Abstract

We propose an “accurate” phase-added stereogram, which can be defined as an improved phase-added stereogram. Generally, the macroblock size transformed by the fast Fourier transform is the same as the segmentation size of the phase-added stereogram. However, the proposed method uses a lager macroblock size than does the conventional method to reduce quantization error in discrete spatial frequencies in the spatial frequency domain. Therefore, even when the fast Fourier transform is used for calculation, the quality of the reconstructed image can be improved to be as clear as the Fresnel hologram.

© 2008 Optical Society of America

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  1. P. St. Hilaire, S. A. Benton, M. Lucente, M. L. Jepsen, J. Kollin, H. Yoshikawa, and J. Underkoffler, “Electronic display system for computational holography,” Proc. SPIE 1212, 174-182 (1990).
  2. K. Taima, H. Ueda, H. Okamoto, T. Kubota, Y. Nakamura, H. Nishida, H. Takahashi, and E. Shimizu, “New approach to the interactive holographic display system,” Proc. SPIE 2176, 23-29 (1994).
  3. J. A. Watlington, M. Lucente, C. J. Sparrell, V. M. Bove, and I. Tamitani, “A hardware architecture for rapid generation of electro-holographic fringe patterns,” Proc. SPIE 2406, 172-183 (1995).
  4. T. Okada, S. Iwata, O. Nishikawa, K. Matsumoto, H. Yoshikawa, K. Sato, and T. Honda, “The fast computation of holograms for the interactive holographic 3D display system,” Proc. SPIE 2577, 33-40 (1995).
  5. M. Lucente, “Interactive computation of hologram using a look-up table,” J. Electron. Imaging 2, 28-34 (1993).
    [CrossRef]
  6. F. Mok, J. Diep, H. Liu, and D. Psaltis, “Real-time computer generated hologram by means of liquid-crystal television spatial light modulator,” Opt. Lett. 11, 748-750 (1986).
    [CrossRef] [PubMed]
  7. T. Shimobaba, A. Shiraki, N. Masuda, and T. Ito, “Electroholographic display unit for three-dimensional display by use of special-purpose computational chip for holography and reflective LCD panel,” Opt. Express 13, 4196-4201 (2005).
    [CrossRef] [PubMed]
  8. H. Yoshikawa and H. Kameyama, “Integral holography,” Proc. SPIE 2406, 226-234 (1995).
  9. H. Yoshikawa and T. Yamaguchi, “Fast hologram calculation for holographic video display,” Proc. SPIE 6027, 561-566 (2006).
  10. T. Yatagai, “Stereoscopic approach to 3-D display using computer-generated holograms,” Appl. Opt. 15, 2722-2729 (1976).
    [CrossRef] [PubMed]
  11. M. Lucente, “Holographic bandwidth compression using spatial subsampling,” Opt. Eng. 35, 1529-1537 (1996).
    [CrossRef]
  12. M. Lucente, “Computational holographic bandwidth compression,” IBM Syst. J. 35, 349-365 (1996).
    [CrossRef]
  13. M. Yamaguchi, H. Hoshino, T. Honda, and N. Ohyama, “Phase-added stereogram: calculation of hologram using computer graphic technique,” Proc. SPIE 1914, 25-33 (1993).
  14. J. Tamai and H. Yoshikawa, “Faster computation of subsampled coherent stereogram,” J. Inst. Telev. Eng. Jpn. 50, 1612-1615 (1996) (in Japanese).
    [CrossRef]
  15. H. Kang, T. Fujii, T. Yamaguchi, and H. Yoshikawa, “A compensated phase-added stereogram for real-time holographic display,” Opt. Eng. 46, 095802 (2007).
    [CrossRef]
  16. J. P. Waters, “Holographic image synthesis utilizing theoretical method,” Appl. Phys. Lett. 9, 405-407 (1966).
    [CrossRef]
  17. M. Lucente, “Optimization of hologram computation for real-time display,” Proc. SPIE 1667, 25-33 (1992).
  18. H. Yoshikawa, “Digital holographic signal processing,” in Proceeding of the TAO First International Symposium on Three Dimensional Image Communication Technologies (Telecommunications Advancement Organization of Japan, 1993), paper S-4-2.
  19. H. Yoshikawa and K. Sasaki, “Image scaling for electro-holographic display,” Proc. SPIE 2176, 12-22 (1994).
  20. M. Lucente, “Diffraction-specific fringe computation for electro-holography,” Ph.D. dissertation (MIT, 1994).
  21. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley-Interscience, 1994).
  22. C. Palmer, Diffraction Grating Handbook (Richardson Grating Laboratory, Newport Corporation, 2002).
  23. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  24. E. Hecht, Optics (Wesley Longman, 2002).

2007 (1)

H. Kang, T. Fujii, T. Yamaguchi, and H. Yoshikawa, “A compensated phase-added stereogram for real-time holographic display,” Opt. Eng. 46, 095802 (2007).
[CrossRef]

2005 (1)

1996 (3)

M. Lucente, “Holographic bandwidth compression using spatial subsampling,” Opt. Eng. 35, 1529-1537 (1996).
[CrossRef]

M. Lucente, “Computational holographic bandwidth compression,” IBM Syst. J. 35, 349-365 (1996).
[CrossRef]

J. Tamai and H. Yoshikawa, “Faster computation of subsampled coherent stereogram,” J. Inst. Telev. Eng. Jpn. 50, 1612-1615 (1996) (in Japanese).
[CrossRef]

1993 (1)

M. Lucente, “Interactive computation of hologram using a look-up table,” J. Electron. Imaging 2, 28-34 (1993).
[CrossRef]

1986 (1)

1976 (1)

1966 (1)

J. P. Waters, “Holographic image synthesis utilizing theoretical method,” Appl. Phys. Lett. 9, 405-407 (1966).
[CrossRef]

Benton, S. A.

P. St. Hilaire, S. A. Benton, M. Lucente, M. L. Jepsen, J. Kollin, H. Yoshikawa, and J. Underkoffler, “Electronic display system for computational holography,” Proc. SPIE 1212, 174-182 (1990).

Bove, V. M.

J. A. Watlington, M. Lucente, C. J. Sparrell, V. M. Bove, and I. Tamitani, “A hardware architecture for rapid generation of electro-holographic fringe patterns,” Proc. SPIE 2406, 172-183 (1995).

Diep, J.

Fujii, T.

H. Kang, T. Fujii, T. Yamaguchi, and H. Yoshikawa, “A compensated phase-added stereogram for real-time holographic display,” Opt. Eng. 46, 095802 (2007).
[CrossRef]

Goodman, J. W.

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

Hecht, E.

E. Hecht, Optics (Wesley Longman, 2002).

Honda, T.

M. Yamaguchi, H. Hoshino, T. Honda, and N. Ohyama, “Phase-added stereogram: calculation of hologram using computer graphic technique,” Proc. SPIE 1914, 25-33 (1993).

T. Okada, S. Iwata, O. Nishikawa, K. Matsumoto, H. Yoshikawa, K. Sato, and T. Honda, “The fast computation of holograms for the interactive holographic 3D display system,” Proc. SPIE 2577, 33-40 (1995).

Hoshino, H.

M. Yamaguchi, H. Hoshino, T. Honda, and N. Ohyama, “Phase-added stereogram: calculation of hologram using computer graphic technique,” Proc. SPIE 1914, 25-33 (1993).

Ito, T.

Iwata, S.

T. Okada, S. Iwata, O. Nishikawa, K. Matsumoto, H. Yoshikawa, K. Sato, and T. Honda, “The fast computation of holograms for the interactive holographic 3D display system,” Proc. SPIE 2577, 33-40 (1995).

Jepsen, M. L.

P. St. Hilaire, S. A. Benton, M. Lucente, M. L. Jepsen, J. Kollin, H. Yoshikawa, and J. Underkoffler, “Electronic display system for computational holography,” Proc. SPIE 1212, 174-182 (1990).

Kameyama, H.

H. Yoshikawa and H. Kameyama, “Integral holography,” Proc. SPIE 2406, 226-234 (1995).

Kang, H.

H. Kang, T. Fujii, T. Yamaguchi, and H. Yoshikawa, “A compensated phase-added stereogram for real-time holographic display,” Opt. Eng. 46, 095802 (2007).
[CrossRef]

Kollin, J.

P. St. Hilaire, S. A. Benton, M. Lucente, M. L. Jepsen, J. Kollin, H. Yoshikawa, and J. Underkoffler, “Electronic display system for computational holography,” Proc. SPIE 1212, 174-182 (1990).

Kubota, T.

K. Taima, H. Ueda, H. Okamoto, T. Kubota, Y. Nakamura, H. Nishida, H. Takahashi, and E. Shimizu, “New approach to the interactive holographic display system,” Proc. SPIE 2176, 23-29 (1994).

Liu, H.

Lucente, M.

M. Lucente, “Holographic bandwidth compression using spatial subsampling,” Opt. Eng. 35, 1529-1537 (1996).
[CrossRef]

M. Lucente, “Computational holographic bandwidth compression,” IBM Syst. J. 35, 349-365 (1996).
[CrossRef]

M. Lucente, “Interactive computation of hologram using a look-up table,” J. Electron. Imaging 2, 28-34 (1993).
[CrossRef]

J. A. Watlington, M. Lucente, C. J. Sparrell, V. M. Bove, and I. Tamitani, “A hardware architecture for rapid generation of electro-holographic fringe patterns,” Proc. SPIE 2406, 172-183 (1995).

P. St. Hilaire, S. A. Benton, M. Lucente, M. L. Jepsen, J. Kollin, H. Yoshikawa, and J. Underkoffler, “Electronic display system for computational holography,” Proc. SPIE 1212, 174-182 (1990).

M. Lucente, “Diffraction-specific fringe computation for electro-holography,” Ph.D. dissertation (MIT, 1994).

M. Lucente, “Optimization of hologram computation for real-time display,” Proc. SPIE 1667, 25-33 (1992).

Masuda, N.

Matsumoto, K.

T. Okada, S. Iwata, O. Nishikawa, K. Matsumoto, H. Yoshikawa, K. Sato, and T. Honda, “The fast computation of holograms for the interactive holographic 3D display system,” Proc. SPIE 2577, 33-40 (1995).

Mok, F.

Nakamura, Y.

K. Taima, H. Ueda, H. Okamoto, T. Kubota, Y. Nakamura, H. Nishida, H. Takahashi, and E. Shimizu, “New approach to the interactive holographic display system,” Proc. SPIE 2176, 23-29 (1994).

Nishida, H.

K. Taima, H. Ueda, H. Okamoto, T. Kubota, Y. Nakamura, H. Nishida, H. Takahashi, and E. Shimizu, “New approach to the interactive holographic display system,” Proc. SPIE 2176, 23-29 (1994).

Nishikawa, O.

T. Okada, S. Iwata, O. Nishikawa, K. Matsumoto, H. Yoshikawa, K. Sato, and T. Honda, “The fast computation of holograms for the interactive holographic 3D display system,” Proc. SPIE 2577, 33-40 (1995).

Ohyama, N.

M. Yamaguchi, H. Hoshino, T. Honda, and N. Ohyama, “Phase-added stereogram: calculation of hologram using computer graphic technique,” Proc. SPIE 1914, 25-33 (1993).

Okada, T.

T. Okada, S. Iwata, O. Nishikawa, K. Matsumoto, H. Yoshikawa, K. Sato, and T. Honda, “The fast computation of holograms for the interactive holographic 3D display system,” Proc. SPIE 2577, 33-40 (1995).

Okamoto, H.

K. Taima, H. Ueda, H. Okamoto, T. Kubota, Y. Nakamura, H. Nishida, H. Takahashi, and E. Shimizu, “New approach to the interactive holographic display system,” Proc. SPIE 2176, 23-29 (1994).

Palmer, C.

C. Palmer, Diffraction Grating Handbook (Richardson Grating Laboratory, Newport Corporation, 2002).

Psaltis, D.

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley-Interscience, 1994).

Sasaki, K.

H. Yoshikawa and K. Sasaki, “Image scaling for electro-holographic display,” Proc. SPIE 2176, 12-22 (1994).

Sato, K.

T. Okada, S. Iwata, O. Nishikawa, K. Matsumoto, H. Yoshikawa, K. Sato, and T. Honda, “The fast computation of holograms for the interactive holographic 3D display system,” Proc. SPIE 2577, 33-40 (1995).

Shimizu, E.

K. Taima, H. Ueda, H. Okamoto, T. Kubota, Y. Nakamura, H. Nishida, H. Takahashi, and E. Shimizu, “New approach to the interactive holographic display system,” Proc. SPIE 2176, 23-29 (1994).

Shimobaba, T.

Shiraki, A.

Sparrell, C. J.

J. A. Watlington, M. Lucente, C. J. Sparrell, V. M. Bove, and I. Tamitani, “A hardware architecture for rapid generation of electro-holographic fringe patterns,” Proc. SPIE 2406, 172-183 (1995).

St. Hilaire, P.

P. St. Hilaire, S. A. Benton, M. Lucente, M. L. Jepsen, J. Kollin, H. Yoshikawa, and J. Underkoffler, “Electronic display system for computational holography,” Proc. SPIE 1212, 174-182 (1990).

Taima, K.

K. Taima, H. Ueda, H. Okamoto, T. Kubota, Y. Nakamura, H. Nishida, H. Takahashi, and E. Shimizu, “New approach to the interactive holographic display system,” Proc. SPIE 2176, 23-29 (1994).

Takahashi, H.

K. Taima, H. Ueda, H. Okamoto, T. Kubota, Y. Nakamura, H. Nishida, H. Takahashi, and E. Shimizu, “New approach to the interactive holographic display system,” Proc. SPIE 2176, 23-29 (1994).

Tamai, J.

J. Tamai and H. Yoshikawa, “Faster computation of subsampled coherent stereogram,” J. Inst. Telev. Eng. Jpn. 50, 1612-1615 (1996) (in Japanese).
[CrossRef]

Tamitani, I.

J. A. Watlington, M. Lucente, C. J. Sparrell, V. M. Bove, and I. Tamitani, “A hardware architecture for rapid generation of electro-holographic fringe patterns,” Proc. SPIE 2406, 172-183 (1995).

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley-Interscience, 1994).

Ueda, H.

K. Taima, H. Ueda, H. Okamoto, T. Kubota, Y. Nakamura, H. Nishida, H. Takahashi, and E. Shimizu, “New approach to the interactive holographic display system,” Proc. SPIE 2176, 23-29 (1994).

Underkoffler, J.

P. St. Hilaire, S. A. Benton, M. Lucente, M. L. Jepsen, J. Kollin, H. Yoshikawa, and J. Underkoffler, “Electronic display system for computational holography,” Proc. SPIE 1212, 174-182 (1990).

Waters, J. P.

J. P. Waters, “Holographic image synthesis utilizing theoretical method,” Appl. Phys. Lett. 9, 405-407 (1966).
[CrossRef]

Watlington, J. A.

J. A. Watlington, M. Lucente, C. J. Sparrell, V. M. Bove, and I. Tamitani, “A hardware architecture for rapid generation of electro-holographic fringe patterns,” Proc. SPIE 2406, 172-183 (1995).

Yamaguchi, M.

M. Yamaguchi, H. Hoshino, T. Honda, and N. Ohyama, “Phase-added stereogram: calculation of hologram using computer graphic technique,” Proc. SPIE 1914, 25-33 (1993).

Yamaguchi, T.

H. Kang, T. Fujii, T. Yamaguchi, and H. Yoshikawa, “A compensated phase-added stereogram for real-time holographic display,” Opt. Eng. 46, 095802 (2007).
[CrossRef]

H. Yoshikawa and T. Yamaguchi, “Fast hologram calculation for holographic video display,” Proc. SPIE 6027, 561-566 (2006).

Yatagai, T.

Yoshikawa, H.

H. Kang, T. Fujii, T. Yamaguchi, and H. Yoshikawa, “A compensated phase-added stereogram for real-time holographic display,” Opt. Eng. 46, 095802 (2007).
[CrossRef]

J. Tamai and H. Yoshikawa, “Faster computation of subsampled coherent stereogram,” J. Inst. Telev. Eng. Jpn. 50, 1612-1615 (1996) (in Japanese).
[CrossRef]

H. Yoshikawa and T. Yamaguchi, “Fast hologram calculation for holographic video display,” Proc. SPIE 6027, 561-566 (2006).

H. Yoshikawa and H. Kameyama, “Integral holography,” Proc. SPIE 2406, 226-234 (1995).

H. Yoshikawa, “Digital holographic signal processing,” in Proceeding of the TAO First International Symposium on Three Dimensional Image Communication Technologies (Telecommunications Advancement Organization of Japan, 1993), paper S-4-2.

H. Yoshikawa and K. Sasaki, “Image scaling for electro-holographic display,” Proc. SPIE 2176, 12-22 (1994).

P. St. Hilaire, S. A. Benton, M. Lucente, M. L. Jepsen, J. Kollin, H. Yoshikawa, and J. Underkoffler, “Electronic display system for computational holography,” Proc. SPIE 1212, 174-182 (1990).

T. Okada, S. Iwata, O. Nishikawa, K. Matsumoto, H. Yoshikawa, K. Sato, and T. Honda, “The fast computation of holograms for the interactive holographic 3D display system,” Proc. SPIE 2577, 33-40 (1995).

Appl. Phys. Lett. (1)

J. P. Waters, “Holographic image synthesis utilizing theoretical method,” Appl. Phys. Lett. 9, 405-407 (1966).
[CrossRef]

Appl. Opt. (1)

IBM Syst. J. (1)

M. Lucente, “Computational holographic bandwidth compression,” IBM Syst. J. 35, 349-365 (1996).
[CrossRef]

J. Electron. Imaging (1)

M. Lucente, “Interactive computation of hologram using a look-up table,” J. Electron. Imaging 2, 28-34 (1993).
[CrossRef]

J. Inst. Telev. Eng. Jpn. (1)

J. Tamai and H. Yoshikawa, “Faster computation of subsampled coherent stereogram,” J. Inst. Telev. Eng. Jpn. 50, 1612-1615 (1996) (in Japanese).
[CrossRef]

Opt. Eng. (1)

M. Lucente, “Holographic bandwidth compression using spatial subsampling,” Opt. Eng. 35, 1529-1537 (1996).
[CrossRef]

Opt. Eng. (1)

H. Kang, T. Fujii, T. Yamaguchi, and H. Yoshikawa, “A compensated phase-added stereogram for real-time holographic display,” Opt. Eng. 46, 095802 (2007).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Other (15)

H. Yoshikawa and H. Kameyama, “Integral holography,” Proc. SPIE 2406, 226-234 (1995).

H. Yoshikawa and T. Yamaguchi, “Fast hologram calculation for holographic video display,” Proc. SPIE 6027, 561-566 (2006).

P. St. Hilaire, S. A. Benton, M. Lucente, M. L. Jepsen, J. Kollin, H. Yoshikawa, and J. Underkoffler, “Electronic display system for computational holography,” Proc. SPIE 1212, 174-182 (1990).

K. Taima, H. Ueda, H. Okamoto, T. Kubota, Y. Nakamura, H. Nishida, H. Takahashi, and E. Shimizu, “New approach to the interactive holographic display system,” Proc. SPIE 2176, 23-29 (1994).

J. A. Watlington, M. Lucente, C. J. Sparrell, V. M. Bove, and I. Tamitani, “A hardware architecture for rapid generation of electro-holographic fringe patterns,” Proc. SPIE 2406, 172-183 (1995).

T. Okada, S. Iwata, O. Nishikawa, K. Matsumoto, H. Yoshikawa, K. Sato, and T. Honda, “The fast computation of holograms for the interactive holographic 3D display system,” Proc. SPIE 2577, 33-40 (1995).

M. Lucente, “Optimization of hologram computation for real-time display,” Proc. SPIE 1667, 25-33 (1992).

H. Yoshikawa, “Digital holographic signal processing,” in Proceeding of the TAO First International Symposium on Three Dimensional Image Communication Technologies (Telecommunications Advancement Organization of Japan, 1993), paper S-4-2.

H. Yoshikawa and K. Sasaki, “Image scaling for electro-holographic display,” Proc. SPIE 2176, 12-22 (1994).

M. Lucente, “Diffraction-specific fringe computation for electro-holography,” Ph.D. dissertation (MIT, 1994).

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley-Interscience, 1994).

C. Palmer, Diffraction Grating Handbook (Richardson Grating Laboratory, Newport Corporation, 2002).

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

E. Hecht, Optics (Wesley Longman, 2002).

M. Yamaguchi, H. Hoshino, T. Honda, and N. Ohyama, “Phase-added stereogram: calculation of hologram using computer graphic technique,” Proc. SPIE 1914, 25-33 (1993).

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Figures (14)

Fig. 1
Fig. 1

Model to calculate the 2D Fresnel hologram.

Fig. 2
Fig. 2

Model to calculate the 2D holographic stereogram.

Fig. 3
Fig. 3

Unitary point and its inverse Fourier transformed image: (a)–(d) unitary points, whose coordinates are from (−1, −1) to (−4, −4); (e)–(h) inverse Fourier transformed image.

Fig. 4
Fig. 4

Stereographic approximation of wavefront reconstruction for digital holograms: (a) Fresnel hologram, (b) conventional holographic stereogram, (c) PAS.

Fig. 5
Fig. 5

Thin transparent plate with periodically varying thickness serves as a diffraction grating.

Fig. 6
Fig. 6

Relation between various numbers of pixels in a segment and the suitable distance between the object and the hologram plane ( Δ p = 10.4 μ m , λ = 633 n m ) .

Fig. 7
Fig. 7

Fresnel hologram and PAS of various segmentation sizes, and its image transformed by the FFT: (a) Fresnel hologram, (b) 16 × 16 PAS, (c) 64 × 64 PAS, (d) 128 × 128 PAS, (e) transformed image of Fresnel hologram, (f) transformed image of 16 × 16 PAS, (g) transformed image of 64 × 64 PAS, (h) transformed image of 128 × 128 PAS.

Fig. 8
Fig. 8

Discrete spatial frequency domain.

Fig. 9
Fig. 9

Relation between macroblock size and segmentation size: (a) conventional method, (b) proposed method.

Fig. 10
Fig. 10

Relation between quantization step and number of pixels in a segment.

Fig. 11
Fig. 11

Schematic of the holographic display system for laser light illumination.

Fig. 12
Fig. 12

Comparison of calculation times between various hologram calculation methods.

Fig. 13
Fig. 13

(a)–(d) Various holograms reconstructed by simulation software and (e)–(h) various holograms displayed by the optical holographic display system: (a) Fresnel hologram, (b) the PAS without the FFT, (c) the PAS with the FFT, (d) the APAS with the FFT, (e) Fresnel hologram, (f) the PAS without the FFT, (g) the PAS with the FFT, (h) the APAS with the FFT.

Fig. 14
Fig. 14

Various holograms displayed bythe optical holographic display system: (a) Fresnel, (b) APAS (16, 64), (c) APAS (16, 128), (d) APAS (32, 128).

Tables (3)

Tables Icon

Table 1 Comparison between the Fresnel Hologram and the Coherent Stereograms

Tables Icon

Table 2 Mean Square Error of Images of Various PASs Reconstructed by Fresnel Diffraction a

Tables Icon

Table 3 Characteristics of the Optical Holographic Display System

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

O ( ξ , η ) = p = 1 N a p r p exp [ j ( k r p + φ p ) ] ,
r p = ( ξ x p ) 2 + ( η y p ) 2 + z 2 .
R ( ξ , η ) = a R exp [ j k ( ξ sin θ ξ + η sin θ η ) ] ,
I F H ( ξ , η ) = | O + R | 2 = | O | 2 + | R | 2 + 2 R e { O R * } ,
I F H b ( ξ , η ) = 2 p = 1 N a p r p cos ( k r p + k ξ sin θ ξ + k η sin θ η + φ p ) .
I H S ( ξ , η ) = p = 1 N a p r p cos { 2 π [ ( ξ ξ c ) f p ξ c + ( η η c ) f p η c ] } ,
f p ξ c = sin θ p ξ c sin θ ξ r e f λ ,
f p η c = sin θ p η c sin θ η r e f λ ,
I P A S ( ξ , η ) = p = 1 N a p r p cos { 2 π [ ( ξ ξ c ) f p ξ c + ( η η c ) f p η c ] + k r p } .
f B S F = 1 T B P = 1 Δ S = 1 Δ p N S ,
ξ i n t = i n t e g e r ( f S F _ ξ f B S F _ ξ ) ,
η i n t = i n t e g e r ( f S F _ η f B S F _ η ) ,
I P A S _ F F T ( ξ , η ) = p = 1 N a p r p cos { 2 π [ ( ξ ξ c ) f p ξ c _ i n t + ( η η c ) f p η c _ i n t ] + k r p } ,
f p ξ c _ i n t = ξ int f B S F _ ξ ,
f p η c _ i n t = η int f B S F _ η .
sin θ q sin θ i = q ( λ / Λ ) ,
A 2 W r e c t ( t 2 W ) A s i n c ( 2 W f ) ,
Δ f 1 2 W = 1 Δ S ,
Δ f = f 1 f 2 = sin θ q 1 sin θ i λ sin θ q 2 sin θ i λ = sin θ q 1 sin θ q 2 λ ,
sin θ = λ / Λ = λ f ,
tan θ = y / D ,
sin θ tan θ θ y / D .
Δ f = sin θ q 1 sin θ q 2 λ y 1 y 2 D λ = Δ S D λ .
D ( Δ S ) 2 λ = ( N S Δ p ) 2 λ ,
1 2 Δ S < Δ f q _ e r r o r 1 2 Δ S .

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