Abstract

The degrees of freedom of an image formed by any real instrument are only a finite number, while those those of the object are an infinite number. Several different objects may correspond to the same image.

It is shown that in the case of coherent illumination a large class of objects corresponding to a given image can be found very easily. Two-point resolution is impossible unless the observer has a priori an infinite amount of information about the object.

© 1955 Optical Society of America

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References

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  1. See, for example, R. K. Luneberg, Mathematical Theory of Optics (Brown University Press, Providence, 1944), p. 390.
  2. S. A. Schelkunoff, Bell System Tech. J. 22, 80 (1943).
    [Crossref]
  3. C. J. Bouwkamp and N. G. De Bruijn, Philips Research Repts. 1, 135 (1946).
  4. G. Toraldo di Francia, Nuovo cimento, Suppl. 9, 426 (1952).
    [Crossref]
  5. A. Maréchal and P. Lacomme, Compt. rend. 234, 1865 (1952).
  6. G. Toraldo di Francia, Atti fond azione “Giorgio Ronchi” 6, 73 (1951); Atti fond azione “Giorgio Ronchi” 8, 203 (1953).
  7. G. W. King and A. G. Emslie, J. Opt. Soc. Am. 41, 405 (1951); J. Opt. Soc. Am. 43, 658, 664 (1953).
    [Crossref] [PubMed]
  8. A. Blanc-Lapierre, Ann. inst. Henri Poincaré 13, 245 (1953). For further literature the reader may be referred to this paper.
  9. G. Toraldo di Francia, Optica Acta (to be published.)
  10. C. E. Shannon, Bell Telephone System Monograph B-1598 (1948), p. 51.
  11. Both of them can be revealed by phase contrast techniques.
  12. If only bright points were considered, the sample values would not be different from those of a uniformly bright image. No resolution would be possible.
  13. P. M. Woodward, Probability and Information Theory (Pergamon Press, London, 1953), p. 33.
  14. S. Goldman, Information Theory (Prentice-Hall, Inc., New York, 1954), p. 69.
  15. D. A. Bell, Information Theory (Pittman and Sons, London, 1953), p. 43.

1953 (1)

A. Blanc-Lapierre, Ann. inst. Henri Poincaré 13, 245 (1953). For further literature the reader may be referred to this paper.

1952 (2)

G. Toraldo di Francia, Nuovo cimento, Suppl. 9, 426 (1952).
[Crossref]

A. Maréchal and P. Lacomme, Compt. rend. 234, 1865 (1952).

1951 (2)

G. Toraldo di Francia, Atti fond azione “Giorgio Ronchi” 6, 73 (1951); Atti fond azione “Giorgio Ronchi” 8, 203 (1953).

G. W. King and A. G. Emslie, J. Opt. Soc. Am. 41, 405 (1951); J. Opt. Soc. Am. 43, 658, 664 (1953).
[Crossref] [PubMed]

1948 (1)

C. E. Shannon, Bell Telephone System Monograph B-1598 (1948), p. 51.

1946 (1)

C. J. Bouwkamp and N. G. De Bruijn, Philips Research Repts. 1, 135 (1946).

1943 (1)

S. A. Schelkunoff, Bell System Tech. J. 22, 80 (1943).
[Crossref]

Bell, D. A.

D. A. Bell, Information Theory (Pittman and Sons, London, 1953), p. 43.

Blanc-Lapierre, A.

A. Blanc-Lapierre, Ann. inst. Henri Poincaré 13, 245 (1953). For further literature the reader may be referred to this paper.

Bouwkamp, C. J.

C. J. Bouwkamp and N. G. De Bruijn, Philips Research Repts. 1, 135 (1946).

De Bruijn, N. G.

C. J. Bouwkamp and N. G. De Bruijn, Philips Research Repts. 1, 135 (1946).

Emslie, A. G.

Goldman, S.

S. Goldman, Information Theory (Prentice-Hall, Inc., New York, 1954), p. 69.

King, G. W.

Lacomme, P.

A. Maréchal and P. Lacomme, Compt. rend. 234, 1865 (1952).

Luneberg, R. K.

See, for example, R. K. Luneberg, Mathematical Theory of Optics (Brown University Press, Providence, 1944), p. 390.

Maréchal, A.

A. Maréchal and P. Lacomme, Compt. rend. 234, 1865 (1952).

Schelkunoff, S. A.

S. A. Schelkunoff, Bell System Tech. J. 22, 80 (1943).
[Crossref]

Shannon, C. E.

C. E. Shannon, Bell Telephone System Monograph B-1598 (1948), p. 51.

Toraldo di Francia, G.

G. Toraldo di Francia, Nuovo cimento, Suppl. 9, 426 (1952).
[Crossref]

G. Toraldo di Francia, Atti fond azione “Giorgio Ronchi” 6, 73 (1951); Atti fond azione “Giorgio Ronchi” 8, 203 (1953).

G. Toraldo di Francia, Optica Acta (to be published.)

Woodward, P. M.

P. M. Woodward, Probability and Information Theory (Pergamon Press, London, 1953), p. 33.

Ann. inst. Henri Poincaré (1)

A. Blanc-Lapierre, Ann. inst. Henri Poincaré 13, 245 (1953). For further literature the reader may be referred to this paper.

Atti fond azione “Giorgio Ronchi” (1)

G. Toraldo di Francia, Atti fond azione “Giorgio Ronchi” 6, 73 (1951); Atti fond azione “Giorgio Ronchi” 8, 203 (1953).

Bell System Tech. J. (1)

S. A. Schelkunoff, Bell System Tech. J. 22, 80 (1943).
[Crossref]

Bell Telephone System Monograph B-1598 (1)

C. E. Shannon, Bell Telephone System Monograph B-1598 (1948), p. 51.

Compt. rend. (1)

A. Maréchal and P. Lacomme, Compt. rend. 234, 1865 (1952).

J. Opt. Soc. Am. (1)

Nuovo cimento, Suppl. (1)

G. Toraldo di Francia, Nuovo cimento, Suppl. 9, 426 (1952).
[Crossref]

Philips Research Repts. (1)

C. J. Bouwkamp and N. G. De Bruijn, Philips Research Repts. 1, 135 (1946).

Other (7)

See, for example, R. K. Luneberg, Mathematical Theory of Optics (Brown University Press, Providence, 1944), p. 390.

G. Toraldo di Francia, Optica Acta (to be published.)

Both of them can be revealed by phase contrast techniques.

If only bright points were considered, the sample values would not be different from those of a uniformly bright image. No resolution would be possible.

P. M. Woodward, Probability and Information Theory (Pergamon Press, London, 1953), p. 33.

S. Goldman, Information Theory (Prentice-Hall, Inc., New York, 1954), p. 69.

D. A. Bell, Information Theory (Pittman and Sons, London, 1953), p. 43.

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Figures (5)

Fig. 1
Fig. 1

Construction of a possible object corresponding to the image f(x).

Fig. 2
Fig. 2

Diffraction image of two point sources a, b placed at a distance λ/2α apart.

Fig. 3
Fig. 3

The same image as that of Fig. 2 can be ascribed to several point sources a, b, c, etc.

Fig. 4
Fig. 4

Image of a point source a.

Fig. 5
Fig. 5

The same image as that of Fig. 4 can be ascribed to several point sources a, b, c, etc.

Equations (6)

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a i ( x ) = - + a ( s ) sin [ 2 π α ( x - s ) / λ ] π ( x - s ) d s .
sin ( 2 π α x / λ ) π x = - α / λ + α / λ exp ( 2 π i f x ) d f ,
N = 8 α 2 ( S / λ 2 ) ,
I i ( x ) = - + I ( s ) sin 2 [ 2 π α ( x - s ) / λ ] π 2 ( x - s ) 2 d s ,
sin 2 ( 2 π α x / λ ) π 2 x 2 = - 2 α / λ + 2 α / λ ( 2 α / λ - f ) exp ( 2 π i f x ) d f .
N = 16 α 2 ( S / λ 2 )