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3D object watermarking by a 3D hidden object

Sherif Kishk and Bahram Javidi

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Abstract

In this paper we present a method to watermark a 3D object with another hidden 3D object using digital holography. The watermark or the hidden information is a 3D object that is embedded in the digital hologram of a 3D host object. The digital holograms are obtained optically by phase shift interferometery. The hologram of the hidden 3D object is double phase encoded before embedding it to the host 3D object hologram. Then, the watermarked hologram is double phase encoded again using different set of codes. The resultant watermarked hologram is very secure because of the multi-key nature of the watermarking process. We discuss the effect of distortion caused by hologram quantization and occlusion of some of the hologram pixels. We present tests to illustrate the effect of using a window of the hologram to reconstruct the hidden 3D object and the host 3D object. Both mathematical analysis and simulations are presented to illustrate the system performance. To the best of our knowledge, this is the first report of embedding a 3D objects within another 3D object.

©2003 Optical Society of America

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References

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  • Publication
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    [Crossref]
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    [Crossref]
  4. S. Kishk and B. Javidi, “Information Hiding Technique using Double Phase Encoding,” Appl. Opt. 41, 5470–5482 (2002).
    [Crossref]
  5. Chuan-Fu Wu and Wen-Shyong Hsieh, “Digital watermarking using zerotree of DCT,” Consumer Electronics, IEEE Transactions on,  46, 87–94 (2000).
    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]

2003 (1)

S. Kishk and B. Javidi “Watermarking of a 3D Object Using Digital Holography,” Opt. Let. 28, 167–169 (2003).
[Crossref]

2002 (3)

R. Ohbuchi, A. ukaiyama, and S. Takahashi “A frequency-domain approach to watermarking 3D shapes,” Computer Graphics Forum 21, 373–382 Sp. Iss. SI(2002).
[Crossref]

S. Kishk and B. Javidi, “Information Hiding Technique using Double Phase Encoding,” Appl. Opt. 41, 5470–5482 (2002).
[Crossref]

T. Naughton, Y. Frauel, B. Javidi, and E. Tajahuerce, “Compression of digital holograms for threedimensional object reconstruction and recognition,” Appl. Opt. 11, 4124–4132 (2002).
[Crossref]

2001 (1)

2000 (2)

Chuan-Fu Wu and Wen-Shyong Hsieh, “Digital watermarking using zerotree of DCT,” Consumer Electronics, IEEE Transactions on,  46, 87–94 (2000).
[Crossref]

G. C. Langelaar, I. Setyawan, and R. L. Lagendijk, “Watermarking Digital Image and Video Data. A state of the Art overview,” IEEE Signal Processing Magazine 17, 20–46 (2000).
[Crossref]

1998 (2)

1997 (1)

1996 (2)

R. K. Wang, I. A. Watson, and C. Chatwin, “Random phase encoding for optical secutity,” Opt. Eng. 35, 2464–2460 (1996).
[Crossref]

W. Bender, D. Gruhl, N. Morimoto, and L. Lu, “Techniques for data hiding,” IBM Systems Journal 35, 313–336 (1996).
[Crossref]

1995 (1)

P. Refregier and B. Javidi, “Optical image encryption using input plane and Fourier plane random encoding,” Opt. Let. 20, 767–769 (1995).
[Crossref]

1994 (1)

1974 (1)

Bender, W.

W. Bender, D. Gruhl, N. Morimoto, and L. Lu, “Techniques for data hiding,” IBM Systems Journal 35, 313–336 (1996).
[Crossref]

Bollaro, F.

Brangaccio, D. J.

Bruninig, J. H.

Caulfield, H. J.

H. J. Caulfield, Handbook of Optical Holography, (Academic press, London, 1979).

Chatwin, C.

R. K. Wang, I. A. Watson, and C. Chatwin, “Random phase encoding for optical secutity,” Opt. Eng. 35, 2464–2460 (1996).
[Crossref]

Duric, Z.

N. F. Johnson, Z. Duric, and S. Jajodia, Information Hiding : Steganography and Watermarking - Attacks and Countermeasures (Advances in Information Security, Volume 1,Kluwer Academic2001).

Frauel, Y.

T. Naughton, Y. Frauel, B. Javidi, and E. Tajahuerce, “Compression of digital holograms for threedimensional object reconstruction and recognition,” Appl. Opt. 11, 4124–4132 (2002).
[Crossref]

Gallagher, J. E.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, (McGraw-Hill, New York, NY, 1996).

Goudail, F.

Gruhl, D.

W. Bender, D. Gruhl, N. Morimoto, and L. Lu, “Techniques for data hiding,” IBM Systems Journal 35, 313–336 (1996).
[Crossref]

Herriott, D. R.

Hosinger, C.

C. Hosinger and M. Rabbani, “Data Embedding using Phase Dispersion,” The International Conference on Information Technology: Coding and Computing (ITCC 2000), Las Vegas, Nev. 27–29 March (2000).

Hsieh, Wen-Shyong

Chuan-Fu Wu and Wen-Shyong Hsieh, “Digital watermarking using zerotree of DCT,” Consumer Electronics, IEEE Transactions on,  46, 87–94 (2000).
[Crossref]

Jajodia, S.

N. F. Johnson, Z. Duric, and S. Jajodia, Information Hiding : Steganography and Watermarking - Attacks and Countermeasures (Advances in Information Security, Volume 1,Kluwer Academic2001).

Javidi, B.

S. Kishk and B. Javidi “Watermarking of a 3D Object Using Digital Holography,” Opt. Let. 28, 167–169 (2003).
[Crossref]

S. Kishk and B. Javidi, “Information Hiding Technique using Double Phase Encoding,” Appl. Opt. 41, 5470–5482 (2002).
[Crossref]

T. Naughton, Y. Frauel, B. Javidi, and E. Tajahuerce, “Compression of digital holograms for threedimensional object reconstruction and recognition,” Appl. Opt. 11, 4124–4132 (2002).
[Crossref]

F. Goudail, F. Bollaro, B. Javidi, and P. Refregier, “Influence of perturbation in a double phase-encoding system,” J. Opt. Soc. Am. A 15, 2629–2638 (1998).
[Crossref]

P. Refregier and B. Javidi, “Optical image encryption using input plane and Fourier plane random encoding,” Opt. Let. 20, 767–769 (1995).
[Crossref]

Javidi, Bahram

Johnson, N. F.

N. F. Johnson, Z. Duric, and S. Jajodia, Information Hiding : Steganography and Watermarking - Attacks and Countermeasures (Advances in Information Security, Volume 1,Kluwer Academic2001).

Jüptner, W. P. O.

Kishk, S.

S. Kishk and B. Javidi “Watermarking of a 3D Object Using Digital Holography,” Opt. Let. 28, 167–169 (2003).
[Crossref]

S. Kishk and B. Javidi, “Information Hiding Technique using Double Phase Encoding,” Appl. Opt. 41, 5470–5482 (2002).
[Crossref]

Kuo, C.-C. Jay

Lagendijk, R. L.

G. C. Langelaar, I. Setyawan, and R. L. Lagendijk, “Watermarking Digital Image and Video Data. A state of the Art overview,” IEEE Signal Processing Magazine 17, 20–46 (2000).
[Crossref]

Langelaar, G. C.

G. C. Langelaar, I. Setyawan, and R. L. Lagendijk, “Watermarking Digital Image and Video Data. A state of the Art overview,” IEEE Signal Processing Magazine 17, 20–46 (2000).
[Crossref]

Lu, L.

W. Bender, D. Gruhl, N. Morimoto, and L. Lu, “Techniques for data hiding,” IBM Systems Journal 35, 313–336 (1996).
[Crossref]

Morimoto, N.

W. Bender, D. Gruhl, N. Morimoto, and L. Lu, “Techniques for data hiding,” IBM Systems Journal 35, 313–336 (1996).
[Crossref]

Naughton, T.

T. Naughton, Y. Frauel, B. Javidi, and E. Tajahuerce, “Compression of digital holograms for threedimensional object reconstruction and recognition,” Appl. Opt. 11, 4124–4132 (2002).
[Crossref]

Ohbuchi, R.

R. Ohbuchi, A. ukaiyama, and S. Takahashi “A frequency-domain approach to watermarking 3D shapes,” Computer Graphics Forum 21, 373–382 Sp. Iss. SI(2002).
[Crossref]

Rabbani, M.

C. Hosinger and M. Rabbani, “Data Embedding using Phase Dispersion,” The International Conference on Information Technology: Coding and Computing (ITCC 2000), Las Vegas, Nev. 27–29 March (2000).

Refregier, P.

F. Goudail, F. Bollaro, B. Javidi, and P. Refregier, “Influence of perturbation in a double phase-encoding system,” J. Opt. Soc. Am. A 15, 2629–2638 (1998).
[Crossref]

P. Refregier and B. Javidi, “Optical image encryption using input plane and Fourier plane random encoding,” Opt. Let. 20, 767–769 (1995).
[Crossref]

Rosen, Joseph

Rosenfeld, D. P.

Schnars, U.

Schwider, J.

J. Schwider, “Advanced Evaluation Techniques in Interferometry,” in Progress in Optics, E. Wolf, ed (North Holland, Amesterdam) XXVIII, 271–359 (1990)

Setyawan, I.

G. C. Langelaar, I. Setyawan, and R. L. Lagendijk, “Watermarking Digital Image and Video Data. A state of the Art overview,” IEEE Signal Processing Magazine 17, 20–46 (2000).
[Crossref]

Su, Po-Chyi

Tajahuerce, E.

T. Naughton, Y. Frauel, B. Javidi, and E. Tajahuerce, “Compression of digital holograms for threedimensional object reconstruction and recognition,” Appl. Opt. 11, 4124–4132 (2002).
[Crossref]

Takahashi, S.

R. Ohbuchi, A. ukaiyama, and S. Takahashi “A frequency-domain approach to watermarking 3D shapes,” Computer Graphics Forum 21, 373–382 Sp. Iss. SI(2002).
[Crossref]

ukaiyama, A.

R. Ohbuchi, A. ukaiyama, and S. Takahashi “A frequency-domain approach to watermarking 3D shapes,” Computer Graphics Forum 21, 373–382 Sp. Iss. SI(2002).
[Crossref]

Wang, Houng-jyh

Wang, R. K.

R. K. Wang, I. A. Watson, and C. Chatwin, “Random phase encoding for optical secutity,” Opt. Eng. 35, 2464–2460 (1996).
[Crossref]

Watson, I. A.

R. K. Wang, I. A. Watson, and C. Chatwin, “Random phase encoding for optical secutity,” Opt. Eng. 35, 2464–2460 (1996).
[Crossref]

White, A. D.

Wu, Chuan-Fu

Chuan-Fu Wu and Wen-Shyong Hsieh, “Digital watermarking using zerotree of DCT,” Consumer Electronics, IEEE Transactions on,  46, 87–94 (2000).
[Crossref]

Yamaguchi, I

Zhang, T

Appl. Opt. (5)

Computer Graphics Forum (1)

R. Ohbuchi, A. ukaiyama, and S. Takahashi “A frequency-domain approach to watermarking 3D shapes,” Computer Graphics Forum 21, 373–382 Sp. Iss. SI(2002).
[Crossref]

Consumer Electronics, IEEE Transactions on (1)

Chuan-Fu Wu and Wen-Shyong Hsieh, “Digital watermarking using zerotree of DCT,” Consumer Electronics, IEEE Transactions on,  46, 87–94 (2000).
[Crossref]

IBM Systems Journal (1)

W. Bender, D. Gruhl, N. Morimoto, and L. Lu, “Techniques for data hiding,” IBM Systems Journal 35, 313–336 (1996).
[Crossref]

IEEE Signal Processing Magazine (1)

G. C. Langelaar, I. Setyawan, and R. L. Lagendijk, “Watermarking Digital Image and Video Data. A state of the Art overview,” IEEE Signal Processing Magazine 17, 20–46 (2000).
[Crossref]

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

Opt. Eng. (1)

R. K. Wang, I. A. Watson, and C. Chatwin, “Random phase encoding for optical secutity,” Opt. Eng. 35, 2464–2460 (1996).
[Crossref]

Opt. Express (1)

Opt. Let. (2)

S. Kishk and B. Javidi “Watermarking of a 3D Object Using Digital Holography,” Opt. Let. 28, 167–169 (2003).
[Crossref]

P. Refregier and B. Javidi, “Optical image encryption using input plane and Fourier plane random encoding,” Opt. Let. 20, 767–769 (1995).
[Crossref]

Opt. Lett. (1)

Other (5)

H. J. Caulfield, Handbook of Optical Holography, (Academic press, London, 1979).

J. W. Goodman, Introduction to Fourier Optics, (McGraw-Hill, New York, NY, 1996).

N. F. Johnson, Z. Duric, and S. Jajodia, Information Hiding : Steganography and Watermarking - Attacks and Countermeasures (Advances in Information Security, Volume 1,Kluwer Academic2001).

C. Hosinger and M. Rabbani, “Data Embedding using Phase Dispersion,” The International Conference on Information Technology: Coding and Computing (ITCC 2000), Las Vegas, Nev. 27–29 March (2000).

J. Schwider, “Advanced Evaluation Techniques in Interferometry,” in Progress in Optics, E. Wolf, ed (North Holland, Amesterdam) XXVIII, 271–359 (1990)

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Figures (14)
Fig. 1.
Fig. 1. Phase shifting interferometer

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Fig. 2.
Fig. 2. Block diagram of the proposed system (a) Transmitter (b) Receiver

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Fig. 3.
Fig. 3. 3D object reconstructed at different distances from the output plane

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Fig. 4.
Fig. 4. The original host and hidden 3D objects.

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Fig. 5.
Fig. 5. (a) The autocorrelation for the watermarked hologram without using the second double phase encoding process. (b) The autocorrelation for the watermarked hologram when using the second double phase encoding process.

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Fig. 6.
Fig. 6. The Reconstructed 3D objects from the double phase encoded watermarked hologram without distortions (a) the 3D host object(b) The 3D hidden object

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Fig. 7.
Fig. 7. The reconstructed 3D host object using uniform quantization (a) 8 bits (b) 2 bits

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Fig. 8.
Fig. 8. The reconstructed 3D hidden object using uniform quantization (a) 8 bits (b) 2 bits

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Fig. 9.
Fig. 9. The reconstructed 3D objects using different portions of the digital holograms using quantized watermarked hologram. (a,b) 1/16 of the hologram area and 8 bits quantization (c,d) 1/4 of the hologram area and 4 bits quantization (e,f) 1/2 of the hologram area and 2 bits quantization

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Fig. 10.
Fig. 10. (a) The transmitted hologram having 50% occlusion and 4 bits quantization (b) The transmitted hologram having 75% occlusion and 4 bits quantization (c) The reconstructed 3D host object using the hologram in a (d) The reconstructed 3D hidden object using the hologram in a (e) The reconstructed 3D host object using the hologram in b (f) The reconstructed 3D hidden object using the hologram in b.

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Fig. 11.
Fig. 11. The effect of number of quantization levels and occluding parts of the transmitted double phase encoded watermarked hologram when using a uniform quantizer

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Fig. 12.
Fig. 12. The reconstructed 3D objects using an optimum quantizer with 4 bits quantization and 25% of the holograms (a) host object (b) hidden object

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Fig. 13.
Fig. 13. The error when using an optimum quantizer compared to the error when using a uniform quantizer, 25% of the hologram is used.

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Fig. 14.
Fig. 14. Effect of blind decoding on the hidden object (a) only the hidden object spatial domain phase code is unknown. (b) only the hidden object Fourier domain phase code is unknown.

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Tables (2)

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Table 1. error when using a uniform quantizer

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Table 2. Error when using only 25% of the hologram and a uniform quantizer

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Equations (48)

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D ( x , y ) = A ( x , y ) exp ( j ϕ ( x , y ) )
R ( x , y ; α ) = A R exp ⌊ j ( ϕ R + α ) ⌋
I ( x , y ; α ) = A 2 ( x , y ) + A R 2 + 2 A ( x , y ) A R cos ⁡ [ ϕ ( x , y ) − ϕ R − α ] .
ϕ O ( x , y ) = arctan ⁡ [ I 4 ( x , y ) − I 2 ( x , y ) I 1 ( x , y ) − I 3 ( x , y ) ] ,
A O ( x , y ) = 1 4 { [ I 1 ( x , y ) − I 3 ( x , y ) ] 2 + [ I 2 ( x , y ) − I 4 ( x , y ) ] 2 } 1 2
H O ( x , y ) = A O ( x , y ) exp [ j ϕ O ( x , y ) ]
H d ( x , y ) = { H ( x , y ) ψ 1 ( x , y ) } ⊗ IFT [ ψ 2 ( ξ , γ ) ]
σ 2 = 1 N . M ∑ y = 0 M − 1 ∑ x = 0 N − 1 H ( x , y ) H * ( x , y )
H w ( x , y ) = { [ H host ( x , y ) + α [ H hidden ( x , y ) ψ 1 ( x , y ) ⊗ ψ 2 ( x , y ) ] ] ψ ′ 1 ( x , y ) } ⊗ ψ ′ 2 ( x , y )
σ w 2 = 1 2 . N . M ∑ y = 0 M − 1 ∑ x = 0 N − 1 H hidden ( x , y ) H hidden * ( x , y )
H ˜ hidden ( x , y ) = α H hidden ( x , y ) + IFT { H ̂ host ( ξ , γ ) ψ * 2 ( ξ , γ ) } ψ * 1 ( x , y )
σ w 2 = 1 N . M ∑ y = 0 M − 1 ∑ x = 0 N − 1 H host ( x , y ) H host * ( x , y )
O ˜ d ( u , v ; d 1 ) = exp [ − j π λ d 1 ( Δ u 2 u 2 + Δ v 2 v 2 ) ] ∑ x = 0 N − 1 ∑ y = 0 M − 1 { H ˜ hidden ( x , y )
exp [ − j π λ d 1 ( Δ x 2 x + Δ y 2 y ) ] exp [ j 2 π ( x u N + y v M ) ] }
Δ u = λ d 1 N Δ x , and
Δ v = λ d 1 M Δ y
O ˜ d ( u , v ; d 1 ) = α O d ( u , v ; d 1 ) + exp [ − j π λ d 1 ( Δ u 2 u 2 + Δ v 2 v 2 ) ] .
∑ x = 0 N − 1 ∑ y = 0 M − 1 { ( IFT { H ˜ host ( ξ , γ ) ψ ′ * 2 ( ξ , γ ) } ψ ′ 1 * ( x , y ) )
exp [ − j π λ d 1 ( Δ x 2 x + Δ y 2 y ) ] exp [ j 2 π ( x u N + y v M ) ] }
O ˜ h ( u , v ; d 2 ) = O h ( u , v ; d 2 ) + exp [ − j π λ d 2 ( Δ u 2 u 2 + Δ v 2 v 2 ) ]
∑ x = 0 N − 1 ∑ = 01 M − 1 { α [ H hidden ( x , y ) ψ * 1 ( x , y ) ⊗ ψ * 2 ( x , y ) ]
exp [ − j π λ d 2 ( Δ x 2 x + Δ y 2 y ) ] exp [ j 2 π ( x u N + y v M ) ] }
σ host 2 = α 2 N . M ∑ y = 0 M − 1 ∑ x = 0 N − 1 H hidden ( x , y ) H hidden * ( x , y )
H ˜ w ( x , y ) = H w ( x , y ) + Δ H w ( x , y )
O ˜ d ( u , v ; d 1 ) = α O d ( u , v ; d 1 ) + exp [ − jπ λ d 1 ( Δ u 2 u 2 + Δ v 2 v 2 ) ] .
{ ∑ x = 0 N − 1 ∑ y = 0 M − 1 ( IFT { H ˜ host ( ξ , γ ) ψ ′ 2 * ( ξ , γ ) } ψ ′ 1 * ( x , y ) ) .
exp [ − j π λ d 1 ( Δ x 2 x + Δ y 2 y ) ] exp [ j 2 π ( x u N + y v M ) ] +
∑ x = 0 N − 1 ∑ y = 0 M − 1 Δ H w ( x , y ) exp [ − j π λ d ( Δ x 2 x + Δ y 2 y ) ] exp [ j 2 π ( x u N + y v M ) ] } .
O ˜ d ( u , v ; d 2 ) = O d ( u , v ; d 2 ) + exp [ − j π λ d 2 ( Δ u 2 u 2 + Δ v 2 v 2 ) ] .
{ ∑ u = 0 N − 1 ∑ v = 0 M − 1 α [ H hidden ( x , y ) ψ 1 ( x , y ) ⊗ ψ 2 ( x , y ) ] .
exp [ − j π λ d 2 ( Δ x 2 x + Δ y 2 y ) ] exp [ j 2 π ( x u N + y v M ) ] +
∑ x = 0 N − 1 ∑ y = 0 M − 1 Δ H w ( x , y ) exp [ − jπ λ d 2 ( Δ x 2 x + Δ y 2 y ) ] exp [ j 2 π ( x u N + y v M ) ] }
σ host 2 = α 2 N . M ∑ y = 0 M − 1 ∑ x = 0 N − 1 H hidden ( x , y ) H hidden * ( x , y ) + Δ 2 12
σ w 2 = 1 N . M ∑ y = 0 M − 1 ∑ x = 0 N − 1 H host ( x , y ) H host * ( x , y ) + Δ 2 12
H w o ( x , y ) = H w ( x , y ) ( 1 − W ( x , y ) ) = H w ( x , y ) − H w ( x , y ) W ( x , y )
O d ( u , v ; d ) = ∫ − ∞ ∞ − exp [ j 2 π λ ( z − d ) ] λ ( z − d ) exp [ j π λ ( z − d ) ( u 2 + v 2 ) ] ·
∫ − ∞ ∞ ∫ O ( x , y , z ) exp [ j π λ ( z − d ) ( x 2 + y 2 ) ]
exp [ j 2 π λ ( z − d ) ( x u + y v ) ] dx dy dz
error = 1 P ∑ p = 1 P 1 N × M ∑ y = 0 M − 1 ∑ x = 0 N − 1 [ O d ( u , v ; d ( p ) ) − O ( u , v ; d ( p ) ) ] 2 O ( u , v ; d ( p ) )
X ( u ) ≡ exp [ − j π λ d ( Δ u 2 u 2 ) ] · ∑ x = 0 N − 1 ( IFT { H ̂ host ( ξ ) ψ ′ 2 ( ξ ) } ψ ′ 1 ( x ) ) exp [ − j π λ d ( Δ x 2 x ) ] exp [ j 2 π ( x u N ) ] .
R x ( u , u ′ ) = E ⌊ X * ( u ) X ( u ′ ) ⌋
R x ( u , u ′ ) = E [ exp [ j π λ d ( Δ u 2 u 2 ) ] ∑ x = 0 N − 1 ∑ ξ = 0 N − 1 1 N H ̂ host * ( ξ ) ψ ′ 2 * ( ξ ) ψ ′ 1 * ( x ) exp [ j π λ d ( Δ x 2 x ) ] exp [ − j 2 π ( x u N ) ] . exp [ − j π λ d ( Δ u 2 u ′ 2 ) ] ∑ x ′ = 0 N − 1 ∑ ξ ′ = 0 N − 1 1 N H ̂ host ( ξ ′ ) ψ ′ 2 ( ξ ′ ) ψ ′ 1 ( x ′ ) exp [ − j π λ d ( Δ x 2 x ′ ) ] exp [ j 2 π ( x ′ u ′ N ) ] ]
R x ( u , u ′ ) = 1 N 2 E [ exp [ j π λ d Δ u 2 ( u 2 − u ′ 2 ) ] ∑ x = 0 N − 1 ∑ x ′ = 0 N − 1 ∑ ξ = 0 N − 1 ∑ ξ ′ N − 1 H * ( ζ ) H ( ξ ′ ) ψ ′ 2 ( ξ ) ψ ′ 2 ( ξ ' ) ψ ′ 1 ( x ) ψ ′ 1 ( x ′ ) exp [ j π λ d Δ x 2 ( x − x ′ ) ] exp [ − j 2 π ( x u N − x ′ u ′ N ) ] ]
R x ( u , u ′ ) = 1 N 2 exp [ j π λ d Δ u 2 ( u 2 − u ′ 2 ) ] ∑ x = 0 N − 1 ∑ x ′ = 0 N − 1 ∑ ξ = 0 N − 1 ∑ ξ ′ N − 1 H * ( ζ ) H ( ξ ′ ) E n 2 [ ψ ′ 2 ( ξ ′ ) ψ ′ 2 ( ξ ′ ) ] E n 1 [ ψ ′ 1 ( x ) ψ ′ 12 ( x ′ ) ]
exp [ j π λ d Δ x 2 ( x − x ′ ) ] exp [ − j 2 π ( x u N − x ′ u ′ N ) ]
R ( u , u ′ ) = 1 N 2 exp [ j π λ d Δ u 2 ( u 2 − u ′ 2 ) ] ∑ x = 0 N − 1 ∑ ζ = 0 N − 1 ∣ H ( ξ ) ∣ 2 exp [ − j 2 π x ( u N − u ′ N ) ] .
R ( u , u ′ ) = 1 N exp [ j π λ d Δ u 2 ( u 2 − u ′ 2 ) ∑ ξ = 0 N − 1 ∣ H ( ξ ) ∣ 2 δ ( u − u ′ ) ]
R ( u , u ′ ) = 1 N ∑ ξ = 0 N − 1 ∣ H ( ξ ) ∣ 2 δ ( u − u ′ )

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