A. Watson, and C. Chatwin, "Random phase encoding for optical security," Opt. Eng. 35, 2464-2468 (1996).
[CrossRef]

Wang, W. C.

M. Z. He, L. Z. Cai, Q. Liu, W. C. Wang, and X. F. Meng, "Multiple image encryption and watermarking by random phase matching," Opt. Commun. 247, 29-37 (2005).
[CrossRef]

Watson, I. A.

R. K. Wang, I. A. Watson, and C. Chatwin, "Random phase encoding for optical security," Opt. Eng. 35, 2464-2468 (1996).
[CrossRef]

Yang, X. L.

Yeh, C. H.

Yu, L.

L. Yu, X. Peng, and L. Cai, "Parameterized multi-dimensional data encryption by digital optics," Opt. Commun. 203, 67-77 (2002).
[CrossRef]

Zhang, J.

G. Situ and J. Zhang, "Image hiding with computer-generated phase codes for optical authentication," Opt. Commun. 245, 55-65 (2005).
[CrossRef]

G. Situ and J. Zhang, "A lensless optical security system based on computer-generated phase only masks," Opt. Commun. 232, 115-122 (2004).
[CrossRef]

G. Situ and J. Zhang, "A cascaded iterative Fourier transform algorithm for optical security applications," Optik (Stuttgart) 114, 473-477 (2003).
[CrossRef]

Zhao, J.

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).
[CrossRef]

Appl. Opt. (12)

S. Kishk and B. Javidi, "Information hiding technique with double phase encoding," Appl. Opt. 41, 5462-5470 (2002).
[CrossRef] [PubMed]

T. Nomura, S. Mikan, Y. Morimoto, and B. Javidi, "Secure optical data storage with random phase key codes by use of a configuration of a joint transform correlator," Appl. Opt. 42, 1508-1514 (2003).
[CrossRef] [PubMed]

E. Tajahuerce and B. Javidi, "Encrypting three dimensional information with digital holography," Appl. Opt. 39, 6595-6601 (2000).
[CrossRef]

N. Takai and Y. Mifune, "Digital watermarking by a holographic technique," Appl. Opt. 41, 865-873 (2002).
[CrossRef] [PubMed]

X. Tan, O. Matoba, Y. Okada-Shudo, M. Ide, T. Shimura, and K. Kuroda, "Secure optical memory system with polarization encryption," Appl. Opt. 40, 2310-2315 (2001).
[CrossRef]

Y. Li, K. Kreske, and J. Rosen, "Security and encryption optical systems based on a correlator with significant output images," Appl. Opt. 39, 5295-5301 (2000).
[CrossRef]

H. T. Chang, W. C. Lu, and C. J. Kuo, "Multiple-phase retrieval for optical security systems by use of random-phase encoding," Appl. Opt. 41, 4825-4834 (2002).
[CrossRef] [PubMed]

J. Rosen and B. Javidi, "Hidden images in halftone pictures," Appl. Opt. 40, 3346-3353 (2001).
[CrossRef]

C. H. Yeh, H. T. Chang, H. C. Chien, and C. J. Kuo, "Design of cascaded phase keys for a hierarchical security system," Appl. Opt. 41, 6128-6134 (2002).
[CrossRef] [PubMed]

J. R. Flenup, "Phase retrieval algorithms: a comparison," Appl. Opt. 21, 2758-2769 (1982).
[CrossRef]

L. Z. Cai, M. Z. He, Q. Liu, and X. L. Yang, "Digital image encryption and watermarking by phase-shifting interferometry," Appl. Opt. 43, 3078-3084 (2004).
[CrossRef] [PubMed]

M. Z. He, L. Z. Cai, Q. Liu, and X. L. Yang, "Phase-only encryption and watermarking based on phase-shifting interferometry," Appl. Opt. 44, 2600-2606 (2005).
[CrossRef] [PubMed]

J. Opt. (2)

X. F. Meng, L. Z. Cai, M. Z. He, G. Y. Dong, and X. X. Shen, "Cross-talk-free double-image encryption and watermarking with amplitude-phase separate modulations," J. Opt. 7, 624-631 (2005).
[CrossRef]

G. S. Spagnolo, C. Simonetti, and L. Cozzella, "Content fragile watermarking based on a computer generated hologram coding technique," J. Opt. 7, 333-342 (2005).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Commun. (6)

M. Z. He, L. Z. Cai, Q. Liu, W. C. Wang, and X. F. Meng, "Multiple image encryption and watermarking by random phase matching," Opt. Commun. 247, 29-37 (2005).
[CrossRef]

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).
[CrossRef]

G. Situ and J. Zhang, "A lensless optical security system based on computer-generated phase only masks," Opt. Commun. 232, 115-122 (2004).
[CrossRef]

H. T. Chang and C. T. Chen, "Asymmetric-image verification for security optical system based on joint transform correlator architecture," Opt. Commun. 239, 43-54 (2004).
[CrossRef]

G. Situ and J. Zhang, "Image hiding with computer-generated phase codes for optical authentication," Opt. Commun. 245, 55-65 (2005).
[CrossRef]

L. Yu, X. Peng, and L. Cai, "Parameterized multi-dimensional data encryption by digital optics," Opt. Commun. 203, 67-77 (2002).
[CrossRef]

Opt. Eng. (5)

T. Nomura, K. Uota, and Y. Morimoto, "Hybrid optical encryption of a 3-D object using a digital holographic technique," Opt. Eng. 43, 2228-2232 (2004).
[CrossRef]

B. Javidi and T. Nomura, "Polarization encoding for optical security systems," Opt. Eng. 39, 2439-2443 (2000).
[CrossRef]

H. T. Chang, "Image encryption using separable amplitude-based virtual image and iteratively retrieved phase information," Opt. Eng. 40, 2165-2171 (2001).
[CrossRef]

R. K. Wang, I. A. Watson, and C. Chatwin, "Random phase encoding for optical security," Opt. Eng. 35, 2464-2468 (1996).
[CrossRef]

D. Abookasis, O. Arazi, J. Rosen, and B. Javidi, "Security optical systems based on a joint transform correlator with significant output images," Opt. Eng. 40, 1584-1589 (2001).
[CrossRef]

Opt. Express (1)

Opt. Lett. (6)

Optik (1)

G. Situ and J. Zhang, "A cascaded iterative Fourier transform algorithm for optical security applications," Optik (Stuttgart) 114, 473-477 (2003).
[CrossRef]

Other (1)

B. Javidi, Optical and Digital Techniques for Information Security (Springer, 2005).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Diagram of the information security system of iterative retrieval in the Fresnel domain by (a) double-phase-mask and (b) multiple-phase-mask encoding techniques.

Fig. 2
Fig. 2

Principle of the PRP technique with a 5 × 5 numerical matrix as an example: (a) 5 × 5 RPM, (b) original pixel positions and randomly permutated positions of the numerical matrix, (c) original numerical values and permuted values of the numerical matrix.

Fig. 3
Fig. 3

Example of the PRP technique applied to a 256 × 256 image (256 gray levels) with use of a 256 × 256 RPM (not shown): (a) original image, (b) image after each pixel in the same row of the original image is randomly permuted according to the RPM, (c) image when each column is further randomly permuted.

Fig. 4
Fig. 4

Simulation results of image retrieval with four phase masks: (a) original host image, (b) original target hidden image, (c)–(f) retrieved target hidden image in the first to the fourth iteration steps with all the correct keys.

Fig. 5
Fig. 5

Simulation results of image retrieval with four phase masks after 16 iterations: (a) retrieved host image, (b) retrieved target hidden image.

Fig. 6
Fig. 6

(a) Retrieved target hidden image when one of the four phase masks is not used. (b) Similar result with one phase mask shifted transversely by 1 pixel. (c) Retrieved hidden image when λ has a relative error of 0.5%. (d) Retrieved hidden image without use of the correct RPM.

Fig. 7
Fig. 7

Robustness of this method against occlusion and noise attack: (a) host image with 25% occlusion, (b) retrieved hidden image from (a), (c) host image with Gaussian noise with a standard deviation of 0.1, (d) retrieved hidden image from (c).

Fig. 8
Fig. 8

Comparison of CCs and MSEs versus iteration number for different phase mask numbers: (a) and (b) CCs and MSEs for double phase masks[23] and four phase masks, respectively; (c) and (d) similar results as (a) and (b) but with multiple phase masks.

Fig. 9
Fig. 9

Variation of CCs and MSEs of a retrieved target hidden image with parameter errors: (a) effect of wavelength error Δλ for the two-phase-mask and four-phase-mask systems, (b) effect of distance error Δzi for the four-phase-mask system, (c) effect of distance error Δzi for the two-phase-mask system.[23]

Fig. 10
Fig. 10

Comparison of CCs and MSEs versus iteration number for different phase mask numbers and different phase levels with different algorithms. Left column, Fresnel-transform method: (a), (c), and (e) are the results when phase distributions are quantized to 32, 16, and 8 phase levels, respectively. Right column, similar results but with the Fourier-transform method.[20]

Tables (2)

Tables Icon

Table 1 Calculation Results of Correlation Coefficients and Mean Square Errors for the Retrieved Host and Hidden Images in Figs. 4, 5, and 7

Tables Icon

Table 2 Comparison of the Performance of the Proposed Method with Different Phase Mask Numbers

Equations (9)

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g ^ k ( x , y ) = g k ( x , y ) exp [ i ϕ k ( x , y ) ] = FrT ( FrT { f k ( x 1 , y 1 ) exp [ i ψ 1     k ( x 1 , y 1 ) ] ; z 1 } exp [ i ψ 2     k ( x 2 , y 2 ) ] ; z 2 ) ,
ψ 2       k + 1 ( x 2 , y 2 ) = angle { IFrT { g ( x , y ) exp [ i ϕ k ( x , y ) ] ; z 2 } FrT { f ( x , y ) exp [ i ψ 1     k ( x 1 , y 1 ) ] ; z 1 } } ,
ψ 1     k + 1 ( x 1 , y 1 ) = angle { IFrT ( IFrT { g ( x , y ) exp [ i ϕ k ( x , y ) ] ; z 2 } exp [ i ψ 2     k + 1 ( x 2 , y 2 ) ] ; z 1 ) f ( x , y ) exp [ i ψ 1     k ( x 1 , y 1 ) ] } ,
MSE = 1 M N i = 1 M j = 1 N | h k h | 2 ,
CC = COV ( h , h k ) σ h σ h k ,
COV ( h , h k ) = E { [ h E ( h ) ] [ h k E ( h k ) ] } ,
g ^ = | g ^ | exp ( i ϕ ) = FrT [ (   FrT { FrT [ f exp ( i ψ 1 ) ; z 1 ] exp ( i ψ 2 ) ; z 2 } ) exp ( i ψ n ) ; z n ] .
ψ n = angle { IFrT [ g exp ( i ϕ ) ; z n ] FrT [ (  FrT { FrT [ f exp ( i ψ 1 ) ; z 1 ] exp ( i ψ 2 ) ; z 2 } ) exp ( i ψ n 1 ) ; z n 1 ] } ,
ψ i = angle { IFrT [ (  IFrT { IFrT [ g exp ( i ϕ ) ; z n ] exp ( i ψ n ) ; z n 1 } ) exp ( i ψ i + 1 ) ; z i ] FrT [ (  FrT { FrT [ f exp ( i ψ 1 ) ; z 1 ] exp ( i ψ 2 ) ; z 2 } ) exp ( i ψ i 1 ) ; z i 1 ] } ,

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