## 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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### Equations (9)

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(1)
$$\begin{array}{ll}g\left(x;c,{b}_{1},{b}_{2}\right)=\hfill & c[{f}^{2}\left(x-{b}_{1}\right)+{f}^{2}\left(x-{b}_{2}\right)\hfill \\ \hfill & +2\rho f\left(x-{b}_{1}\right)f\left(x-{b}_{2}\right)].\hfill \end{array}$$
(2)
$$J\phantom{\rule{0.2em}{0ex}}\left(c,{b}_{1},{b}_{2}\right)={\displaystyle \sum _{n}{d}^{2}}\left({x}_{n};c,{b}_{1},{b}_{2}\right),$$
(3)
$$c={\displaystyle \sum _{n}{w}_{n}h\left({x}_{n};{b}_{1},{b}_{2}\right)}/{\displaystyle \sum _{n}{h}^{2}}\left({x}_{n};{b}_{1},{b}_{2}\right),$$
(4)
$${w}_{n}=g\left({x}_{n};\gamma ,{\beta}_{1},{\beta}_{2}\right)+\upsilon ,$$
(5)
$$\begin{array}{lll}-2{\displaystyle \sum _{n}{d}_{n}\frac{\partial {g}_{n}}{\partial c}},\hfill & -2{\displaystyle \sum _{n}{d}_{n}\frac{\partial {g}_{n}}{\partial {b}_{k}}};\hfill & k=\mathrm{1,2},\hfill \end{array}$$
(6)
$$2c\left(1+\rho \right){f}^{2}\left({x}_{n}-b\right),$$
(7)
$${H}_{2}=\text{diag}\left({H}_{1}\phantom{\rule{0.2em}{0ex}}\mu \right),$$
(8)
$$\mu =-4\stackrel{\u02c6}{c}\left[\frac{1-\rho}{1+\rho}\phantom{\rule{0.2em}{0ex}}{\displaystyle \sum _{n}{d}_{n}{\left({f}_{n}^{\left(1\right)}\right)}^{2}}+{\displaystyle \sum _{n}{d}_{n}\phantom{\rule{0.2em}{0ex}}{f}_{n}\phantom{\rule{0.2em}{0ex}}{f}_{n}^{\left(2\right)}}\right],$$
(9)
$$-\frac{8\stackrel{\u02c6}{c}}{1+\rho}\phantom{\rule{0.2em}{0ex}}{\displaystyle \sum _{n}{d}_{n}}{\left({f}_{n}^{\left(1\right)}\right)}^{2}.$$