Abstract

Earlier results on parametric model-based optical resolution by the first author [ J. Opt. Soc. Am. A 4, 1402 ( 1987)] concerned two incoherent sources of known intensity. These results are generalized to sources with any degree of coherence and unknown intensity. A further generalization is that the model underlying the observations and that chosen by the experimenter need not be the same. It is shown how, for a chosen model, observations from which the sources can be resolved can be distinguished from observations from which they cannot.

© 1996 Optical Society of America

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References

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  1. A. van den Bos, “Optical resolution: an analysis based on catastrophe theory,” J. Opt. Soc. Am. A 4, 1402–1406 (1987).
    [CrossRef]
  2. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 326–328.
  3. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

1987 (1)

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 326–328.

van den Bos, A.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

J. Opt. Soc. Am. A (1)

Other (2)

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 326–328.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

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Figures (1)

Fig. 1
Fig. 1

Contours of the least-squares criterion for fitting a bi-Gaussian model to observations described by Eq. (4) with different values of υ. The criteria corresponding to the contours in the top panels have two symmetrically located absolute minima separated by a saddle point at the origin. At the absolute minima the locations b1 and b2 of the components are different; then resolution is possible. The criteria corresponding to the contours in the bottom panels have a minimum at the origin only. Therefore locations b1 and b2 of the components coincide; then resolution is not possible.

Equations (9)

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g ( x ; c , b 1 , b 2 ) = c [ f 2 ( x b 1 ) + f 2 ( x b 2 ) + 2 ρ f ( x b 1 ) f ( x b 2 ) ] .
J ( c , b 1 , b 2 ) = n d 2 ( x n ; c , b 1 , b 2 ) ,
c = n w n h ( x n ; b 1 , b 2 ) / n h 2 ( x n ; b 1 , b 2 ) ,
w n = g ( x n ; γ , β 1 , β 2 ) + υ ,
2 n d n g n c , 2 n d n g n b k ; k = 1,2 ,
2 c ( 1 + ρ ) f 2 ( x n b ) ,
H 2 = diag ( H 1 μ ) ,
μ = 4 c ˆ [ 1 ρ 1 + ρ n d n ( f n ( 1 ) ) 2 + n d n f n f n ( 2 ) ] ,
8 c ˆ 1 + ρ n d n ( f n ( 1 ) ) 2 .

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