Abstract

The maximum number of degrees of freedom of an image, which is usually derived by a rough application of the sampling theorem, has a precise mathematical meaning. This is established by applying the theory of prolate spheroidal functions, as discussed by Slepian and Pollack and other authors. Both coherent and incoherent illumination are considered. The number of degrees of freedom of the image can be substantially different in both cases. A dramatic example is represented by the thin-ring pupil, which is virtually unidimensional in the former case and two dimensional in the latter case.

© 1969 Optical Society of America

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References

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  1. G. Toraldo di Francia, J. Opt. Soc. Am. 45, 497 (1955).
    [Crossref]
  2. G. Toraldo di Francia, Trans. IRE AP-4, 473 (1956).
  3. J. L. Harris, J. Opt. Soc. Am. 54, 931 (1964).
    [Crossref]
  4. H. Wolter, in Progress in Optics, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1961), p. 157.
  5. Y. T. Lo, J. Appl. Phys. 32, 2052 (1961).
    [Crossref]
  6. J I. Khurgin and V. P. Jakovlev, Metody Teorii Celyth Funkciy v Radiofizike, Teorii Svjaziy i Optiki (Moscow, 1962).
  7. W. Lukosz, J. Opt. Soc. Am. 56, 1463 (1966).
    [Crossref]
  8. A. Bachl and W. Lukosz, J. Opt. Soc. Am. 57, 163 (1967).
    [Crossref]
  9. W. Lukosz, J. Opt. Soc. Am. 57, 932 (1967).
    [Crossref]
  10. D. Slepian and H. O. Pollack, Bell System Tech. J. 40, 43 (1961).
    [Crossref]
  11. H. J. Landau and H. O. Pollack, Bell System Tech. J. 40, 65 (1961).
    [Crossref]
  12. H. J. Landau and H. O. Pollack, Bell System Tech. J. 41, 1295 (1962).
    [Crossref]
  13. D. Slepian, Bell System Tech. J. 43, 3009 (1964).
    [Crossref]
  14. D. Slepian, J. Math. and Phys. 44, 99 (1965).
  15. D. Slepian and E. Sonnenblick, Bell System Tech. J. 44, 1745 (1965).
    [Crossref]
  16. C. W. Barnes, J. Opt. Soc. Am. 56, 575 (1966).
    [Crossref]
  17. G. J. Buck and J. J. Gustincic, IEEE Trans. AP-15, 376 (1967).
    [Crossref]
  18. B. R. Frieden, J. Opt. Soc. Am. 57, 1013 (1967).
    [Crossref]
  19. C. K. Rushforth and R. W. Harris, J. Opt. Soc. Am. 58, 539 (1968).
    [Crossref]
  20. D. Gabor, in ICO Symposium on the Applications of Coherent Light (Florence, 1968).
  21. J. P. Wild, Proc. Roy. Soc. (London) A262, 84 (1961).
  22. A. M. L. Carter and J. P. Wild, Proc. Roy. Soc. (London) A282, 252 (1964).
  23. B. Y. Mills and A. G. Little, Austr. J. Phys. 6, 272, (1953).
    [Crossref]
  24. A. E. Covington and N. W. Broten, IRE Trans. AP-5, 247 (1957).
  25. C. J. Drane and G. B. Parrent, IRE Trans. AP-10, 126 (1962).
  26. M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), p. 166.
  27. G. Toraldo di Francia, Electromagnetic Waves (Wiley–Inter-science, Inc., New York, 1956), p. 223.
  28. G. Toraldo di Francia, La diffrazione della luce (Ediz. Scient. Einaudi, 1958), p. 168.
  29. J. C. Heurtley, in Proceedings of the Symposium on Quasi-Optics, Jerome Fox, Ed. (Polytechnic Press, Brooklyn, N. Y., 1964), p. 367.
  30. J. C. Heurtley and W. Streifer, J. Opt. Soc. Am. 55, 1472 (1965).
    [Crossref]
  31. G. W. Stroke (private communication).

1968 (1)

1967 (4)

1966 (2)

1965 (3)

D. Slepian, J. Math. and Phys. 44, 99 (1965).

D. Slepian and E. Sonnenblick, Bell System Tech. J. 44, 1745 (1965).
[Crossref]

J. C. Heurtley and W. Streifer, J. Opt. Soc. Am. 55, 1472 (1965).
[Crossref]

1964 (3)

A. M. L. Carter and J. P. Wild, Proc. Roy. Soc. (London) A282, 252 (1964).

D. Slepian, Bell System Tech. J. 43, 3009 (1964).
[Crossref]

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

1962 (2)

H. J. Landau and H. O. Pollack, Bell System Tech. J. 41, 1295 (1962).
[Crossref]

C. J. Drane and G. B. Parrent, IRE Trans. AP-10, 126 (1962).

1961 (4)

J. P. Wild, Proc. Roy. Soc. (London) A262, 84 (1961).

Y. T. Lo, J. Appl. Phys. 32, 2052 (1961).
[Crossref]

D. Slepian and H. O. Pollack, Bell System Tech. J. 40, 43 (1961).
[Crossref]

H. J. Landau and H. O. Pollack, Bell System Tech. J. 40, 65 (1961).
[Crossref]

1957 (1)

A. E. Covington and N. W. Broten, IRE Trans. AP-5, 247 (1957).

1956 (1)

G. Toraldo di Francia, Trans. IRE AP-4, 473 (1956).

1955 (1)

1953 (1)

B. Y. Mills and A. G. Little, Austr. J. Phys. 6, 272, (1953).
[Crossref]

Bachl, A.

Barnes, C. W.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), p. 166.

Broten, N. W.

A. E. Covington and N. W. Broten, IRE Trans. AP-5, 247 (1957).

Buck, G. J.

G. J. Buck and J. J. Gustincic, IEEE Trans. AP-15, 376 (1967).
[Crossref]

Carter, A. M. L.

A. M. L. Carter and J. P. Wild, Proc. Roy. Soc. (London) A282, 252 (1964).

Covington, A. E.

A. E. Covington and N. W. Broten, IRE Trans. AP-5, 247 (1957).

Drane, C. J.

C. J. Drane and G. B. Parrent, IRE Trans. AP-10, 126 (1962).

Frieden, B. R.

Gabor, D.

D. Gabor, in ICO Symposium on the Applications of Coherent Light (Florence, 1968).

Gustincic, J. J.

G. J. Buck and J. J. Gustincic, IEEE Trans. AP-15, 376 (1967).
[Crossref]

Harris, J. L.

Harris, R. W.

Heurtley, J. C.

J. C. Heurtley and W. Streifer, J. Opt. Soc. Am. 55, 1472 (1965).
[Crossref]

J. C. Heurtley, in Proceedings of the Symposium on Quasi-Optics, Jerome Fox, Ed. (Polytechnic Press, Brooklyn, N. Y., 1964), p. 367.

Jakovlev, V. P.

J I. Khurgin and V. P. Jakovlev, Metody Teorii Celyth Funkciy v Radiofizike, Teorii Svjaziy i Optiki (Moscow, 1962).

Khurgin, J I.

J I. Khurgin and V. P. Jakovlev, Metody Teorii Celyth Funkciy v Radiofizike, Teorii Svjaziy i Optiki (Moscow, 1962).

Landau, H. J.

H. J. Landau and H. O. Pollack, Bell System Tech. J. 41, 1295 (1962).
[Crossref]

H. J. Landau and H. O. Pollack, Bell System Tech. J. 40, 65 (1961).
[Crossref]

Little, A. G.

B. Y. Mills and A. G. Little, Austr. J. Phys. 6, 272, (1953).
[Crossref]

Lo, Y. T.

Y. T. Lo, J. Appl. Phys. 32, 2052 (1961).
[Crossref]

Lukosz, W.

Mills, B. Y.

B. Y. Mills and A. G. Little, Austr. J. Phys. 6, 272, (1953).
[Crossref]

Parrent, G. B.

C. J. Drane and G. B. Parrent, IRE Trans. AP-10, 126 (1962).

Pollack, H. O.

H. J. Landau and H. O. Pollack, Bell System Tech. J. 41, 1295 (1962).
[Crossref]

H. J. Landau and H. O. Pollack, Bell System Tech. J. 40, 65 (1961).
[Crossref]

D. Slepian and H. O. Pollack, Bell System Tech. J. 40, 43 (1961).
[Crossref]

Rushforth, C. K.

Slepian, D.

D. Slepian, J. Math. and Phys. 44, 99 (1965).

D. Slepian and E. Sonnenblick, Bell System Tech. J. 44, 1745 (1965).
[Crossref]

D. Slepian, Bell System Tech. J. 43, 3009 (1964).
[Crossref]

D. Slepian and H. O. Pollack, Bell System Tech. J. 40, 43 (1961).
[Crossref]

Sonnenblick, E.

D. Slepian and E. Sonnenblick, Bell System Tech. J. 44, 1745 (1965).
[Crossref]

Streifer, W.

Stroke, G. W.

G. W. Stroke (private communication).

Toraldo di Francia, G.

G. Toraldo di Francia, Trans. IRE AP-4, 473 (1956).

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

G. Toraldo di Francia, Electromagnetic Waves (Wiley–Inter-science, Inc., New York, 1956), p. 223.

G. Toraldo di Francia, La diffrazione della luce (Ediz. Scient. Einaudi, 1958), p. 168.

Wild, J. P.

A. M. L. Carter and J. P. Wild, Proc. Roy. Soc. (London) A282, 252 (1964).

J. P. Wild, Proc. Roy. Soc. (London) A262, 84 (1961).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), p. 166.

Wolter, H.

H. Wolter, in Progress in Optics, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1961), p. 157.

Austr. J. Phys. (1)

B. Y. Mills and A. G. Little, Austr. J. Phys. 6, 272, (1953).
[Crossref]

Bell System Tech. J. (5)

D. Slepian and E. Sonnenblick, Bell System Tech. J. 44, 1745 (1965).
[Crossref]

D. Slepian and H. O. Pollack, Bell System Tech. J. 40, 43 (1961).
[Crossref]

H. J. Landau and H. O. Pollack, Bell System Tech. J. 40, 65 (1961).
[Crossref]

H. J. Landau and H. O. Pollack, Bell System Tech. J. 41, 1295 (1962).
[Crossref]

D. Slepian, Bell System Tech. J. 43, 3009 (1964).
[Crossref]

IEEE Trans. (1)

G. J. Buck and J. J. Gustincic, IEEE Trans. AP-15, 376 (1967).
[Crossref]

IRE Trans. (2)

A. E. Covington and N. W. Broten, IRE Trans. AP-5, 247 (1957).

C. J. Drane and G. B. Parrent, IRE Trans. AP-10, 126 (1962).

J. Appl. Phys. (1)

Y. T. Lo, J. Appl. Phys. 32, 2052 (1961).
[Crossref]

J. Math. and Phys. (1)

D. Slepian, J. Math. and Phys. 44, 99 (1965).

J. Opt. Soc. Am. (9)

Proc. Roy. Soc. (London) (2)

J. P. Wild, Proc. Roy. Soc. (London) A262, 84 (1961).

A. M. L. Carter and J. P. Wild, Proc. Roy. Soc. (London) A282, 252 (1964).

Trans. IRE (1)

G. Toraldo di Francia, Trans. IRE AP-4, 473 (1956).

Other (8)

H. Wolter, in Progress in Optics, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1961), p. 157.

J I. Khurgin and V. P. Jakovlev, Metody Teorii Celyth Funkciy v Radiofizike, Teorii Svjaziy i Optiki (Moscow, 1962).

D. Gabor, in ICO Symposium on the Applications of Coherent Light (Florence, 1968).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959), p. 166.

G. Toraldo di Francia, Electromagnetic Waves (Wiley–Inter-science, Inc., New York, 1956), p. 223.

G. Toraldo di Francia, La diffrazione della luce (Ediz. Scient. Einaudi, 1958), p. 168.

J. C. Heurtley, in Proceedings of the Symposium on Quasi-Optics, Jerome Fox, Ed. (Polytechnic Press, Brooklyn, N. Y., 1964), p. 367.

G. W. Stroke (private communication).

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Figures (2)

Fig. 1
Fig. 1

Image formation in coherent light.

Fig. 2
Fig. 2

The eigenvalue λi as a function of i.

Equations (31)

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F ( ω ) = - + f ( x ) e - i ω x d x .
f ¯ ( x ) = 1 2 π - Ω + Ω F ( ω ) e i ω x d ω .
S = X Ω / π
f ¯ ( x ) = - X / 2 + X / 2 sin Ω ( x - s ) π ( x - s ) f ( s ) d s .
f ( x ) = i a i ψ i ( x ) .
f ¯ ( x ) = i λ i a i ψ i ( x ) ,
f ¯ ( x ) = i A i ψ i ( x ) ,
A i = λ i a i .
F ( ω , χ ) = - + - + f ( x , y ) exp [ - i ( ω x + χ y ) ] d x d y .
f ( x , y ) = 1 4 π 2 - + - + F ( ω , χ ) e i ( ω χ + χ y ) d ω d χ ,
x = r cos ϕ , y = r sin ϕ , ω = ρ cos θ , χ = ρ sin θ ,
f ( r , ϕ ) = - + N     f N ( r ) e i N ϕ ,
F ( ρ , θ ) = - + N     F N ( ρ ) e i N θ .
F ( ρ , θ ) = 0 R r d r 0 2 π d ϕ f ( r , ϕ ) exp [ - i r ρ cos ( θ - ϕ ) ] = - + N ( - i ) N e i N θ 0 R J N ( r ρ ) r d r 0 2 π f ( r , ϕ ) e - i N ϕ d ϕ ,
F N ( ρ ) = 2 π ( - i ) N 0 R f N ( r ) J N ( r ρ ) r d r .
f ¯ ( r , ϕ ) = 1 4 π 2 0 Ω ρ d ρ 0 2 π d θ F ( ρ , θ ) exp [ i r ρ cos ( θ - ϕ ) ] = 1 4 π 2 - + N     i N e i N ϕ 0 Ω J N ( r ρ ) ρ d ρ × 0 2 π F ( ρ , θ ) e - i N θ d θ .
f ¯ ( r , ϕ ) = - + f ¯ N ( r ) e i N ϕ ,
f ¯ N ( r ) = i N 2 π 0 Ω F N ( ρ ) J N ( r ρ ) ρ d ρ ,
γ N , n ϕ N , n ( x ) = 0 1 J N ( c x x ) ( c x x ) 1 2 ϕ N , n ( x ) d x .
( R x ) 1 2 f N ( R x ) = 0 n     a N , n ϕ N , n ( x ) .
( R x ) 1 2 f ¯ N ( R x ) = c n γ N , n 2 a N , n ϕ N , n ( x ) ,
c = R Ω .
λ N , n = c γ N , n 2 ,
S = R 2 Ω 2 / π 2 .
f ¯ N ( r ) = ( i N / 2 π ) F N ( Ω ) J N ( Ω r ) Ω .
F N ( Ω ) = 2 π ( - 1 ) N R Ω 1 2 n γ N , n a N , n ϕ N , n ( 1 ) .
S = 2 R Ω / π .
g ( x , y ) = P exp [ i ( ω x + χ y ) ] d ω d χ ,
f ¯ ( x , y ) = f ( x , y ) * g ( x , y ) 2 .
F ¯ ( ω , χ ) = F ( ω , χ ) G ( ω , χ ) .
f ¯ ( x , y ) = 1 4 π 2 P * P F ( ω , χ ) G ( ω , χ ) × exp i ( ω x + χ y ) d ω d χ .