## Resolution of coherent and incoherent imaging systems reconsidered - Classical criteria and a statistical alternative

Optics Express, Vol. 14, Issue 9, pp. 3830-3839 (2006)

http://dx.doi.org/10.1364/OE.14.003830

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### Abstract

The resolution of coherent and incoherent imaging systems is usually evaluated in terms of classical resolution criteria, such as Rayleigh’s. Based on these criteria, incoherent imaging is generally concluded to be ‘better’ than coherent imaging. However, this paper reveals some misconceptions in the application of the classical criteria, which may lead to wrong conclusions. Furthermore, it is shown that classical resolution criteria are no longer appropriate if images are interpreted quantitatively instead of qualitatively. Then one needs an alternative criterion to compare coherent and incoherent imaging systems objectively. Such a criterion, which relates resolution to statistical measurement precision, is proposed in this paper. It is applied in the field of electron microscopy, where the question whether coherent high resolution transmission electron microscopy (HRTEM) or incoherent annular dark field scanning transmission electron microscopy (ADF STEM) is preferable has been an issue of considerable debate.

© 2006 Optical Society of America

## 1. Introduction

## 2. Classical resolution criteria

*t*(

**r**) the coherent point spread function, δ(

**r**) the Dirac delta function representing the scattering distribution of a point object,

**r**= (

*x,y*) a two-dimensional vector in the image plane,

**β**

_{1}and

**β**

_{2}the positions of the components, ϕ the relative phase between the two components, and * the convolution operator. Using the same imaging system in the incoherent mode, the model becomes:

*t*(

**r**)|

^{2}.

*phase distribution*associated with these sources. When the sources are in quadrature (ϕ =

*π*/2), the image intensity distribution is identical to that resulting from incoherent point sources. When the sources are in phase (ϕ = 0), the dip in the image intensity distribution is absent and therefore the points are not as well resolved as for incoherent illumination. Finally, when the two point sources are in phase opposition (ϕ =

*π*), the dip is greater than 19%, and the two point sources are resolved better with coherent illumination than with incoherent illumination.

8. P. D. Nellist and S. J. Pennycook, “Accurate structure determination from image reconstruction in ADF STEM,” J. Microsc. **190**, 159–170 (1998). [CrossRef]

9. D. Van Dyck and M. Op de Beeck, “A simple intuitive theory for electron diffraction,” Ultramicroscopy **64**, 99–107 (1996). [CrossRef]

10. S. Van Aert, A. J. den Dekker, A. van den Bos, and D. Van Dyck, “Statistical experimental design for quantitative atomic resolution transmission electron microscopy,” in *Advances in Imaging and Electron Physics*P. W. Hawkes, ed. (Academic Press, San Diego, 2004), Vol. 130, pp. 1–164. [CrossRef]

11. A. J. den Dekker, S. Van Aert, D. Van Dyck, A. van den Bos, and P. Geuens, “Does a monochromator improve the precision in quantitative HRTEM?,” Ultramicroscopy **89**, 275–290 (2001). [CrossRef]

*n*

_{c}represents the total number of atom columns being imaged, the function φ

_{1s,n}(

**r**-

**β**

_{n}) is the lowest energy bound state of the nth atom column located at position

**β**

_{n},

*t*(

**r**) is the point spread function of the electron microscope,

*a*

_{n}is a complex coefficient depending among other parameters on the thickness

*z*of the material. From the comparison of Eq. (3) with Eq. (1), it may be concluded that the Dirac delta function is now replaced with φ

_{1s,n}(

**r**), which is a function of finite size. This expresses the fact that atoms may not be considered to be point scatterers. The most important difference is the presence of the term ‘1’ in Eq. (3), which is absent in Eq. (1). It represents an unscattered wave, which in classical terms means that part of the incident electrons will not scatter in the material, but may interfere with the scattered electrons. Let us now consider ADF STEM assuming the same material being imaged using a microscope with the same lens characteristics as in HRTEM. If the atom column approximation is made, the model is given by [10

10. S. Van Aert, A. J. den Dekker, A. van den Bos, and D. Van Dyck, “Statistical experimental design for quantitative atomic resolution transmission electron microscopy,” in *Advances in Imaging and Electron Physics*P. W. Hawkes, ed. (Academic Press, San Diego, 2004), Vol. 130, pp. 1–164. [CrossRef]

12. S. Van Aert, A. J. den Dekker, D. Van Dyck, and A. van den Bos, “Optimal experimental design of STEM measurement of atom column positions,” Ultramicroscopy **90**, 273–289 (2002). [CrossRef] [PubMed]

13. S. J. Pennycook, B. Rafferty, and P. D. Nellist, “Z-contrast imaging in an aberration-corrected scanning transmission electron microscope,” Microsc. Microanal. **6**, 343–352 (2000). [PubMed]

*A*

_{n}a factor depending among other parameters on the thickness z and on the detector geometry. From the comparison of Eq. (4) with Eq. (2), it may be concluded that these models are equivalent apart from the replacement of the Dirac delta function with a function of finite size. Note that in the derivation of Eq. (4), only elastic scattering has been considered and not thermal diffuse, inelastic scattering. The incoherent characteristics of the model are therefore not caused by incoherent scattering events but are purely created by the detector geometry [13

13. S. J. Pennycook, B. Rafferty, and P. D. Nellist, “Z-contrast imaging in an aberration-corrected scanning transmission electron microscope,” Microsc. Microanal. **6**, 343–352 (2000). [PubMed]

14. O. Scherzer, “The theoretical resolution limit of the electron microscope,” J. Appl. Phys. **20**, 20–28 (1949). [CrossRef]

14. O. Scherzer, “The theoretical resolution limit of the electron microscope,” J. Appl. Phys. **20**, 20–28 (1949). [CrossRef]

## 3. Attainable precision

*attainable*precision can be adequately quantified in the form of the so-called

*Cramér-Rao lower bound*(CRLB). This is a lower bound on the variance of any unbiased estimator of a parameter. The meaning of this lower bound is as follows. One can use different parameter estimation methods in order to estimate unknown parameters, such as the least squares or the maximum likelihood (ML) estimator. The precision of an estimator is represented by the variance or by its square root, the standard deviation. Generally, different estimators will have different precisions. It can be shown, however, that the variance of unbiased estimators will never be lower than the CRLB. Fortunately, there exists a class of estimators (including the ML estimator) that achieves this bound at least asymptotically, that is, for the number of observations going to infinity. A summary of the different steps involved to compute the CRLB is given below. A more detailed description may be found in [16

16. A. van den Bos and A. J. den Dekker, “Resolution reconsidered - Conventional approaches and an alternative,” in *Advances in Imaging and Electron Physics*P. W. Hawkes, ed. (Academic Press, San Diego, 2001), Vol. 117, pp. 241–360. [CrossRef]

17. A. J. den Dekker, S. Van Aert, A. van den Bos, and D. Van Dyck, “Maximum likelihood estimation of structure parameters from high resolution electron microscopy images. Part I: A theoretical framework,” Ultramicroscopy **104**, 83–106 (2005). [CrossRef]

*P*(

*ω*;

*θ*) that a set of observations

*w*= (

*w*

_{1}…

*w*

_{M})

^{T}is equal to

*ω*= (

*ω*

_{1}…

*ω*

_{M})

^{T}is given by:

*λ*the expectation of the observation

_{m}*w*. In Eq. (5), it is supposed that these expectations are described by an expectation model, that is, a physical model, which contains the parameters

_{m}*θ*to be estimated, such as the

*x*- and

*y*- coordinates of the positions

**β**

_{1}and

**α**

_{2}of components or atom columns. As follows from section 2, such a model exists for coherent and incoherent imaging systems. It is given by:

*I*(

**r**) given by Eqs. (1)–(4),

**r**

_{m},

*m*= 1,…,M the measurement points,

*S*,

_{m}*m*= 1,…,

*M*the area of these measurement points,

*N*the total number of detected counts in an image, and C a normalization factor so that the integral of the function

*I*(

**r**)/

*C*is equal to one.

*Fisher information matrix F*with respect to the elements of the

*T*× 1 parameter vector

*θ*= (

*θ*

_{1}…

*θ*

_{T})

^{T}is introduced. It is defined as the

*T*×

*T*matrix

*P*(

*ω*;

*θ*) is the joint probability density function of the observations

*w*= (

*w*

_{1}…

*w*

_{M})

^{T}. The expression between square brackets represents the Hessian matrix of ln

*P*, for which the (

*r, s*)th element is defined by

*∂*

^{2}ln

*P*(

*ω*;

*θ*)/

*∂*

*θ*∂

_{r}*θ*. For independent, Poisson distributed observations, where

_{s}*P*(

*ω*;

*θ*) is given by Eq. (5), it follows that the (

*r,s*)th element of

*F*is equal to:

*x*- and

*y*-coordinates of the positions

*β*

_{1}and

**β**

_{2}. Assuming that these position coordinates are the only parameters to be estimated, the parameter vector

*θ*is given by

*θ*= (

*β*

_{x1}

*β*

_{x2}

*β*

_{y1}

*β*

_{y2})

^{T}. An expression for the elements of the Fisher information matrix is found by substitution of the expectation model given by Eq. (6) and its derivatives with respect to the unknown parameters into Eq. (8):

*I*(

**r**

_{m}) the image model given by Eqs. (1)–(4). Recall that the model I(rm) depends, among other things, on the relative phase between two points and on the spherical aberration constant and defocus determining the point spread function. Furthermore, the number of detected counts

*N*depends on the brightness of the source, the source diameter, and the recording time. Therefore, the elements

*F*

_{rs}, given by Eq. (9), will depend on all these microscope settings as well. Explicit numbers for these elements are obtained by substituting values of a given set of microscope settings and position coordinates of the components or atom columns into Eq. (9).

*cov*(

*θ*̂) of any unbiased estimator

*θ*̂ of

*θ*satisfies:

*cov*(

*θ*̂) and

*F*

^{-1}is positive semidef-inite. Since the diagonal elements of

*cov*(

*θ*̂) represent the variances of

*θ*̂

_{1},&,

*θ*̂

_{T}and since the diagonal elements of a positive semidefinite matrix are nonnegative, these variances are larger than or equal to the corresponding diagonal elements of

*F*

^{-1}:

*r*= 1,…,

*T*and [

*F*

^{-1}]

_{rr}is the (

*r,r*)th element of the inverse of the Fisher information matrix. In this sense,

*F*

^{-1}represents a lower bound to the variances of all unbiased

*θ*̂. The matrix

*F*

^{-1}is called the CRLB on the variance of

*θ*̂.

*γ*(

*θ*) = (

*γ*

_{1}(θ)…

*γ*(θ))

_{c}^{T}be such a vector and let

*γ*̂be an unbiased estimator of

*γ*(

*θ*). Then, it can be shown that

*∂*

_{γ}/∂

*θ*

^{T}is the

*C*×

*T*Jacobian matrix defined by its (

*r,s*)th element

*∂*

*γ*

_{r}/

*∂*

*θ*

_{s}. The right-hand member of this inequality is the CRLB on the variance of

*γ*Į. It can be used to compute the scalar valued CRLB on the variance of unbiased estimators of the distance

*x*- and

*y*-coordinates of the positions

**β**

_{1}and

**β**

_{2}. Equation (12) is then equal to:

*N*. This means that the distance

*δ*can be estimated more precisely if the dose increases. The final expression defining the CRLB on the variance of unbiased estimators of the distance

*δ*, given by the right-hand member of Eq. (13), can be considered as an alternative, nowadays more meaningful, criterion of resolution.

## 4. Conclusions

*π*, coherent imaging is preferred. In terms of precision the conclusion is just the contrary. Moreover, we applied both Rayleigh’s resolution criterion and the precision based alternative in the field of electron microscopy, comparing coherent HRTEM and incoherent ADF STEM. In terms of Rayleigh we found that depending on the material thickness, HRTEM may be preferable even in the absence of a phase difference between neighboring atom columns. In terms of precision we found that HRTEM is usually preferable, except for fields of view smaller than a few squared nanometers.

## Acknowledgments

## References and links

1. | L. Rayleigh, “Wave theory of light,” in |

2. | J. W. Goodman, |

3. | A. J. den Dekker and A. van den Bos, “Resolution: A survey,” J. Opt. Soc. Am. A |

4. | V. Ronchi, “Resolving power of calculated and detected images,” J. Opt. Soc. Am. |

5. | L. Rayleigh, “On the theory of optical images, with special reference to the microscope,” in |

6. | J. C. H. Spence, |

7. | S. J. Pennycook and Y. Yan, “Z-contrast imaging in the scanning transmission electron microscope,” in Progress in transmission electron microscopy 1 - Concepts and techniques,X.-F. Zhang and Z. Zhang, eds. (Springer-Verlag, Berlin, 2001), pp. 81–111. |

8. | P. D. Nellist and S. J. Pennycook, “Accurate structure determination from image reconstruction in ADF STEM,” J. Microsc. |

9. | D. Van Dyck and M. Op de Beeck, “A simple intuitive theory for electron diffraction,” Ultramicroscopy |

10. | S. Van Aert, A. J. den Dekker, A. van den Bos, and D. Van Dyck, “Statistical experimental design for quantitative atomic resolution transmission electron microscopy,” in |

11. | A. J. den Dekker, S. Van Aert, D. Van Dyck, A. van den Bos, and P. Geuens, “Does a monochromator improve the precision in quantitative HRTEM?,” Ultramicroscopy |

12. | S. Van Aert, A. J. den Dekker, D. Van Dyck, and A. van den Bos, “Optimal experimental design of STEM measurement of atom column positions,” Ultramicroscopy |

13. | S. J. Pennycook, B. Rafferty, and P. D. Nellist, “Z-contrast imaging in an aberration-corrected scanning transmission electron microscope,” Microsc. Microanal. |

14. | O. Scherzer, “The theoretical resolution limit of the electron microscope,” J. Appl. Phys. |

15. | L. J. van Vliet, F. R. Boddeke, D. Sudar, and I. T. Young, “Image detectors for digital image microscopy,” in |

16. | A. van den Bos and A. J. den Dekker, “Resolution reconsidered - Conventional approaches and an alternative,” in |

17. | A. J. den Dekker, S. Van Aert, A. van den Bos, and D. Van Dyck, “Maximum likelihood estimation of structure parameters from high resolution electron microscopy images. Part I: A theoretical framework,” Ultramicroscopy |

**OCIS Codes**

(000.2690) General : General physics

(000.5490) General : Probability theory, stochastic processes, and statistics

(030.1640) Coherence and statistical optics : Coherence

(030.4280) Coherence and statistical optics : Noise in imaging systems

(350.5730) Other areas of optics : Resolution

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: January 27, 2006

Revised Manuscript: March 16, 2006

Manuscript Accepted: April 19, 2006

Published: May 1, 2006

**Virtual Issues**

Vol. 1, Iss. 6 *Virtual Journal for Biomedical Optics*

**Citation**

Sandra Van Aert, Dirk Van Dyck, and Arnold J. den Dekker, "Resolution of coherent and incoherent imaging systems reconsidered - Classical criteria and a statistical alternative," Opt. Express **14**, 3830-3839 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-9-3830

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### References

- L. Rayleigh, "Wave theory of light," in Scientific papers by Lord Rayleigh, John William Strutt, (Cambridge University Press, Cambridge, 1902), Vol. 3, pp. 47-189.
- J. W. Goodman, Introduction to fourier optics (McGraw-Hill, San Francisco, 1968).
- A. J. den Dekker and A. van den Bos, "Resolution: A survey," J. Opt. Soc. Am. A 14,547-557 (1997). [CrossRef]
- V. Ronchi, "Resolving power of calculated and detected images," J. Opt. Soc. Am. 51,458-460 (1961). [CrossRef]
- L. Rayleigh, "On the theory of optical images, with special reference to the microscope," in Scientific papers by Lord Rayleigh, John William Strutt, (Cambridge University Press, Cambridge, 1903), Vol. 4, pp. 235-260.
- J. C. H. Spence, High-resolution electron microscopy, 3rd edition (Oxford University Press, New York, 2003).
- S. J. Pennycook and Y. Yan, "Z-contrast imaging in the scanning transmission electron microscope," in Progress in transmission electron microscopy 1 - Concepts and techniques, X.-F. Zhang and Z. Zhang, eds. (Springer-Verlag, Berlin, 2001), pp. 81-111.
- P. D. Nellist and S. J. Pennycook, "Accurate structure determination from image reconstruction in ADF STEM," J. Microsc. 190,159-170 (1998). [CrossRef]
- D. Van Dyck and M. Op de Beeck, "A simple intuitive theory for electron diffraction," Ultramicroscopy 64,99-107 (1996). [CrossRef]
- S. Van Aert, A. J. den Dekker, A. van den Bos, and D. Van Dyck, "Statistical experimental design for quantitative atomic resolution transmission electron microscopy," in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic Press, San Diego, 2004), Vol. 130, pp. 1-164. [CrossRef]
- A. J. den Dekker, S. Van Aert, D. Van Dyck, A. van den Bos, and P. Geuens, "Does a monochromator improve the precision in quantitative HRTEM?," Ultramicroscopy 89,275-290 (2001). [CrossRef]
- S. Van Aert, A. J. den Dekker, D. Van Dyck, and A. van den Bos, "Optimal experimental design of STEM measurement of atom column positions," Ultramicroscopy 90,273-289 (2002). [CrossRef] [PubMed]
- S. J. Pennycook, B. Rafferty, and P. D. Nellist, "Z-contrast imaging in an aberration-corrected scanning transmission electron microscope," Microsc. Microanal. 6,343-352 (2000). [PubMed]
- O. Scherzer, "The theoretical resolution limit of the electron microscope," J. Appl. Phys. 20,20-28 (1949). [CrossRef]
- L. J. van Vliet, F. R. Boddeke, D. Sudar, and I. T. Young, "Image detectors for digital image microscopy," in Digital image analysis of microbes; Imaging, morphometry, fluorometry and motility techniques and applications, modern microbiological methods, M. H. F.Wilkinson and F. Schut, eds. (JohnWiley and Sons, Chichester (UK), 1998), pp. 37-64.
- A. van den Bos and A. J. den Dekker, "Resolution reconsidered - Conventional approaches and an alternative," in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic Press, San Diego, 2001), Vol. 117, pp. 241-360. [CrossRef]
- A. J. den Dekker, S. Van Aert, A. van den Bos, and D. Van Dyck, "Maximum likelihood estimation of structure parameters from high resolution electron microscopy images. Part I: A theoretical framework," Ultramicroscopy 104,83-106 (2005). [CrossRef]

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