Abstract

The optical transfer function (OTF) for image degradation by nonharmonic vibrations is analyzed. Previous analyses of vibration image degradation were restricted to sinusoidal vibrations only or to general vibrations determined by the displacement function. However, most of the real-life vibrations are nonharmonic and in general are determined only by the power density spectrum envelope. We present a method to calculate the OTF or the expected OTF for any high-frequency vibration determined by its power spectral density. The calculation method is practical for analysis and design of imaging systems that are subject to real-life vibrations and for use in restoration of vibrated images.

© 1999 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. N. S. Kopeika, A System Engineering Approach to Imaging (SPIE, Bellingham, Wash., 1998), Chap. 14, pp. 411–440.
  2. G. C. Holst, Electro-Optical Imaging System Performance (SPIE, Bellingham, Wash., 1995), Chap. 4, pp. 64–75; Chap. 6, pp. 110–118.
  3. Ref. 1, Chap. 18, pp. 517–521.
  4. M. Sonka, V. Hlavac, R. Boyle, Image Processing, Analysis and Machine Vision (Chapman & Hall, London, 1993), Chaps. 7 and 14.
  5. O. Hadar, Z. Adar, A. Cotter, N. S. Kopeika, “Restoration of images degraded by extreme mechanical vibrations,” Opt. Laser Technol. 29, 171–177 (1997).
    [CrossRef]
  6. A. Stern, N. S. Kopeika, “General restoration filter for vibrated image restoration,” Appl. Opt. 37, 7596–7603 (1998).
    [CrossRef]
  7. O. Hadar, I. Dror, N. S. Kopeika, “Image resolution limits resulting from mechanical vibration. IV. Real time numerical calculation of optical transfer functions and experimental verification,” Opt. Eng. 33, 566–578 (1994).
    [CrossRef]
  8. A. Stern, N. S. Kopeika, “Analytical method to calculate optical transfer function for image motion and vibration using moments,” J. Opt. Soc. Am. A 14, 388–396 (1997).
    [CrossRef]
  9. N. Jensen, Optical and Photographic Reconnaissance System (Wiley, New York, 1968), pp. 116–124.
  10. R. E. D. Bishop, D. C. Johnson, The Mechanics of Vibration (Cambridge U. Press, London, 1960), Chap. 11, pp. 543–564.
  11. J. T. Broch, Application of B&K Equipment to Mechanical Vibration and Shock Measurements (Brüel and Kjaer, Naerum, Denmark, 1972), Chap. 2, pp. 14–31.
  12. L. Levi, Applied Optics (Wiley, New York, 1980), Chap. 19, p. 725.
  13. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1987), p. 272.
  14. M. R. Spiegel, Mathematical Handbook of Formulas and Tables (McGraw-Hill, New York, 1968), p. 136, Eq. (24.5).
  15. Ref. 14, p. 17, Eq. (5.70).
  16. Ref. 14, p. 17, Eq. (5.72).

1998

1997

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

O. Hadar, Z. Adar, A. Cotter, N. S. Kopeika, “Restoration of images degraded by extreme mechanical vibrations,” Opt. Laser Technol. 29, 171–177 (1997).
[CrossRef]

1994

O. Hadar, I. Dror, N. S. Kopeika, “Image resolution limits resulting from mechanical vibration. IV. Real time numerical calculation of optical transfer functions and experimental verification,” Opt. Eng. 33, 566–578 (1994).
[CrossRef]

Adar, Z.

O. Hadar, Z. Adar, A. Cotter, N. S. Kopeika, “Restoration of images degraded by extreme mechanical vibrations,” Opt. Laser Technol. 29, 171–177 (1997).
[CrossRef]

Bishop, R. E. D.

R. E. D. Bishop, D. C. Johnson, The Mechanics of Vibration (Cambridge U. Press, London, 1960), Chap. 11, pp. 543–564.

Boyle, R.

M. Sonka, V. Hlavac, R. Boyle, Image Processing, Analysis and Machine Vision (Chapman & Hall, London, 1993), Chaps. 7 and 14.

Broch, J. T.

J. T. Broch, Application of B&K Equipment to Mechanical Vibration and Shock Measurements (Brüel and Kjaer, Naerum, Denmark, 1972), Chap. 2, pp. 14–31.

Cotter, A.

O. Hadar, Z. Adar, A. Cotter, N. S. Kopeika, “Restoration of images degraded by extreme mechanical vibrations,” Opt. Laser Technol. 29, 171–177 (1997).
[CrossRef]

Dror, I.

O. Hadar, I. Dror, N. S. Kopeika, “Image resolution limits resulting from mechanical vibration. IV. Real time numerical calculation of optical transfer functions and experimental verification,” Opt. Eng. 33, 566–578 (1994).
[CrossRef]

Hadar, O.

O. Hadar, Z. Adar, A. Cotter, N. S. Kopeika, “Restoration of images degraded by extreme mechanical vibrations,” Opt. Laser Technol. 29, 171–177 (1997).
[CrossRef]

O. Hadar, I. Dror, N. S. Kopeika, “Image resolution limits resulting from mechanical vibration. IV. Real time numerical calculation of optical transfer functions and experimental verification,” Opt. Eng. 33, 566–578 (1994).
[CrossRef]

Hlavac, V.

M. Sonka, V. Hlavac, R. Boyle, Image Processing, Analysis and Machine Vision (Chapman & Hall, London, 1993), Chaps. 7 and 14.

Holst, G. C.

G. C. Holst, Electro-Optical Imaging System Performance (SPIE, Bellingham, Wash., 1995), Chap. 4, pp. 64–75; Chap. 6, pp. 110–118.

Jensen, N.

N. Jensen, Optical and Photographic Reconnaissance System (Wiley, New York, 1968), pp. 116–124.

Johnson, D. C.

R. E. D. Bishop, D. C. Johnson, The Mechanics of Vibration (Cambridge U. Press, London, 1960), Chap. 11, pp. 543–564.

Kopeika, N. S.

A. Stern, N. S. Kopeika, “General restoration filter for vibrated image restoration,” Appl. Opt. 37, 7596–7603 (1998).
[CrossRef]

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

O. Hadar, Z. Adar, A. Cotter, N. S. Kopeika, “Restoration of images degraded by extreme mechanical vibrations,” Opt. Laser Technol. 29, 171–177 (1997).
[CrossRef]

O. Hadar, I. Dror, N. S. Kopeika, “Image resolution limits resulting from mechanical vibration. IV. Real time numerical calculation of optical transfer functions and experimental verification,” Opt. Eng. 33, 566–578 (1994).
[CrossRef]

N. S. Kopeika, A System Engineering Approach to Imaging (SPIE, Bellingham, Wash., 1998), Chap. 14, pp. 411–440.

Levi, L.

L. Levi, Applied Optics (Wiley, New York, 1980), Chap. 19, p. 725.

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1987), p. 272.

Sonka, M.

M. Sonka, V. Hlavac, R. Boyle, Image Processing, Analysis and Machine Vision (Chapman & Hall, London, 1993), Chaps. 7 and 14.

Spiegel, M. R.

M. R. Spiegel, Mathematical Handbook of Formulas and Tables (McGraw-Hill, New York, 1968), p. 136, Eq. (24.5).

Stern, A.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Eng.

O. Hadar, I. Dror, N. S. Kopeika, “Image resolution limits resulting from mechanical vibration. IV. Real time numerical calculation of optical transfer functions and experimental verification,” Opt. Eng. 33, 566–578 (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, 171–177 (1997).
[CrossRef]

Other

N. Jensen, Optical and Photographic Reconnaissance System (Wiley, New York, 1968), pp. 116–124.

R. E. D. Bishop, D. C. Johnson, The Mechanics of Vibration (Cambridge U. Press, London, 1960), Chap. 11, pp. 543–564.

J. T. Broch, Application of B&K Equipment to Mechanical Vibration and Shock Measurements (Brüel and Kjaer, Naerum, Denmark, 1972), Chap. 2, pp. 14–31.

L. Levi, Applied Optics (Wiley, New York, 1980), Chap. 19, p. 725.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1987), p. 272.

M. R. Spiegel, Mathematical Handbook of Formulas and Tables (McGraw-Hill, New York, 1968), p. 136, Eq. (24.5).

Ref. 14, p. 17, Eq. (5.70).

Ref. 14, p. 17, Eq. (5.72).

N. S. Kopeika, A System Engineering Approach to Imaging (SPIE, Bellingham, Wash., 1998), Chap. 14, pp. 411–440.

G. C. Holst, Electro-Optical Imaging System Performance (SPIE, Bellingham, Wash., 1995), Chap. 4, pp. 64–75; Chap. 6, pp. 110–118.

Ref. 1, Chap. 18, pp. 517–521.

M. Sonka, V. Hlavac, R. Boyle, Image Processing, Analysis and Machine Vision (Chapman & Hall, London, 1993), Chaps. 7 and 14.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

(a) Single-sided PSD of the vibration given by Eq. (3) with A1=1 mrad, A2=2 mrad, f1=350 Hz, f2=400 Hz, and arbitrary phases ϕ1 and ϕ2. (b) Possible displacement function during the exposure period (te=1/60 s). (c) MTF calculated with Eq. (10) (continuous curve) and calculated numerically from the motion function (dotted curve).

Fig. 2
Fig. 2

(a) Single-sided PSD of vibrations composed of two components at 200 and 400 Hz with angular amplitudes A1=1 mrad and A2=1.5 mrad, respectively. (b) Two possible displacement functions that have the PSD of (a). (c) Examples of MTF’s (dotted curves) of vibration with a PSD as in (a). The MTF’s depend on the exact displacement function. The solid curve is the absolute value of the mean OTF calculated by Eq. (11). (d) Solid curve, the mean OTF calculated by Eq. (11); dashed curve, the mean OTF calculated by averaging 30 numerical OTF’s appropriate to 30 displacement functions with a PSD as in (a).

Fig. 3
Fig. 3

(a) Single-sided PSD of vibrations composed of three components at 300, 400, and 500 Hz with angular amplitudes A1=0.04 mrad, A2=0.05 mrad, and A3=0.03 mrad, respectively. (b) Possible displacement function that has the PSD of (a). (c) Examples of MTF’s (dotted curves) of vibration with a PSD as in (a) and the absolute value of the mean OTF calculated from Eq. (12) (solid curve). (d) Solid line, the mean OTF calculated by Eq. (12); dashed curve, the mean OTF calculated by averaging 30 numerical OTF’s appropriate to 30 displacement function with a PSD as in (a).

Fig. 4
Fig. 4

(a) Continuous PSD approximated by (b) a discrete PSD.

Fig. 5
Fig. 5

(a) Flat PSD approximated by (b) a PSD of N harmonic vibrations.

Fig. 6
Fig. 6

(a) Flat PSD. (b) One possible displacement function with the PSD as in (a). (c) Solid curve, mean OTF calculated by Eq. (19); dotted curves, several OTF’s appropriate to several displacement functions with the PSD as in (a); dashed curve, mean OTF calculated by averaging 30 numerical OTF’s appropriate to 30 displacement functions with a PSD as in (a).

Fig. 7
Fig. 7

(a) Parameter a [in relation (21)] versus S0σ for the PSD’s shown in (b): 16 Gaussian PSD’s (combination of S0 in the range 0.8–1.2 mrad and σ from 0.2 to 0.4 Hz). (c) Parameter a [in relation (21)] versus S0σ for the PSD’s shown in (d): 16 Gaussian PSD’s (combination of S0 in range 0.005–0.02 mrad and σ from 50 to 200 Hz). Parameter a was calculated by fitting a curve of the form of relation (21) to the OTF calculated by Eq. (14). The slope in (a) is -1.128±0.038 and in (c) is -1.137±0.044.

Equations (44)

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

OTFHF=J0(Dω),
OTF(ω)=-fx(x)exp(-jωx)dx,
OTF(ω)=exp(-05σ2ω2),
x(t)=x1(t)+x2(t),
x1(t)=A1 sin(2πf1t+ϕ1),
x2(t)=A2 sin(2πf2t+ϕ2),
f2nf1,n=1, 2, 3,
OTF(ω)=n=0 mnn! (-jω)n,
mn=1te txtx+te[x1(t)+x2(t)]ndt=1te txtx+tex1n(t)+n1x1n-1(t)x2(t)+n2x1n-2(t)x22(t)+ x2n(t)dt.
limte 1te txtx+te[A1 sin(2πf1t+ϕ1]l[A2 sin(2πf2t+ϕ2]kdt=0lorkoddA1l(l)!2l[(l/2)!]2 A2k(k)!2l[(l/2)!]2l andkeven,f1nf2,n=1, 2, 3 .(7a)(7b)
mn=1te txtx+tex1n(t)+n1x1n-1(t)x2(t)+n2x1n-2(t)x22(t)+ x2n(t)dt=A1n2n n![(n/2)!]2+n!(n-2)!2! A1n-22n-2 (n-2)![(n/2-1)!]2 A2222 2![1!]2+n!(n-4)!4! A1n-42n-4 (n-4)![(n/2-2)!]2 A2224n 4![2!]2++A1n2n n![(n/2)!]2=n!k=0n/2 A1n-2kA22k2nn2-k!2[(2k)!]2
n=0, 2, 4, 6,te1f1, 1f2.
OTF(ω)=n=0(-jω)nk=0n/2 A1n-2kA22k2nn2-k!2[(2k)!]2,
OTFx(ω)=J0(A1ω)J0(A2ω)=OTFx1(ω)OTFx2(ω),
f2nf1,n=1, 2, 3 .
E[OTFx(ω)]=J0(A1ω)J0(A2ω)=OTFx1(ω)OTFx2(ω),f1,f2,
E[OTFN(ω)]=i=1NJ0(Aiω),te1/fi,
A2(f )/2=Sx(f )Δf.
E[OTFN(ω)]=limΔf0k=1B/ΔfJ0(ω2ΔfSx(f0+kΔf )),
Sa(f )=(2πf )4Sx(f ),
E[OTFN(ω)]
=limΔf0k=1B/ΔfJ0ω2Δf Sa(f0+kΔf )[2π(f0+kΔf )]4.
E[OTFflat(ω)]=limNk=0NJ0ω2 BN S0=limNJ0ω2 BN S0N,
E[OTFflat(ω)]=limN1-2BS04N ω2N.
E[OTFflat(ω)]=exp-BS02 ω2.
Sx(f )=S0 exp-(f-f0)22σ2.
OTFGPSD(ω)exp(-aω2),
a=1.13S0σ.
limte1te 0te sinl(ω1t+ϕ1)sink(ω2t+ϕ2)dt
=limte1te 0te (-1)(l-1)/22l-1 sin{l[(ω1t+ϕ1)]}-l1sin{(l-2)[(ω1t+ϕ1)]}++(-1)(l-1)/2l(l-1)/2sin(ω1t+ϕ1)A(t)dt,
limte1te 0te sinl(ω1t+ϕ1)sink(ω2t+ϕ2)dt=limte1te 0tell/2 12l+A(t)kk/2 12k+B(t)dt,
limte1te 0te sinl(ω1t+ϕ1)sink(ω2t+ϕ2)dt
=12l ll/2 12k kk/2=121+k l![(l/2)!]2 k![(k/2)!]2,
l, keven,ω1nω2,n=1, 2, 3,
limte1te 0te (-1)(l-1)/22l-1 lssin[(l-s)(ω1t+ϕ1)]kr
  ×sin[(k-r)(ω2t+ϕ2)]dt,
sl,rk=1, 2, 3,
Elimte1te 0te sinl(ω1t+ϕ1)sink(ω2t+ϕ2)dt
=02π02πfϕ1ϕ2(ϕ1, ϕ2)limte1te 0te(-1)(l-1)/22l-1 sin{l[(ω1t+ϕ1)]}-l1sin{(l-2)[(ω1t+ϕ1)]}+sin {k[(ω1t+ϕ1)]}-k1sin{(k-2)[(ω1t+ϕ1)]}+dtdϕ1dϕ2,
Elimte1te 0te sinl(ω1t+ϕ1)sink(ω2t+ϕ2)dt
=121+k l![(l/2)!]2 k![(k/2)!]2,l,keven,ω1=nω2orω2=nω1,n=1, 2, 3 .
J0(A1ω)J0(A2ω)
=1+(-jA1ω)222+(-jA1ω)4242!++(-jA1ω)2k22k(k!)2+×1+(-jA2ω)222+(-jA2ω)4242!++(-jA2ω)2k22k(k!)2+
=1+(jω)2A1222+A2222+(jω)4A1424(2!)2+A12A2222(1!)222(1!)2+A2424(2!)2++(jω)kA12k22k(k!)2+A12k-2A2222(k-1)[(k-1)!]222(1!)2+A12k-4A2222(k-2)[(k-2)!]224(1!)2++A2424(2!)2+ .

Metrics