Abstract

A transfer theory is developed that determines the image space, and three-dimensional image spectrum, of a 3-D object. For both incoherent and coherent illumination, the image is found to obey convolution, transfer, and sampling theorems that resemble the familiar results of ordinary 2-D theory. A 3-D transfer function is related to the pupil function of the image-forming optical system. One result of the theory is that with incoherent illumination, the object image space contains no more than 1/(λ<sup>3</sup>ƒ/<i>no.</i><sup>4</sup>) degrees of freedom/unit volume, where λ is the wavelength of light. The transfer theory is based on the existence of volumes of stationarity, termed “isotomes;” into which the object must be partitioned. Isotomicity is shown to be approximated, over sufficiently small volumes, in the diffraction-limited case.

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  1. B. R. Frieden, J. Opt. Soc. Am. 56, 1495 (1966).
  2. P. M. Duffieux, L'integrale de Fourier et ses Applications a l'Optique, Besançon (1947), privately printed.
  3. See, e.g., H. H. Hopkins, in Proceedings of the Conference on Optical Instruments and Techniques, K. J. Habell, Ed. (John Wiley & Sons, Inc., New York, 1963), pp. 483, 484.
  4. M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), p. 481.
  5. From the Greek words "iso" and "tomos," meaning "equal" and "volume section," respectively.
  6. P. Dumontet, Opt. Acta 2, 53 (1955).
  7. Ref. 4, p. 752.
  8. R. M. Bracewell, The Fourier Transform and its Applications (McGraw-Hill Book Co., New York, 1965), p. 7.
  9. Reference 4, p. 382.
  10. The orientation of axes α′β′γ′ is as described by B. R. A. Nijboer, Thesis, Groningen, 1948. The approximate coincidence of axes α, β, γ and α′, β′, γ′ is implicitly assumed in, e.g., Ref. 4, p. 480.
  11. Reference 4, p. 484.
  12. Reference 4; p. 754, Eq. (10) and p. 756, Eq. (23).
  13. H. H. Hopkins, Opt. Acta 2, 23 (1955).
  14. The corresponding one-dimensional formulae are derived by S. Goldman, Information Theory (Prentice-Hall, Inc., New York, 1955), p. 82.
  15. A very similar concept was previously used by C. W. Mc-Cutchen, J. Opt. Soc. Am. 54, 240 (1964).
  16. Reference 8, pp. 11, 91.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), p. 481.

Bracewell, R. M.

R. M. Bracewell, The Fourier Transform and its Applications (McGraw-Hill Book Co., New York, 1965), p. 7.

Duffieux, P. M.

P. M. Duffieux, L'integrale de Fourier et ses Applications a l'Optique, Besançon (1947), privately printed.

Dumontet, P.

P. Dumontet, Opt. Acta 2, 53 (1955).

Frieden, B. R.

B. R. Frieden, J. Opt. Soc. Am. 56, 1495 (1966).

Goldman, S.

The corresponding one-dimensional formulae are derived by S. Goldman, Information Theory (Prentice-Hall, Inc., New York, 1955), p. 82.

Hopkins, H. H.

H. H. Hopkins, Opt. Acta 2, 23 (1955).

See, e.g., H. H. Hopkins, in Proceedings of the Conference on Optical Instruments and Techniques, K. J. Habell, Ed. (John Wiley & Sons, Inc., New York, 1963), pp. 483, 484.

Mc-Cutchen, C. W.

A very similar concept was previously used by C. W. Mc-Cutchen, J. Opt. Soc. Am. 54, 240 (1964).

Nijboer, B. R. A.

The orientation of axes α′β′γ′ is as described by B. R. A. Nijboer, Thesis, Groningen, 1948. The approximate coincidence of axes α, β, γ and α′, β′, γ′ is implicitly assumed in, e.g., Ref. 4, p. 480.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), p. 481.

Other

B. R. Frieden, J. Opt. Soc. Am. 56, 1495 (1966).

P. M. Duffieux, L'integrale de Fourier et ses Applications a l'Optique, Besançon (1947), privately printed.

See, e.g., H. H. Hopkins, in Proceedings of the Conference on Optical Instruments and Techniques, K. J. Habell, Ed. (John Wiley & Sons, Inc., New York, 1963), pp. 483, 484.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), p. 481.

From the Greek words "iso" and "tomos," meaning "equal" and "volume section," respectively.

P. Dumontet, Opt. Acta 2, 53 (1955).

Ref. 4, p. 752.

R. M. Bracewell, The Fourier Transform and its Applications (McGraw-Hill Book Co., New York, 1965), p. 7.

Reference 4, p. 382.

The orientation of axes α′β′γ′ is as described by B. R. A. Nijboer, Thesis, Groningen, 1948. The approximate coincidence of axes α, β, γ and α′, β′, γ′ is implicitly assumed in, e.g., Ref. 4, p. 480.

Reference 4, p. 484.

Reference 4; p. 754, Eq. (10) and p. 756, Eq. (23).

H. H. Hopkins, Opt. Acta 2, 23 (1955).

The corresponding one-dimensional formulae are derived by S. Goldman, Information Theory (Prentice-Hall, Inc., New York, 1955), p. 82.

A very similar concept was previously used by C. W. Mc-Cutchen, J. Opt. Soc. Am. 54, 240 (1964).

Reference 8, pp. 11, 91.

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