## Abstract

We revisit the notion of resolution of an imaging system in the light of a probabilistic concept, the Cramér-Rao bound (CRB). We show that the CRB provides a simple quantitative estimation of the accuracy one can expect in measuring an unknown parameter from a scattering experiment. We then investigate the influence of multiple scattering on the CRB for the estimation of the interdistance between two objects in a typical two-sphere scattering experiments. We show that, contrarily to a common belief, the occurence of strong multiple scattering does not automatically lead to a resolution enhancement.

©2007 Optical Society of America

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### Equations (12)

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(1)
$$E\left[{\left(\hat{\alpha}-\alpha \right)}^{2}\right]\ge {\left(E{\left[{\partial}_{\alpha}L\left(\alpha \right)\right]}^{2}\right)}^{-1},$$
(2)
$${I}_{j}^{m}={I}_{j}^{t}+{N}_{j},j=1,..,N,$$
(3)
$${N}_{j}~N\left(0,{I}_{0}^{2}\right),E\left[{N}_{i}{N}_{j}\right]={I}_{0}^{2}{\delta}_{\mathrm{ij}},$$
(4)
$$\mathrm{CRB}={I}_{0}^{2}{\left(\sum _{j=1}^{N}{\left[{\partial}_{\alpha}{I}_{j}^{t}\right]}^{2}\right)}^{-1}$$
(5)
$${I}_{j}^{m}={I}_{j}^{t}{N}_{j},j=1,..,N.$$
(6)
$${N}_{j}~\Gamma (\mu ,L),E\left[{N}_{i}{N}_{j}\right]=\frac{{\mu}^{2}}{L}{\delta}_{\mathrm{ij}}+{\mu}^{2}.$$
(7)
$$\mathrm{CRB}=\frac{{\mu}^{2}}{L}{\left(\sum _{j=1}^{N}{\left[\frac{{\partial}_{\alpha}{I}_{j}^{t}}{{I}_{j}^{t}}\right]}^{2}\right)}^{-1}$$
(8)
$$\mathrm{Resolution}\left(\alpha \right)=\sqrt{\frac{\mathrm{CRB}}{{\alpha}^{2}}}$$
(9)
$$I\left(\hat{\mathbf{K}}\right)=2{I}_{s}\left(\hat{\mathbf{K}}\right)\left(1+\mathrm{cos}\left[\alpha \Phi \left(\hat{\mathbf{K}}\right)\right]\right)$$
(10)
$${\mathrm{CRB}}^{-1}\simeq \mathrm{const}\times {\int}_{{\Phi}_{min}}^{{\Phi}_{max}}\frac{{\Phi}^{2}{\mathrm{sin}}^{2}\left(\alpha \Phi \right)}{{\mid 1+\mathrm{cos}\left(\alpha \Phi \right)\mid}^{2}}\mathrm{d\Phi}$$
(11)
$${\mathrm{CRB}}^{-1}\simeq \mathrm{const}\times {I}_{s}{\int}_{{\Phi}_{min}}^{{\Phi}_{max}}{\Phi}^{2}{\mathrm{sin}}^{2}\left(\alpha \Phi \right)\mathrm{d\Phi}$$
(12)
$$I\left(\hat{\mathbf{K}}\right)~{\mid \hat{\chi}\left(\mathbf{K}-{\mathbf{K}}_{0}\right)\mid}^{2},$$