Abstract

The existence of an ultimate absolute limit for resolving power is investigated utilizing the ambiguous image 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.

© 1964 Optical Society of America

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References

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  1. B. P. Ramsya, E. L. Cleveland, and O. T. Kappins, J. Opt. Soc. Am. 31, 26 (1941).
    [Crossref]
  2. C. Sparrow, Astrophys. J. 44, 76 (1916).
    [Crossref]
  3. R. Barakat, J. Opt. Soc. Am. 52, 276 (1962).
    [Crossref]
  4. J. L. Harris, Scripps Inst. Oceano. Contrib.63-10 (1963).
  5. H. S. Coleman and M. F. Coleman, J. Opt. Soc. Am. 37, 572 (1947).
    [Crossref] [PubMed]
  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).
    [Crossref]
  8. V. Ronchi, J. Opt. Soc. Am. 51, 458 (L) (1961).
    [Crossref]
  9. J. L. Harris, J. Opt. Soc. Am. 54, 606 (1964).
    [Crossref]
  10. Corrections have been accomplished by subjecting the image to Fourier analysis, operating on each spatial frequency component 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., Engle-wood 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.

1964 (1)

1963 (2)

Corrections have been accomplished by subjecting the image to Fourier analysis, operating on each spatial frequency component 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)].

J. L. Harris, Scripps Inst. Oceano. Contrib.63-10 (1963).

1962 (1)

1961 (1)

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

1955 (1)

1947 (1)

1941 (1)

1916 (1)

C. Sparrow, Astrophys. J. 44, 76 (1916).
[Crossref]

Barakat, R.

Cleveland, E. L.

Coleman, H. S.

Coleman, M. F.

Goldman, S.

S. Goldman, Information Theory (Prentice-Hall, Inc., Engle-wood 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, J. Opt. Soc. Am. 54, 606 (1964).
[Crossref]

J. L. Harris, Scripps Inst. Oceano. Contrib.63-10 (1963).

Corrections have been accomplished by subjecting the image to Fourier analysis, operating on each spatial frequency component 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)].

Kappins, O. T.

Korn,

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

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

Ramsya, B. P.

Ronchi, V.

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

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).
[Crossref]

Toraldo di Francia, G.

Watson,

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

Whittaker,

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

Astrophys. J. (1)

C. Sparrow, Astrophys. J. 44, 76 (1916).
[Crossref]

J. Opt. Soc. Am. (6)

Scripps Inst. Oceano. Contrib. (2)

Corrections have been accomplished by subjecting the image to Fourier analysis, operating on each spatial frequency component 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)].

J. L. Harris, Scripps Inst. Oceano. Contrib.63-10 (1963).

Other (5)

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

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., Engle-wood 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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Figures (4)

F. 1
F. 1

A hypothetical spatial frequency spectrum showing arbitrary extensions of the spectrum beyond the cutoff frequency, fc. Where no restrictions are placed on the class of possible objects, the extensions define distinctly different objects having identical images.

F. 2
F. 2

A pictorial representation of the sampling theorem in the spatial frequency domain. The solid line is that portion of the object spectrum which falls within the passband of the optical system. For an object of angular size w or smaller, the spectrum of the object is completely determined for all frequencies by specifying the complex spectrum at the sample frequencies (S,P). Cutoff frequency: F.

F. 3
F. 3

The composite image formed by two monochromatic mutually incoherent point sources. The vertical lines show the position of the two points. The image plane distance has been normalized to the Rayleigh criterion distance, so that Δx is 0.2 of the Rayleigh criterion separation.

F. 4
F. 4

Showing the restoration of the image of Fig. 3. The dashed curve is the representation of the two point sources which is achieved by using the first four terms of a Fourier series over the interval ±0.15. The similar solid curve is the restoration accomplished in the illustrative example. The broad, solid curve is the original image of Fig. 3 replotted to the scale of the image restoration.

Equations (23)

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N ( x , y ) = + + G ( f x , f y ) e i 2 π ( f x x + f y y ) d f x d f y ,
G ( f x , f y ) = + + N ( x , y ) e i 2 π ( f x x + f y y ) d x d y .
N ( x , y ) = m = + n = + G m n e i 2 π [ ( m x / X ) + ( n y / Y ) ] ,
G m n = 1 X Y Y / 2 + Y / 2 X / 2 + X / 2 N ( x , y ) e i 2 π [ ( m x / X ) + ( n y / Y ) ] d x d y .
G ( f x , f y ) = Y / 2 + Y / 2 X / 2 + X / 2 [ m = + n = + G m n e i 2 π [ ( m x / X ) + ( n y / Y ) ] × e i 2 π ( f x x + f y y ) d x d y ,
G ( f x , f y ) = m = + n = + G m n Y / 2 + Y / 2 X / 2 + X / 2 exp { i 2 π [ ( m X f x ) x + ( n Y f y ) y ] } d x d y .
G ( f x , f y ) = m = + n = + G m n [ exp { i π [ ( m / X ) f x ] X } exp { + i π [ ( m / X ) f x ] X } i 2 π [ ( m / X ) f x ] ] × [ exp { i π [ ( n / Y ) f y ] Y } exp { + i π [ ( n / Y ) f y ] Y } i 2 π ( n / Y f y ) ] ,
G ( f x , f y ) = X Y m = + n = + G m n sin π [ ( m / X ) f x ] X π [ ( m / X ) f x ] X × sin π [ ( n / Y ) f y ] Y π [ ( n / Y ) f y ] Y .
H ( x ) = K sin 2 π x / ( π x ) 2 ,
H ( x ) = 1 2 { sin 2 π ( x + 0.1 ) [ π ( x + 0.1 ) ] 2 + sin 2 π ( x 0.1 ) [ π ( x 0.1 ) ] 2 } .
G ( f ) = 1 2 [ e i 2 π f ( 0.1 ) + e i 2 π f ( 0.1 ) ] ,
G ( f ) = cos 2 π ( 0.1 ) f ,
G ( 0 ) = 1.0 , G ( 1 3 ) = cos 12 ° , G ( 2 3 ) = cos 24 ° , G ( 1 ) = cos 36 ° .
X = 0.3 .
G ( 0 ) = X G 0 ,
G 0 = 1 / 0.3 .
cos 12 ° = 0.3 { 1 0.3 sin 0.1 π 0.1 π + m = 1 3 G m × [ sin π ( m 0.1 ) π ( m 0.1 ) + sin π ( m + 0.1 ) π ( m + 0.1 ) ] } ,
cos 24 ° = 0.3 { 1 0.3 sin 0.2 π 0.2 π + m = 1 3 G m × [ sin π ( m 0.2 ) π ( m 0.2 ) + sin π ( m + 0.2 ) π ( m + 0.2 ) ] } ,
cos 36 ° = 0.3 { 1 0.3 sin 0.3 π 0.3 π + m = 1 3 G m × [ sin π ( m 0.3 ) π ( m 0.3 ) + sin π ( m + 0.3 ) π ( m + 0.3 ) ] } .
G 1 = 1.68 , G 2 = 1.88 , G 3 = + 2.63 ;
G 0 = + 3.33 .
G 1 = 1.667 , G 2 = 1.667 , G 3 = + 3.33 ,
G 0 = + 3.33 .