## Influence of multiple scattering on the resolution of an imaging system: a Cramér-Rao analysis

Optics Express, Vol. 15, Issue 3, pp. 1340-1347 (2007)

http://dx.doi.org/10.1364/OE.15.001340

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### 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

## 1. Introduction

*α*between the spheres centers. The variance of the estimate

*α*̂ obtained from the data of the diffracted intensity is calculated for several values of

*α*, radius and permittivity of the spheres, the latter being chosen to generate more or less interaction between the spheres.

## 2. The Cramér-Rao bound

**I**

*=[*

^{m}*I*

^{m}_{1}, ..

*I*] the vector of the

_{N}^{m}*N*measured far-field intensities at different angles in one realization of the experiment. We denote

*α*the unknown parameter of the scattering object to be estimated. In our experiment

*α*is the center-to-center distance of two homogeneous identical dielectric spheres of given diameter

*d*and optical index

*n*. The Cramér-Rao analysis also applies to the multi-parameter case, but for sake of simplicity we will restrict ourselves to a single parameter. Now, the measurement of the scattered intensities

**I**

*has limited accuracy since it is affected by various sources of noise. We denote*

^{m}**I**

*=[*

^{t}*I*

_{1}

*, ..*

^{t}*I*] the “true” values of the intensity, that is the values that would be measured in a noiseless experimental system. An estimator

_{N}^{t}*α*̂ for the unknown parameter can only be derived on the basis of the noisy data. Hence, the variance of this estimator depends drastically on the level of noise in the measurement. The Cramér-Rao inequality sets an absolute lower bound for this variance in the case of an unbiased estimator:

*L*(

*α*), which is the logarithm of the density of probability of measuring the values

**I**

*for a given value of the parameter*

^{m}*α*,and its derivative

*∂*with respect to the parameter

_{α}L*α*. We see from Eq. (1) that the CRB is a measure of the sensitivity of the experiment to a parameter. Indeed, a sharp variation of the detected intensity, as the parameter

*α*varies, results in large derivative of the log-likelihood function and a small CRB. More explicit calculation requires a model of noise. We will adopt some reasonable assumptions that render the analysis tractable.

**Gaussian additive noise**The source-independent background noise is often modelled as a Gaussian additive noise. We therefore write the measures

*I*as:

^{m}_{j}*I*

_{0}is a constant intensity and

*δ*the Kronecker symbol. With these hypotheses the CRB can be easily derived:

_{ij}**Multiplicative noise**Another cause of noise in an optical or micro-wave experiment is the fluctuation of the source. This is in general modelled by a Gamma distributed multiplicative noise:

*α*as the square root of the relative minimal variance predicted by the Cramér-Rao bound:

## 3. A scattering experiment

*d*and optical index

*n*are placed in vacuum. They are separated by a distance

*α*and aligned along the

**x**̂- or

**z**̂-axis. An incoming plane wave with wave vector

**K**

_{0}=2

*π*/

*λ*

**z**̂ illuminates the spheres. The plane of observation is (

**x**̂,

**z**̂) and the scattering angle

*θ*defines the direction of observation

_{j}**K**̂

*=cos*

_{j}*θ*

_{j}**z**̂-sin

*θ*

_{j}**x**̂. This incident wave is polarized along

**x**̂ or

**y**̂. The scattered intensity is recorded in a 30 degrees aperture cone around the forward or around the backward direction. The different configurations with their nomenclatures are depicted on Fig. 1. These configurations have been chosen to enhance or disminish the role of single scattering in the estimation of

*α*.

### 3.1. Single scattering analysis

**K**̂ direction,

*I*(

**K**̂), is given by the classical interference formula:

*I*(

_{s}**K**̂) is the scattering intensity of one single sphere,

**u**̂ is the direction of alignment of the spheres. Assuming a large number of angular observations, so that the discrete summation can be replaced by a continous one, we obtain a simple analytical expression for the CRB in the case of a multiplicative noise:

*I*is cancelled out in the calculation. A similar formula can be derived in the additive case, assuming in addition that

_{s}*I*is quasi-constant in the angular region of observation (this is actually the case for small spheres):

_{s}*d*=0.06

*λ*). Figure 2 shows the evolution of the inverse relative CRB as a function of the normalized interdistance

*α*/

*λ*for the three scattering configurations depicted on Fig. 1 (

*C*

_{1}and

*C*

_{2}give almost identical results) and the two types of noise. It appears from these plots that the configurations B,C and A listed in this order correspond to an increasing degree of resolution. This is in agreement with the widely accepted rule that finer details of the scatterer can be obtained as the Ewald vector

**K**-

**K**

_{0}is increased, since this corresponds to the spatial frequency probed by the interrogating wave in the Rayleigh or Born regime. Indeed we recall that:

*α*=

*λ*/4+

*pλ*/2, where

*p*is an integer, for which the scattered intensity in the backward

*z*direction is null due to destructive interference. Thus, the noise is also null for this direction.

9. Y.L. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. **34**,4573–4588 (1995). [CrossRef] [PubMed]

### 3.2. Multiple scattering analysis

*μ*

^{2}/

*L*has been set to 1. We consider two incident polarizations, orthogonal (polarization 1, (3a)) or parallel (polarization 2, (3b)) to the plane of observation. Not surprisingly, we observe that as

*α*/

*λ*is increased, the CRB obtained in the rigorous experiments rejoins that obtained when the interaction between the spheres is neglected. On the contrary, when the coupling between the spheres becomes important, the CRB gets lower than that given by the single scattering analysis. We recall that, for a given

*α*, the electromagnetic interaction between the spheres is more important in polarization 1 than in polarization 2 [10

10. P.C. Chaumet and M. Nieto-Vesperinas, “Optical binding of particles with or without the presence of a flat dielectric surface,” Phys. Rev. B **64**,035422–035429 (2001). [CrossRef]

*d*=0.3

*λ*). The analysis of the CRB in this case (Fig. 5) stresses even more the influence of the configurations. Indeed, in configuration A (Fig. 5b) the amelioration of the resolution due to multiple scattering is patent. Moreover, we notice by comparing Fig. (4b) and (5b) that the estimation of

*α*can be more accurate for larger spheres than for smaller ones. This result confirms the study of Belkebir et al [3

3. K Belkebir, A Sentenac, and PC Chaumet, “Influence of multiple scattering on three-dimensional imaging with optical diffraction tomography,” J. Opt. Soc. Am. A **23**,586–595 (2006). [CrossRef]

## 4. Conclusion

## Acknowledgements

## References and links

1. | F. Simonetti, “Multiple scattering: The key to unravel the subwavelength world from the far-field pattern of a scattered wave,” Phys. Rev. E |

2. | F.C. Chen and W.C. Chew, “Experimental verification of super resolution in nonlinear inverse scattering,” Appl. Phys. Lett. |

3. | K Belkebir, A Sentenac, and PC Chaumet, “Influence of multiple scattering on three-dimensional imaging with optical diffraction tomography,” J. Opt. Soc. Am. A |

4. | P. Blomgren and G. Papanicolaou, “Super-resolution in time-reversal acoustics,” J. Acoust. Soc. Am. |

5. | C. Prada and J.L. Thomas, “Experimental subwavelength localization of scatterers by decomposition of time-reversal operator,” J. Acoust. Soc. Am. |

6. | M. Shahram and P. Milanfar, “Imaging below the diffraction limit: A statistical analysis,” IEEE Trans. Image Proc. |

7. | S. Van Aert, D. Van Dyck, and A.J. den Dekker, “Resolution of coherent and incoherent imaging systems reconsidered - classical criteria and a statistical alternative,” Opt. Express |

8. | Refregier P., “Noise theory and application to physics from fluctuation to information,” chapter statistical estimation,167–205, springer 2004. |

9. | Y.L. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. |

10. | P.C. Chaumet and M. Nieto-Vesperinas, “Optical binding of particles with or without the presence of a flat dielectric surface,” Phys. Rev. B |

**OCIS Codes**

(110.0180) Imaging systems : Microscopy

(290.4020) Scattering : Mie theory

(290.4210) Scattering : Multiple scattering

**ToC Category:**

Scattering

**History**

Original Manuscript: October 27, 2006

Manuscript Accepted: December 6, 2006

Published: February 5, 2007

**Virtual Issues**

Vol. 2, Iss. 3 *Virtual Journal for Biomedical Optics*

**Citation**

Anne Sentenac, Charles-Antoine Guérin, Patrick C. Chaumet, Filip Drsek, Hugues Giovannini, Nicolas Bertaux, and Matthias Holschneider, "Influence of multiple scattering on the resolution of an imaging system: a Cramér-Rao analysis," Opt. Express **15**, 1340-1347 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-3-1340

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### References

- F. Simonetti, "Multiple scattering: The key to unravel the subwavelength world from the far-field pattern of a scattered wave," Phys. Rev. E 73, (2006). [CrossRef]
- F. C. Chen and W. C. Chew, "Experimental verification of super resolution in nonlinear inverse scattering," Appl. Phys. Lett. 72, 3080-3082 (1998). [CrossRef]
- K. Belkebir, A. Sentenac, and P. C. Chaumet, "Influence of multiple scattering on three-dimensional imaging with optical diffraction tomography," J. Opt. Soc. Am. A 23, 586-595 (2006). [CrossRef]
- P. Blomgren and G. Papanicolaou, "Super-resolution in time-reversal acoustics," J. Acoust. Soc. Am. 111, 230 - 248 (2002). [CrossRef] [PubMed]
- C. Prada and J. L. Thomas, "Experimental subwavelength localization of scatterers by decomposition of timereversal operator," J. Acoust. Soc. Am. 114, 235 - 243 (2003). [CrossRef] [PubMed]
- M. Shahram and P. Milanfar, "Imaging below the diffraction limit: A statistical analysis," IEEE Trans. Image Process 13, 677 - 689 (2004). [CrossRef]
- S. Van Aert, D. Van Dyck, and A. J. den Dekker, "Resolution of coherent and incoherent imaging systems reconsidered - classical criteria and a statistical alternative," Opt. Express 14, 3830-3839 (2006). [CrossRef] [PubMed]
- P. Refregier, "Noise theory and application to physics from fluctuation to information," chapter statistical estimation, (Springer 2004) pp.167 205.
- Y. L. Xu, "Electromagnetic scattering by an aggregate of spheres," Appl. Opt. 34, 4573-4588 (1995). [CrossRef] [PubMed]
- P. C. Chaumet and M. Nieto-Vesperinas, "Optical binding of particles with or without the presence of a flat dielectric surface," Phys. Rev. B 64, 035422-035429 (2001). [CrossRef]

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