Abstract

A method is derived for digitally reconstructing any two-dimensional, partially coherent, polychromatic object from experimental knowledge of the image and point spread function. In the absence of noise, the reconstruction is perfect. The object must lie wholly within a known region of the object plane. The optics may be generally coated and tilted, and may have any aberrations. As an illustration, the reconstruction process is applied to the problem of resolving double stars. The reconstruction scheme is also used to correct the output of a conventional spectrometer for instrument broadening, and to correct the output of a Fourier-transform spectroscope for finite extent of the interferogram. Practical use of the method requires the calculation of prolate spheroidal wavefunctions and eigenvalues. The effect of noise upon the accuracy of reconstruction is analytically computed. It is shown that periodic noise and piecewise-continuous noise both cause zero error at all points in the reconstruction except at the sampling points, where the error is (theoretically) infinite. Finally, bandwidth-limited noise is shown to be indistinguishable from the object.

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  1. E. L. O'Neill, Introduction to Statistical Optics (Addison-Wesley Publ. Co., Reading, Mass., 1963), p. 22.
  2. Reference 1, p. 125.
  3. Reference 1, p. 85.
  4. H. Wolter, in Progress in Optics I, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1961), Ch. V, Sec. 4.6.
  5. H. Lass, Elements of Pure and Applied Mathematics (McGraw-Hill Book Co., New York, 1957), pp. 150, 151.
  6. J. L. Harris, J. Opt. Soc. Am. 54, 931 (1964).
  7. C. W. Barnes, J. Opt. Soc. Am. 56, 575 (1966). This article also presents the most up-to-date summary of past work in the field.
  8. D. Slepian and H. O. Pollak, Bell System Tech. J. 40, 43 (1961).
  9. H. J. Landau and H. O. Pollak, Bell System Tech. J. 40, 65 (1961).
  10. C. Flammer, Spheroidal Wave Functions (Stanford University Press, Stanford, Calif., 1957).
  11. M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), p. 756.
  12. We use the notation of M. J. Beran and G. B. Parrent, Jr., in Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964), pp. 103–107.
  13. Reference 11, p. 384.
  14. Reference 12, p. 106; and Ref. 1.
  15. Reference 8, p. 45, result ii.
  16. R. E. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964).
  17. D. L. Fried, J. Opt. Soc. Am. 56, 1372 (1966).
  18. E. H. Linfoot, Fourier Methods in Optical Image Evaluation (The Focal Press, London, 1964), p. 45.
  19. P. Jacquinot and B. Roizen-Dossier, in Progress in Optics III, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1964), p. 147,
  20. Reference 19, p. 163.
  21. Reference 19, p. 122 et seq.
  22. B. R. Frieden, J. Opt. Soc. Am. 56, 1356 (1966), Eqs. (3.2) and (3.4).
  23. F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill Book Co., New York, 1956), p. 319 et seq.
  24. A. Maréchal and P. Croce, Compt. Rend. 237, 706 (1953).
  25. R. V. Churchill, Fourier Series and Boundary Value Problems (McGraw-Hill Book Co., New York, 1941).
  26. J. J. Downing, Modulation Systems and Noise (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964), p. 51.
  27. W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill Book Co., New York, 1958).
  28. S. Goldman, Information Theory (Prentice-Hall, Inc., Englewood-Cliffs, N. J., 1955), p. 121.

Barnes, C. W.

C. W. Barnes, J. Opt. Soc. Am. 56, 575 (1966). This article also presents the most up-to-date summary of past work in the field.

Beran, M. J.

We use the notation of M. J. Beran and G. B. Parrent, Jr., in Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964), pp. 103–107.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), p. 756.

Churchill, R. V.

R. V. Churchill, Fourier Series and Boundary Value Problems (McGraw-Hill Book Co., New York, 1941).

Croce, P.

A. Maréchal and P. Croce, Compt. Rend. 237, 706 (1953).

Davenport, W. B.

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill Book Co., New York, 1958).

Downing, J. J.

J. J. Downing, Modulation Systems and Noise (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964), p. 51.

Flammer, C.

C. Flammer, Spheroidal Wave Functions (Stanford University Press, Stanford, Calif., 1957).

Fried, D. L.

D. L. Fried, J. Opt. Soc. Am. 56, 1372 (1966).

Frieden, B. R.

B. R. Frieden, J. Opt. Soc. Am. 56, 1356 (1966), Eqs. (3.2) and (3.4).

Goldman, S.

S. Goldman, Information Theory (Prentice-Hall, Inc., Englewood-Cliffs, N. J., 1955), p. 121.

Harris, J. L.

J. L. Harris, J. Opt. Soc. Am. 54, 931 (1964).

Hildebrand, F. B.

F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill Book Co., New York, 1956), p. 319 et seq.

Hufnagel, R. E.

R. E. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964).

Jacquinot, P.

P. Jacquinot and B. Roizen-Dossier, in Progress in Optics III, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1964), p. 147,

Landau, H. J.

H. J. Landau and H. O. Pollak, Bell System Tech. J. 40, 65 (1961).

Lass, H.

H. Lass, Elements of Pure and Applied Mathematics (McGraw-Hill Book Co., New York, 1957), pp. 150, 151.

Linfoot, E. H.

E. H. Linfoot, Fourier Methods in Optical Image Evaluation (The Focal Press, London, 1964), p. 45.

Maréchal, A.

A. Maréchal and P. Croce, Compt. Rend. 237, 706 (1953).

O’Neill, E. L.

E. L. O'Neill, Introduction to Statistical Optics (Addison-Wesley Publ. Co., Reading, Mass., 1963), p. 22.

Parrent, Jr., G. B.

We use the notation of M. J. Beran and G. B. Parrent, Jr., in Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964), pp. 103–107.

Pollak, H. O.

D. Slepian and H. O. Pollak, Bell System Tech. J. 40, 43 (1961).

H. J. Landau and H. O. Pollak, Bell System Tech. J. 40, 65 (1961).

Roizen-Dossier, B.

P. Jacquinot and B. Roizen-Dossier, in Progress in Optics III, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1964), p. 147,

Root, W. L.

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill Book Co., New York, 1958).

Slepian, D.

D. Slepian and H. O. Pollak, Bell System Tech. J. 40, 43 (1961).

Stanley, N. R.

R. E. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), p. 756.

Wolter, H.

H. Wolter, in Progress in Optics I, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1961), Ch. V, Sec. 4.6.

Other (28)

E. L. O'Neill, Introduction to Statistical Optics (Addison-Wesley Publ. Co., Reading, Mass., 1963), p. 22.

Reference 1, p. 125.

Reference 1, p. 85.

H. Wolter, in Progress in Optics I, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1961), Ch. V, Sec. 4.6.

H. Lass, Elements of Pure and Applied Mathematics (McGraw-Hill Book Co., New York, 1957), pp. 150, 151.

J. L. Harris, J. Opt. Soc. Am. 54, 931 (1964).

C. W. Barnes, J. Opt. Soc. Am. 56, 575 (1966). This article also presents the most up-to-date summary of past work in the field.

D. Slepian and H. O. Pollak, Bell System Tech. J. 40, 43 (1961).

H. J. Landau and H. O. Pollak, Bell System Tech. J. 40, 65 (1961).

C. Flammer, Spheroidal Wave Functions (Stanford University Press, Stanford, Calif., 1957).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), p. 756.

We use the notation of M. J. Beran and G. B. Parrent, Jr., in Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964), pp. 103–107.

Reference 11, p. 384.

Reference 12, p. 106; and Ref. 1.

Reference 8, p. 45, result ii.

R. E. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964).

D. L. Fried, J. Opt. Soc. Am. 56, 1372 (1966).

E. H. Linfoot, Fourier Methods in Optical Image Evaluation (The Focal Press, London, 1964), p. 45.

P. Jacquinot and B. Roizen-Dossier, in Progress in Optics III, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1964), p. 147,

Reference 19, p. 163.

Reference 19, p. 122 et seq.

B. R. Frieden, J. Opt. Soc. Am. 56, 1356 (1966), Eqs. (3.2) and (3.4).

F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill Book Co., New York, 1956), p. 319 et seq.

A. Maréchal and P. Croce, Compt. Rend. 237, 706 (1953).

R. V. Churchill, Fourier Series and Boundary Value Problems (McGraw-Hill Book Co., New York, 1941).

J. J. Downing, Modulation Systems and Noise (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964), p. 51.

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill Book Co., New York, 1958).

S. Goldman, Information Theory (Prentice-Hall, Inc., Englewood-Cliffs, N. J., 1955), p. 121.

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