Abstract

Mechanical vibrations are often the principal cause of image degradation. Low temporal-frequency mechanical vibrations involve random image degradation that depends on the instant of exposure. Exact restoration requires the calculation of a specific filter unique to each vibrated image. To calculate the restoration filter for each image, one needs the specific optical transfer function unique to the motion in the image. Therefore the instant of exposure and the motion function have to be measured or estimated by some other means. We develop a restoration filter for individual images blurred randomly by low-frequency mechanical vibrations. The filter is independent of the instant of exposure. The filter is designed to give its best performance averaged over a complete ensemble of vibrated images. Although when applying the new filter to any vibrated image the restoration achieved is slightly poorer than that achieved with an exact filter unique to the specific motion function, the new filter has the advantage of simplicity.

© 1998 Optical Society of America

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References

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  1. N. S. Kopeika, A System Engineering Approach to Imaging, Vol. PM38 of SPIE Press Monographs and Handbooks (SPIE, Bellingham, Wash., 1998), Chap. 14, pp. 411–438.
  2. O. Hadar, Z. Adar, A. Cotter, N. S. Kopeika, “Restoration of images degraded by extreme mechanical vibrations,” Opt. Laser Technol. 29, 161–177 (1997).
    [CrossRef]
  3. D. Wulich, N. S. Kopeika, “Image resolution limits resulting from mechanical vibration,” Opt. Eng. 26, 529–533 (1987).
    [CrossRef]
  4. O. Hadar, I. Dror, N. S. Kopeika, “Image resolution limits resulting from mechanical vibration. Part IV: real time numerical calculation of optical transfer functions and experimental verification,” Opt. Eng. 33, 566–577 (1994).
    [CrossRef]
  5. I. M. Sezan, L. Lagendijk, Motion Analysis and Image Sequence Processing (Kluwer, Boston, Mass., 1993), Chap. 6, pp. 152–165.
  6. O. Hadar, M. Robbins, Y. Novogrozky, D. Kaplan, “Image motion restoration from a sequence of images,” Opt. Eng. 35, 2898–2904 (1996).
    [CrossRef]
  7. O. Hadar, S. R. Rotman, N. S. Kopeika, “Thermal imaging target acquisition probabilities in the presence of vibration,” Infrared Phys. Technol. 36, 691–702 (1995).
    [CrossRef]
  8. O. Hadar, N. S. Kopeika, S. R. Rotman, “Target acquisition modeling of forward motion considerations for airborne reconnaissance over hostile territory,” Opt. Eng. 33, 3106–3117 (1994).
    [CrossRef]
  9. O. Hadar, D. Sadot, N. S. Kopeika, “Contrast-limited target acquisition: atmospheric and motion effects,” in Target and Backgrounds: Characterization and Representation, W. R. Watkins, D. Clement, eds., Proc. SPIE2469, 530–567 (1995).
  10. O. Hadar, A. Mandelblatt, R. Sabath, N. S. Kopeika, S. R. Rotman, “Influence of sensor motion in infrared target acquisition,” Infrared Phys. Technol. 38(6), 373–382 (1997).
    [CrossRef]
  11. L. Guan, R. K. Ward, “Restoration of randomly blurred images by the Wiener filter,” IEEE Trans. Acoust. Speech Signal Process. 37, 589–592 (1989).
    [CrossRef]
  12. K. R. Castelman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1979), Chap. 14, p. 282.
  13. A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989), pp. 290–291.
  14. R. K. Ward, B. E. A. Saleh, “Restoration of images distorted by systems of random impulse response,” J. Opt. Soc. Am. A 2, 1254–1259 (1985).
    [CrossRef]
  15. A. Stern, N. S. Kopeika, “Analytical method to calculate the optical transfer function for image motion and vibration using moments,” J. Opt. Soc. Am. A 14, 388–396 (1997).
    [CrossRef]
  16. K. C. Tan, H. Lim, B. T. G. Tan, “Restoration of real-world motion blurred images,” CVGIP: Graphic. Models Image Process. 53, 291–299 (1991).
    [CrossRef]

1997

O. Hadar, Z. Adar, A. Cotter, N. S. Kopeika, “Restoration of images degraded by extreme mechanical vibrations,” Opt. Laser Technol. 29, 161–177 (1997).
[CrossRef]

O. Hadar, A. Mandelblatt, R. Sabath, N. S. Kopeika, S. R. Rotman, “Influence of sensor motion in infrared target acquisition,” Infrared Phys. Technol. 38(6), 373–382 (1997).
[CrossRef]

A. Stern, N. S. Kopeika, “Analytical method to calculate the optical transfer function for image motion and vibration using moments,” J. Opt. Soc. Am. A 14, 388–396 (1997).
[CrossRef]

1996

O. Hadar, M. Robbins, Y. Novogrozky, D. Kaplan, “Image motion restoration from a sequence of images,” Opt. Eng. 35, 2898–2904 (1996).
[CrossRef]

1995

O. Hadar, S. R. Rotman, N. S. Kopeika, “Thermal imaging target acquisition probabilities in the presence of vibration,” Infrared Phys. Technol. 36, 691–702 (1995).
[CrossRef]

1994

O. Hadar, N. S. Kopeika, S. R. Rotman, “Target acquisition modeling of forward motion considerations for airborne reconnaissance over hostile territory,” Opt. Eng. 33, 3106–3117 (1994).
[CrossRef]

O. Hadar, I. Dror, N. S. Kopeika, “Image resolution limits resulting from mechanical vibration. Part IV: real time numerical calculation of optical transfer functions and experimental verification,” Opt. Eng. 33, 566–577 (1994).
[CrossRef]

1991

K. C. Tan, H. Lim, B. T. G. Tan, “Restoration of real-world motion blurred images,” CVGIP: Graphic. Models Image Process. 53, 291–299 (1991).
[CrossRef]

1989

L. Guan, R. K. Ward, “Restoration of randomly blurred images by the Wiener filter,” IEEE Trans. Acoust. Speech Signal Process. 37, 589–592 (1989).
[CrossRef]

1987

D. Wulich, N. S. Kopeika, “Image resolution limits resulting from mechanical vibration,” Opt. Eng. 26, 529–533 (1987).
[CrossRef]

1985

Adar, Z.

O. Hadar, Z. Adar, A. Cotter, N. S. Kopeika, “Restoration of images degraded by extreme mechanical vibrations,” Opt. Laser Technol. 29, 161–177 (1997).
[CrossRef]

Castelman, K. R.

K. R. Castelman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1979), Chap. 14, p. 282.

Cotter, A.

O. Hadar, Z. Adar, A. Cotter, N. S. Kopeika, “Restoration of images degraded by extreme mechanical vibrations,” Opt. Laser Technol. 29, 161–177 (1997).
[CrossRef]

Dror, I.

O. Hadar, I. Dror, N. S. Kopeika, “Image resolution limits resulting from mechanical vibration. Part IV: real time numerical calculation of optical transfer functions and experimental verification,” Opt. Eng. 33, 566–577 (1994).
[CrossRef]

Guan, L.

L. Guan, R. K. Ward, “Restoration of randomly blurred images by the Wiener filter,” IEEE Trans. Acoust. Speech Signal Process. 37, 589–592 (1989).
[CrossRef]

Hadar, O.

O. Hadar, A. Mandelblatt, R. Sabath, N. S. Kopeika, S. R. Rotman, “Influence of sensor motion in infrared target acquisition,” Infrared Phys. Technol. 38(6), 373–382 (1997).
[CrossRef]

O. Hadar, Z. Adar, A. Cotter, N. S. Kopeika, “Restoration of images degraded by extreme mechanical vibrations,” Opt. Laser Technol. 29, 161–177 (1997).
[CrossRef]

O. Hadar, M. Robbins, Y. Novogrozky, D. Kaplan, “Image motion restoration from a sequence of images,” Opt. Eng. 35, 2898–2904 (1996).
[CrossRef]

O. Hadar, S. R. Rotman, N. S. Kopeika, “Thermal imaging target acquisition probabilities in the presence of vibration,” Infrared Phys. Technol. 36, 691–702 (1995).
[CrossRef]

O. Hadar, I. Dror, N. S. Kopeika, “Image resolution limits resulting from mechanical vibration. Part IV: real time numerical calculation of optical transfer functions and experimental verification,” Opt. Eng. 33, 566–577 (1994).
[CrossRef]

O. Hadar, N. S. Kopeika, S. R. Rotman, “Target acquisition modeling of forward motion considerations for airborne reconnaissance over hostile territory,” Opt. Eng. 33, 3106–3117 (1994).
[CrossRef]

O. Hadar, D. Sadot, N. S. Kopeika, “Contrast-limited target acquisition: atmospheric and motion effects,” in Target and Backgrounds: Characterization and Representation, W. R. Watkins, D. Clement, eds., Proc. SPIE2469, 530–567 (1995).

Jain, A. K.

A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989), pp. 290–291.

Kaplan, D.

O. Hadar, M. Robbins, Y. Novogrozky, D. Kaplan, “Image motion restoration from a sequence of images,” Opt. Eng. 35, 2898–2904 (1996).
[CrossRef]

Kopeika, N. S.

O. Hadar, A. Mandelblatt, R. Sabath, N. S. Kopeika, S. R. Rotman, “Influence of sensor motion in infrared target acquisition,” Infrared Phys. Technol. 38(6), 373–382 (1997).
[CrossRef]

O. Hadar, Z. Adar, A. Cotter, N. S. Kopeika, “Restoration of images degraded by extreme mechanical vibrations,” Opt. Laser Technol. 29, 161–177 (1997).
[CrossRef]

A. Stern, N. S. Kopeika, “Analytical method to calculate the optical transfer function for image motion and vibration using moments,” J. Opt. Soc. Am. A 14, 388–396 (1997).
[CrossRef]

O. Hadar, S. R. Rotman, N. S. Kopeika, “Thermal imaging target acquisition probabilities in the presence of vibration,” Infrared Phys. Technol. 36, 691–702 (1995).
[CrossRef]

O. Hadar, N. S. Kopeika, S. R. Rotman, “Target acquisition modeling of forward motion considerations for airborne reconnaissance over hostile territory,” Opt. Eng. 33, 3106–3117 (1994).
[CrossRef]

O. Hadar, I. Dror, N. S. Kopeika, “Image resolution limits resulting from mechanical vibration. Part IV: real time numerical calculation of optical transfer functions and experimental verification,” Opt. Eng. 33, 566–577 (1994).
[CrossRef]

D. Wulich, N. S. Kopeika, “Image resolution limits resulting from mechanical vibration,” Opt. Eng. 26, 529–533 (1987).
[CrossRef]

N. S. Kopeika, A System Engineering Approach to Imaging, Vol. PM38 of SPIE Press Monographs and Handbooks (SPIE, Bellingham, Wash., 1998), Chap. 14, pp. 411–438.

O. Hadar, D. Sadot, N. S. Kopeika, “Contrast-limited target acquisition: atmospheric and motion effects,” in Target and Backgrounds: Characterization and Representation, W. R. Watkins, D. Clement, eds., Proc. SPIE2469, 530–567 (1995).

Lagendijk, L.

I. M. Sezan, L. Lagendijk, Motion Analysis and Image Sequence Processing (Kluwer, Boston, Mass., 1993), Chap. 6, pp. 152–165.

Lim, H.

K. C. Tan, H. Lim, B. T. G. Tan, “Restoration of real-world motion blurred images,” CVGIP: Graphic. Models Image Process. 53, 291–299 (1991).
[CrossRef]

Mandelblatt, A.

O. Hadar, A. Mandelblatt, R. Sabath, N. S. Kopeika, S. R. Rotman, “Influence of sensor motion in infrared target acquisition,” Infrared Phys. Technol. 38(6), 373–382 (1997).
[CrossRef]

Novogrozky, Y.

O. Hadar, M. Robbins, Y. Novogrozky, D. Kaplan, “Image motion restoration from a sequence of images,” Opt. Eng. 35, 2898–2904 (1996).
[CrossRef]

Robbins, M.

O. Hadar, M. Robbins, Y. Novogrozky, D. Kaplan, “Image motion restoration from a sequence of images,” Opt. Eng. 35, 2898–2904 (1996).
[CrossRef]

Rotman, S. R.

O. Hadar, A. Mandelblatt, R. Sabath, N. S. Kopeika, S. R. Rotman, “Influence of sensor motion in infrared target acquisition,” Infrared Phys. Technol. 38(6), 373–382 (1997).
[CrossRef]

O. Hadar, S. R. Rotman, N. S. Kopeika, “Thermal imaging target acquisition probabilities in the presence of vibration,” Infrared Phys. Technol. 36, 691–702 (1995).
[CrossRef]

O. Hadar, N. S. Kopeika, S. R. Rotman, “Target acquisition modeling of forward motion considerations for airborne reconnaissance over hostile territory,” Opt. Eng. 33, 3106–3117 (1994).
[CrossRef]

Sabath, R.

O. Hadar, A. Mandelblatt, R. Sabath, N. S. Kopeika, S. R. Rotman, “Influence of sensor motion in infrared target acquisition,” Infrared Phys. Technol. 38(6), 373–382 (1997).
[CrossRef]

Sadot, D.

O. Hadar, D. Sadot, N. S. Kopeika, “Contrast-limited target acquisition: atmospheric and motion effects,” in Target and Backgrounds: Characterization and Representation, W. R. Watkins, D. Clement, eds., Proc. SPIE2469, 530–567 (1995).

Saleh, B. E. A.

Sezan, I. M.

I. M. Sezan, L. Lagendijk, Motion Analysis and Image Sequence Processing (Kluwer, Boston, Mass., 1993), Chap. 6, pp. 152–165.

Stern, A.

Tan, B. T. G.

K. C. Tan, H. Lim, B. T. G. Tan, “Restoration of real-world motion blurred images,” CVGIP: Graphic. Models Image Process. 53, 291–299 (1991).
[CrossRef]

Tan, K. C.

K. C. Tan, H. Lim, B. T. G. Tan, “Restoration of real-world motion blurred images,” CVGIP: Graphic. Models Image Process. 53, 291–299 (1991).
[CrossRef]

Ward, R. K.

L. Guan, R. K. Ward, “Restoration of randomly blurred images by the Wiener filter,” IEEE Trans. Acoust. Speech Signal Process. 37, 589–592 (1989).
[CrossRef]

R. K. Ward, B. E. A. Saleh, “Restoration of images distorted by systems of random impulse response,” J. Opt. Soc. Am. A 2, 1254–1259 (1985).
[CrossRef]

Wulich, D.

D. Wulich, N. S. Kopeika, “Image resolution limits resulting from mechanical vibration,” Opt. Eng. 26, 529–533 (1987).
[CrossRef]

CVGIP: Graphic. Models Image Process.

K. C. Tan, H. Lim, B. T. G. Tan, “Restoration of real-world motion blurred images,” CVGIP: Graphic. Models Image Process. 53, 291–299 (1991).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process.

L. Guan, R. K. Ward, “Restoration of randomly blurred images by the Wiener filter,” IEEE Trans. Acoust. Speech Signal Process. 37, 589–592 (1989).
[CrossRef]

Infrared Phys. Technol.

O. Hadar, S. R. Rotman, N. S. Kopeika, “Thermal imaging target acquisition probabilities in the presence of vibration,” Infrared Phys. Technol. 36, 691–702 (1995).
[CrossRef]

O. Hadar, A. Mandelblatt, R. Sabath, N. S. Kopeika, S. R. Rotman, “Influence of sensor motion in infrared target acquisition,” Infrared Phys. Technol. 38(6), 373–382 (1997).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Eng.

O. Hadar, M. Robbins, Y. Novogrozky, D. Kaplan, “Image motion restoration from a sequence of images,” Opt. Eng. 35, 2898–2904 (1996).
[CrossRef]

O. Hadar, N. S. Kopeika, S. R. Rotman, “Target acquisition modeling of forward motion considerations for airborne reconnaissance over hostile territory,” Opt. Eng. 33, 3106–3117 (1994).
[CrossRef]

D. Wulich, N. S. Kopeika, “Image resolution limits resulting from mechanical vibration,” Opt. Eng. 26, 529–533 (1987).
[CrossRef]

O. Hadar, I. Dror, N. S. Kopeika, “Image resolution limits resulting from mechanical vibration. Part IV: real time numerical calculation of optical transfer functions and experimental verification,” Opt. Eng. 33, 566–577 (1994).
[CrossRef]

Opt. Laser Technol.

O. Hadar, Z. Adar, A. Cotter, N. S. Kopeika, “Restoration of images degraded by extreme mechanical vibrations,” Opt. Laser Technol. 29, 161–177 (1997).
[CrossRef]

Other

N. S. Kopeika, A System Engineering Approach to Imaging, Vol. PM38 of SPIE Press Monographs and Handbooks (SPIE, Bellingham, Wash., 1998), Chap. 14, pp. 411–438.

I. M. Sezan, L. Lagendijk, Motion Analysis and Image Sequence Processing (Kluwer, Boston, Mass., 1993), Chap. 6, pp. 152–165.

O. Hadar, D. Sadot, N. S. Kopeika, “Contrast-limited target acquisition: atmospheric and motion effects,” in Target and Backgrounds: Characterization and Representation, W. R. Watkins, D. Clement, eds., Proc. SPIE2469, 530–567 (1995).

K. R. Castelman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1979), Chap. 14, p. 282.

A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989), pp. 290–291.

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Figures (7)

Fig. 1
Fig. 1

Motion functions during exposure of a low-frequency vibrating-image platform for constant vibration parameters, i.e., constant amplitude D and vibration period T 0, and a given exposure time t e . The PSF depends1,4 on the instant of the exposure t x , which is random.

Fig. 2
Fig. 2

(a) Image blurred by low-frequency vertical vibration (D = 30 pixels, t e = 0.1T 0, and t x = 0T 0). (b) Image (a) after passage through the modified Wiener filter [Eq. (11)].

Fig. 3
Fig. 3

(a) Original image (the vertical bar). (b) Images taken at different instants of exposure (t x 1 t x 5 ) exhibit different blurs and different shifts. (c) The centralized (nonshifted) blurred images.

Fig. 4
Fig. 4

Amplitude frequency response of the general restoration filter [Eq. (21)] (upper solid curve) and of the vibration Wiener filter [Eq. (18)] (lower solid curve), with the mean centralized OTF (dashed curve) and the blur-noise power spectrum (dashed–dotted curve) for a vibration amplitude of D = 25 pixels and an exposure time of t e = 0.15T 0.

Fig. 5
Fig. 5

Images blurred by vertical vibration with an amplitude of D = 25 pixels, an exposure time of t e = 0.15T 0, and instants of exposure t x of (a1) 0, (b1) 0.12T 0, (c1) 0.24T 0, and (d1) 0.38T 0. (a2)–(d2) The restored images of (a1)–(d1), respectively.

Fig. 6
Fig. 6

Images blurred by vertical vibration with an amplitude of D = 30 pixels, an exposure time of t e = 0.1T 0, and instants of exposure t x of (a1) 0 and (b1) 0.12T 0. (a2) Restored image of (a1). (b2) Restored image of (b1).

Fig. 7
Fig. 7

(a) Image blurred by vertical vibration with an amplitude of D = 30 pixels, an exposure time of t e = 0.1T 0, and an instant of exposure of t x = 0.26T 0. Image of (a) restored (b) by use of the general restoration filter, (c) by use of the Wiener filter and a value of γ = 0.01, and (d) by use of the Wiener filter and a value of γ = 0.05. The ghosting effect appears in the Wiener filter restorations but not in the image restored with the general restoration filter.

Equations (28)

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

x t = D   sin ω 0 t ,
g ξ 1 ,   ξ 2 = h ξ 1 ,   ξ 2   * *   f ξ 1 ,   ξ 2 + n 2 ξ 1 ,   ξ 2 ,
h ξ 1 ,   ξ 2 = h ¯ ξ 1 ,   ξ 2 + n 1 ξ 1 ,   ξ 2 ,
g ξ 1 ,   ξ 2 = h ¯ ξ 1 ,   ξ 2   * *   f ξ 1 ,   ξ 2 + n ξ 1 ,   ξ 2 ,
n ξ 1 ,   ξ 2 = n 1 ξ 1 ,   ξ 2   * *   f ξ 1 ,   ξ 2 + n 2 ξ 1 ,   ξ 2 .
G ω 1 ,   ω 2 = H ¯ ω 1 ,   ω 2 F ω 1 ,   ω 2 + N ω 1 ,   ω 2 ,
N ω 1 ,   ω 2 = N 1 ω 1 ,   ω 2 F ω 1 ,   ω 2 + N 2 ω 1 ,   ω 2 ,
min ( E f ξ 1 ,   ξ 2 - f ˆ ξ 1 ,   ξ 2 ) ,
M ω 1 ,   ω 2 = H ¯ * ω 1 ,   ω 2 | H ¯ ω 1 ,   ω 2 | 2 + S n 1 ω 1 ,   ω 2 + S n 2 ω 1 ,   ω 2 | F ω 1 ,   ω 2 | 2 ,
min ( E f ξ 1 ,   ξ 2 - f ˆ c ξ 1 ,   ξ 2 2 ) ,
g c ξ 1 ,   ξ 2 = g ξ 1 - μ 1 ,   ξ 2 - μ 2 ,
μ 1 = - -   ξ 1 h ξ 1 ,   ξ 2 ;   t x d ξ 1 d ξ 2 , μ 2 = - -   ξ 2 h ξ 1 ,   ξ 2 ;   t x d ξ 1 d ξ 2 ,
g c ξ 1 ,   ξ 2 ;   t x = h c ξ 1 ,   ξ 2 ;   t x   * *   f ξ 1 ,   ξ 2 + n 2 ξ 1 - μ 1 ,   ξ 2 - μ 2 ;   t x ,
h c ξ 1 ,   ξ 2 ;   t x = h ξ 1 - μ 1 ,   ξ 2 - μ ;   t x .
h c ξ 1 ,   ξ 2 ;   t x = h ¯ c ξ 1 ,   ξ 2 ;   t x + n c 1 ξ 1 ,   ξ 2 ,
h ¯ c ξ 1 ,   ξ 2 ;   t x = 1 T 0 0 T 0   h c ξ 1 - μ 1 t x ,   ξ 2 - μ 2 t x d t x = E h c .
H ¯ c ω 1 ,   ω 2 = E H ω 1 ,   ω 2 exp j ω 1 μ 1 + ω 2 μ 2 .
M 1 ω 1 ,   ω 2 = H ¯ c * ω 1 ,   ω 2 | H ¯ c ω 1 ,   ω 2 | 2 + S c n 1 ω 1 ,   ω 2 + S n 2 ω 1 ,   ω 2 | F ω 1 ,   ω 2 | 2 ,
M ω 1 ,   ω 2 = M 2 ω 1 ,   ω 2 α M 1 ω 1 ,   ω 2 1 - α ,
M 1 ω 1 ,   ω 2 = H ¯ c * ω 1 ,   ω 2 | H ¯ c ω 1 ,   ω 2 | 2 .
S c n 1 ω 1 ,   ω 2     S n 2 ω 1 ,   ω 2 | F ω 1 ,   ω 2 | 2 ,
M ω 1 ,   ω 2 = H ¯ c * ω 1 ,   ω 2 | H ¯ c ω 1 ,   ω 2 | 2 α H ¯ c * ω 1 ,   ω 2 | H ¯ c ω 1 ,   ω 2 | 2 + S c n 1 ω 1 ,   ω 2 1 - α .
H ω = n = 0 m n n ! - j ω n , m n t x ;   D ,   ω 0 ,   t e = 1 t e t x t x + t e D   sin   ω 0 t n d t ,
S c n 1 ω 1 ,   ω 2 = | H c ω 1 ,   ω 2 - H ¯ c ω 1 ,   ω 2 | 2 .
M 3 ω 1 ,   ω 2 = H ¯ c * ω 1 ,   ω 2 | H ¯ c ω 1 ,   ω 2 | 2 + γ .
M ω 1 ,   ω 2 = H ¯ c * ω 1 ,   ω 2 | H ¯ c ω 1 ,   ω 2 | 2 α H ¯ c * ω 1 ,   ω 2 | H ¯ c ω 1 ,   ω 2 | 2 + S c n 1 ω 1 ,   ω 2 + S n 2 ω 1 ,   ω 2 | F ω 1 ,   ω 2 | 2 1 - α = H * ω 1 ,   ω 2 | H ω 1 ,   ω 2 | 2 α H * ω 1 ,   ω 2 | H ω 1 ,   ω 2 | 2 + S n 2 ω 1 ,   ω 2 | F ω 1 ,   ω 2 | 2 1 - α ,
M ω 1 ,   ω 2 = H * ω 1 ,   ω 2 α | H ω 1 ,   ω 2 | 2 α H * ω 1 ,   ω 2 1 - α | H ω 1 ,   ω 2 | 2 1 - α + 1 - α S n 2 ω 1 ,   ω 2 | F ω 1 ,   ω 2 | 2 | H ω 1 ,   ω 2 | 2 - α + H * ω 1 ,   ω 2 | H ω 1 ,   ω 2 | 2 + 1 - α S n 2 ω 1 ,   ω 2 | F ω 1 ,   ω 2 | 2 .
M ω 1 ,   ω 2 = H * ω 1 ,   ω 2 | H ω 1 ,   ω 2 | 2 + γ NSR ω 1 ,   ω 2 ,

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