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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 16, Iss. 9 — Sep. 1, 1999
  • pp: 2200–2208

Optical transfer function analysis of images blurred by nonharmonic vibrations characterized by their power spectrum density

A. Stern and N. S. Kopeika  »View Author Affiliations

JOSA A, Vol. 16, Issue 9, pp. 2200-2208 (1999)

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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

OCIS Codes
(110.2990) Imaging systems : Image formation theory
(110.4100) Imaging systems : Modulation transfer function
(110.4850) Imaging systems : Optical transfer functions

Original Manuscript: November 3, 1998
Revised Manuscript: March 26, 1999
Manuscript Accepted: March 26, 1999
Published: September 1, 1999

A. Stern and N. S. Kopeika, "Optical transfer function analysis of images blurred by nonharmonic vibrations characterized by their power spectrum density," J. Opt. Soc. Am. A 16, 2200-2208 (1999)

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  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).

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