Abstract

The existence of an ultimate absolute limit for resolving power is investigated utilizing the <i>ambiguous image</i> concept, viz., different objects cannot be distinguished if they have identical images. Any absolute limit to the resolving power of an optical system must be based upon the existence of ambiguous images rather than on an arbitrary specification of the precision of image measurement, since precision can always be improved, even at the photon-counting limit. It is shown that for all objects of finite angular size, the image spectrum within the passband of the optical system contains the information necessary to determine the object spectrum throughout the entire frequency domain. Knowledge of the object spectrum implies knowledge of the object. It is shown that two distinctly different objects of finite size cannot have identical images, so that no ambiguous image exists for such objects. Therefore, diffraction limits resolving power in the sense of only the lack of precision of image measurement imposed by the system noise. Equations are derived which describe processing procedures by means of which object detail can be extracted from diffraction images. An illustrative example shows the successful processing of the image of two monochromatic point sources separated by 0.2 of the Rayleigh criterion distance.

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  1. B. P. Ramsya, E. L. Cleveland, and O. T. Kappins, J. Opt. Soc. Am. 31, 26 (1941).
  2. C. Sparrow, Astrophys. J. 44, 76 (1916).
  3. R. Barakat, J. Opt. Soc. Am. 52, 276 (1962).
  4. J. L. Harris, Scripps Inst. Oceano. Contrib. 63โ€“10 (1963).
  5. H. S. Coleman and M. F. Coleman, J. Opt. Soc. Am. 37, 572 (1947).
  6. V. Ronchi, Optics, The Science of Vision (New York University Press, New York, 1957), p. 205.
  7. G. Toraldo di Francia, J. Opt. Soc. Am. 45, 497 (1955).
  8. V. Ronchi, J. Opt. Soc. Am. 51, 458 (L) (1961).
  9. J. L. Harris, J. Opt. Soc. Am. 54, 606 (1964).
  10. Corrections have been accomplished by subjecting the image to Fourier analysis, operating on each spatial frequency comnponent in a manner which exactly compensates for the attenuation and phase shift indicated by the optical transfer function, and then performing an inverse Fourier transformation to obtain the restored image [See J. L. Harris, Scripps Inst. Oceano. (Contrib., 63-10 (1963); Proceedings National Aerospace Electronics Conference, 403 (1963)].
  11. Whittaker and Watson, Modern Analysis (Cambridge University Press, Cambridge, England, 1935), p. 67.
  12. E. A. Guillemin, The Mathematics of Circuit Analysis (John Wiley & Sons, Inc., New York, 1951), pp. 288, 290.
  13. S. Goldman, Information Theory (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1954), pp. 67, 81.
  14. Korn and Korn, Mathematical handbook for Scientists and Engineers (McGraw-Hill Book Company, Inc., New York, 1961), p. 139.

Barakat, R.

R. Barakat, J. Opt. Soc. Am. 52, 276 (1962).

Cleveland, E. L.

B. P. Ramsya, E. L. Cleveland, and O. T. Kappins, J. Opt. Soc. Am. 31, 26 (1941).

Coleman, H. S.

H. S. Coleman and M. F. Coleman, J. Opt. Soc. Am. 37, 572 (1947).

Coleman, M. F.

H. S. Coleman and M. F. Coleman, J. Opt. Soc. Am. 37, 572 (1947).

di Francia, G. Toraldo

G. Toraldo di Francia, J. Opt. Soc. Am. 45, 497 (1955).

Goldman, S.

S. Goldman, Information Theory (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1954), pp. 67, 81.

Guillemin, E. A.

E. A. Guillemin, The Mathematics of Circuit Analysis (John Wiley & Sons, Inc., New York, 1951), pp. 288, 290.

Harris, J. L.

J. L. Harris, Scripps Inst. Oceano. Contrib. 63โ€“10 (1963).

J. L. Harris, J. Opt. Soc. Am. 54, 606 (1964).

Kappins, O. T.

B. P. Ramsya, E. L. Cleveland, and O. T. Kappins, J. Opt. Soc. Am. 31, 26 (1941).

Ramsya, B. P.

B. P. Ramsya, E. L. Cleveland, and O. T. Kappins, J. Opt. Soc. Am. 31, 26 (1941).

Ronchi, V.

V. Ronchi, J. Opt. Soc. Am. 51, 458 (L) (1961).

V. Ronchi, Optics, The Science of Vision (New York University Press, New York, 1957), p. 205.

Sparrow, C.

C. Sparrow, Astrophys. J. 44, 76 (1916).

Other

B. P. Ramsya, E. L. Cleveland, and O. T. Kappins, J. Opt. Soc. Am. 31, 26 (1941).

C. Sparrow, Astrophys. J. 44, 76 (1916).

R. Barakat, J. Opt. Soc. Am. 52, 276 (1962).

J. L. Harris, Scripps Inst. Oceano. Contrib. 63โ€“10 (1963).

H. S. Coleman and M. F. Coleman, J. Opt. Soc. Am. 37, 572 (1947).

V. Ronchi, Optics, The Science of Vision (New York University Press, New York, 1957), p. 205.

G. Toraldo di Francia, J. Opt. Soc. Am. 45, 497 (1955).

V. Ronchi, J. Opt. Soc. Am. 51, 458 (L) (1961).

J. L. Harris, J. Opt. Soc. Am. 54, 606 (1964).

Corrections have been accomplished by subjecting the image to Fourier analysis, operating on each spatial frequency comnponent in a manner which exactly compensates for the attenuation and phase shift indicated by the optical transfer function, and then performing an inverse Fourier transformation to obtain the restored image [See J. L. Harris, Scripps Inst. Oceano. (Contrib., 63-10 (1963); Proceedings National Aerospace Electronics Conference, 403 (1963)].

Whittaker and Watson, Modern Analysis (Cambridge University Press, Cambridge, England, 1935), p. 67.

E. A. Guillemin, The Mathematics of Circuit Analysis (John Wiley & Sons, Inc., New York, 1951), pp. 288, 290.

S. Goldman, Information Theory (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1954), pp. 67, 81.

Korn and Korn, Mathematical handbook for Scientists and Engineers (McGraw-Hill Book Company, Inc., New York, 1961), p. 139.

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