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1252 S. Van Aert et al.

addition, the atom columns were spotted at different locations in the course

of time.

The ML method is, of course, also applicable to other types of

experiments for the estimation of unknown parameters. An interesting

application to EELS spectra can be found in Verbeeck and Van Aert

(2004).

5.4 Statistical Experimental Design

In this section, it will be shown how the statistical model-based reso-

lution of a quantitative atomic resolution TEM experiment may be

improved by optimizing its design.

Usually, instrumental developments in microscopy, telescopy, or

even spectrometry aim at improving the Rayleigh resolution or

Rayleigh-like resolution criteria. Examples of such developments in

electron microscopy are the spherical aberration corrector (Rose, 1990;

Haider et al., 1998) and the monochromator (Mook and Kruit, 1999). A

resolution (in the sense of Rayleigh) beyond 1 Å allows microscopists

to visually distinguish atom columns of solids oriented along a main

zone axis. In addition, electron microscopes now have increased ver-

satility of microscope settings, which will preferably be computer con-

trollable in the future. Electron microscopists can choose between TEM

or STEM, between imaging or diffraction techniques, focus, accelerat-

ing voltage, spherical aberration, energy spread of incoming electrons,

beam tilt, and crystal tilt. The main constraints of the experiment will

be the radiation sensitivity of the object or the specimen drift. The

question then arises as to which microscope and which settings are

optimal in keeping the incident electron dose per square ångstrom or

the recording time subcritical. In this section this question will be

addressed using statistical experimental design. Statistical experimen-

tal design can be deﬁ ned as the selection of free experimental settings

in an experiment to improve the precision with which unknown

parameters can be estimated. For quantitative atomic resolution TEM

experiments, these parameters are the atom or atom column positions

in particular. The optimal design is then given by the combination of

experimental settings for which the precision of the atom or atom

column positions is highest. In the optimization of the experimental

design, the CRLB is taken as the optimal criterion. This criterion thus

aims at an improvement of statistical model-based resolution rather

than an improvement of Rayleigh resolution or Rayleigh-like resolution

criteria. Note that the optimal statistical experimental design may be

different for different objects under study.

In Section 5.2, it was shown how from the joint probability density

function, which was introduced in Section 5.1, the elements of the

Fisher information matrix may be calculated explicitly. From the latter,

the CRLB on the variance of the parameters of the expectation model

and on the variance of functions of these parameters may be computed

from the right-hand side of Eqs. (27) and (29), respectively. The diago-

nal elements of the CRLB give a lower bound on the variance of any

Chapter 20 The Notion of Resolution 1253

unbiased estimator of the parameters. Since the joint probability density

function is a function of the experimental settings, the CRLB is a func-

tion of these settings as well. Therefore, the CRLB may be used to

evaluate and to optimize the experimental design in terms of precision.

However, simultaneous minimization of the diagonal elements of the

CRLB, that is, the right-hand side of Eq. (28), is usually impossible.

Therefore, statistical parameter estimation theory provides different

optimality criteria, which are functions of the elements of the CRLB.

These are scalar measures. Experimenters may choose one of these

criteria or may produce their own criterion, reﬂ ecting their purpose.

The interested reader may ﬁ nd a selection of such criteria in Fedorov

(1972), Pázman (1986), and Van Aert et al. (2004).

Statistical experimental design can be illustrated as follows. Suppose

the microscope is able to visualize individual atoms or atom columns

so that the structure can be resolved. In other words, neighboring

atoms or atom columns can be discriminated in the images. Further-

more, it will be assumed that the expectations of the Poisson distrib-

uted image pixel values can be modeled as a summation of Gaussian

peaks centered at the atom or atom column positions. The position

coordinates of these atoms or atom columns are the unknown param-

eters in the expectation model. It then follows from Eq. (31) that the

lower bound on the standard deviation of these coordinate estimates,

that is, the square root of the CRLB, is approximately given by

22N . In this expression, ρ

represents the Rayleigh resolution,

which depends on the Rayleigh resolution of all different subchannels

contributing to the imaging process as follows from Eq. (7) and N rep-

resents the number of detected electrons per atom or atom column. It

is now clear that it is not only the Rayleigh resolution that matters but

the electron dose as well. If the Rayleigh resolution can be improved

only at the expense of a decrease in the number of detected electrons,

both effects have to be balanced under the existing physical constraints

so as to produce the highest precision. Furthermore, it follows from Eq.

(7) that below a certain value for ρ

, it will be useless to further improve

the Rayleigh resolution of the electron microscope since scattering is

dominant or, equivalently, ρ

is almost equal to the width of the elec-

trostatic potential of an atom, that is, ρ

. This result is meaningful in

practice. For example, in STEM experiments, further narrowing of the

probe, which represents the point spread function of the electron

microscope, is not so beneﬁ cial in terms of precision since the width

of the probe is currently almost equal to the width of an atom (Krivanek

et al., 2002).

In the following examples, the main results obtained from the

numerical optimization and evaluation of the design of quantitative

atomic resolution TEM experiments will be presented. Also the possi-

ble beneﬁ ts of new developments in instruments will be discussed. In

these examples, expectation models with a solid physical base, instead

of Gaussian peaks, have been assumed and the observations are

assumed to be independent and Poisson distributed. The results

obtained may intuitively be interpreted using the rules for the CRLB,

which have been obtained from Gaussian peaked expectation models.

1254 S. Van Aert et al.

Comprehensive descriptions of these results can be found in den

Dekker et al. (1999, 2001), Van Aert and Van Dyck (2001), and Van Aert

et al. (2002a–c, 2004).

Example 5 (Statistical Experimental Design of Scanning Transmis-

sion Electron Microscopy) The expectation model that has been used in

the evaluation and optimization of the experimental design of STEM experi-

ments is based on the simpliﬁ ed channelling theory to describe the dynamic,

elastic scattering of the electrons with atom columns (Broeckx et al., 1995;

Geuens and Van Dyck, 2002; Pennycook and Jesson, 1992; Van Aert et al.,

2002b; Van Dyck and Op de Beeck, 1996). Therefore, it has been assumed that

the dynamic motion of an electron in a column may be primarily expressed in

terms of so-called tightly bound 1s states, which are peaked at the atom column

positions. Furthermore, in this model, temporal incoherence due to chromatic

aberration has not been taken into account since STEM imaging seems to be

robust to chromatic aberration (Batson, et al., 2002; Krivanek et al., 2002;

Nellist and Pennycook, 1998, 2000). Thermal diffuse, inelastic scattering has

not been taken into account for the following reasons. Thermal diffuse scattered

electrons are predominantly collected in the detector at high angles (Treacy,

1982). Therefore, increasing the inner detector angle of an annular detector

has the effect of increasing thermal diffuse, inelastic scattering relative to

elastic scattering (Wang, 2001). Usually, the inner detector angle is chosen to

be relatively large since the detected signal then strongly depends on the atomic

number Z, hence the name, Z-contrast imaging. The disadvantage of increas-

ing this detector angle, however, is the accompanied decrease of dose efﬁ ciency,

which leads to a decrease in signal-to-noise ratio (SNR). It can be shown that

as a result of this decrease in SNR, the optimal inner collection angle in terms

of precision is small compared to the angles where thermal diffuse scattering

is important. This justiﬁ es the fact that thermal diffuse scattering has not been

taken into account. The details of the expectation model can be found in Van

Aert et al. (2002b, 2004).

It has been shown in Van Aert and Van Dyck (2001) and Van Aert et al.

(2002b, 2004) that the optimal aperture radius in terms of precision may be

considerably smaller than the aperture radius at the Scherzer conditions for

incoherent imaging, which is referred to as being optimal in terms of direct

visual interpretability (Scherzer, 1949; Pennycook and Jesson, 1991). This

indicates that the optimal width of diffraction-limited probes, for which the

width is inversely proportional to the aperture radius, is not as small as pos-

sible. The optimal objective aperture radius, which determines the size of a

diffraction-limited probe, has been found to be mainly determined by the object

under study. More speciﬁ cally, for isolated atom columns, it is proportional to

the depth of the two-dimensional electrostatic potential of the atom column

projected along the column direction, that is, the difference between the

maximum and minimum potential energy. In terms of the column-speciﬁ c 1s

state, this means that the main lobe of the optimal probe is broader than the

1s state. This is shown in Figure 20–11, where both the 1s state and the ampli-

tude of the optimal probe are shown for a silicon and a gold [100] atom column.

However, in the presence of neighboring columns, it has been found that the

optimal aperture radius may increase so as to avoid strong overlap of neighbor-

ing columns in the image.

Chapter 20 The Notion of Resolution 1255

In the evaluation of the experimental design, annular and axial detector types

have been compared. Apart from some exceptions, an annular detector usually

results in higher precision than an axial one. The optimal inner radius of an

annular detector has been found to be equal to the optimal objective aperture

radius, whereas in a conventional approach the radius of the hole is usually

taken to be much larger than the aperture radius (Pennycook et al., 1995; Hartel

et al., 1996). The optimal outer radius of an axial detector is usually slightly

smaller than the optimal aperture radius. However, if this detector leads to very

low contrast of the image, the optimal detector radius decreases.

Moreover, it has been found that a spherical aberration corrector improves

the precision. The accompanied gain depends on the object under study. Cor-

rection of the spherical aberration is more useful in terms of precision for heavy

than for light atom columns. This is shown in Figures 20–12 and 20–13, where

the ratio of the lower bound on the standard deviation of the position coordi-

nates for a given spherical aberration constant to the lower bound for a spheri-

cal aberration constant of 0 mm is shown as a function of the spherical

aberration constant. This is done for an isolated silicon and gold [100] atom

column, respectively. Moreover, the evaluation is done for an annular as well

as an axial detector. For each spherical aberration constant considered, the

objective aperture radius is set to its optimal value. Furthermore, the detector

radius is taken to be equal to this optimal objective aperture radius. For silicon,

which is a light atom column, it follows from Figure 20–12 that the precision

that is gained by reducing the spherical aberration constant from 0.5 mm to

0 mm is a factor of 1.0009 and 1.0011 for an annular and axial detector,

respectively. These gains are negligible. For gold, which is a heavy atom

column, it follows from Figure 20–13 that the precision that is gained by

reducing the spherical aberration constant from 0.5 mm to 0 mm is a factor of

1.21 and 1.39 for an annular and axial detector, respectively, which is still

-10 -5 0 5 10

0.0

0.4

0.8

1.2

probe

silic o n [100]

am plitude

spatial coordinate (Å)

-10 -5 0 5 10

0.0

0.8

1.6

2.4

3.2

spatial coordinate (Å)

probe

gold [100]

Figure 20–11. The dashed curves represent the 1s state for a silicon [100] (left)

and a gold [100] (right) atom column, respectively. The solid curves represent

the amplitude of their associated optimal STEM probes for C

equal to

0.5 mm.

1256 S. Van Aert et al.

relatively small. This makes it questionable whether such a corrector is needed

to obtain a prespeciﬁ ed precision of the atom column positions.

Example 6 (Statistical Experimental Design of High-Resolution

Transmission Electron Microscopy) The optimal experimental design of

HRTEM experiments has been reconsidered in terms of statistical model-based

resolution (den Dekker et al., 1999, 2001; Van Aert et al., 2004). The expecta-

0.0 0.2 0.4 0.6 0.8 1.0

1.0000

1.0025

1.0050

annular detector

axial detector

ratioofstandard deviations of position coordinate

spherical aberration constant (mm )

Figure 20–12. Ratio of the lower bound on the standard deviation of the posi-

tion coordinates for a given spherical aberration constant to the lower bound

for a spherical aberration constant of 0 mm for an isolated silicon [100] atom

column as a function of the spherical aberration constant for an annular as

well as for an axial STEM detector.

0.0 0.2 0.4 0.6 0.8 1.0

1.0

1.2

1.4

1.6

1.8

annular detector

axial detector

ratioofstandarddevia tio ns of po s ition co ordin ates

spherical aberration constant (mm )

Figure 20–13. Ratio of the lower bound on the standard deviation of the posi-

tion coordinates for a given spherical aberration constant to the lower bound

for a spherical aberration constant of 0 mm for an isolated gold [100] atom

column as a function of the spherical aberration constant for an annular as

well as for an axial STEM detector.

Chapter 20 The Notion of Resolution 1257

tion model that has been used in the evaluation and optimization of HRTEM

experiments is based on the simpliﬁ ed channeling theory (Geuens and Van

Dyck, 2002; Van Dyck and Op de Beeck, 1996). For microscopes operating at

intermediate accelerating voltages of the order of 300 kV, it has been shown

that use of a spherical aberration corrector or a chromatic aberration corrector

is only of limited value and that the use of a monochromator usually is not

worthwhile in terms of precision, presuming that specimen drift places a prac-

tical constraint on the experiment. These results follow from Figure 20–14,

where the lower bound on the standard deviation of the position coordinates

of a gold atom column is evaluated as a function of the spherical aberration

constant. The solid curve corresponds to a microscope without correction for

chromatic aberration, that is, a microscope without a chromatic aberration

corrector and monochromator. The dashed curve corresponds to a microscope

with a chromatic aberration corrector, for which the chromatic aberration

constant is equal to 0 mm. The dotted curve corresponds to a microscope with

a monochromator, for which the energy spread corresponds to a typical full

width at half maximum height of 200 meV (Batson, 1999). In Figure 20–14 it

is assumed that the specimen drift is the relevant physical constraint. Hence,

the recording time is kept constant. Consequently, the total number of detected

electrons is smaller in the presence of a monochromator. The values for the

other microscope settings, that is, for the original, nonoptimized microscope

settings, are in accordance with values that are typical for today’s electron

microscopes. Therefore, it also follows from Figure 20–14 that for today’s

electron microscopes it is in principle possible to obtain a precision of the order

of 0.01 Å, even with a microscope that is not corrected for spherical and chro-

matic aberration.

However, for amorphous instead of crystalline structures, the conclusions

are different since amorphous structures are very sensitive to radiation damage,

Figure 20–14. The lower bound on the standard deviation of the position

coordinates of a gold [100] atom column as a function of the spherical aberra-

tion constant for a microscope operating at 300 kV equipped with or without

a chromatic aberration corrector or monochromator. In this evaluation, the

recording time is kept constant.

1258 S. Van Aert et al.

so that radiation sensitivity rather than specimen drift will be the limiting

factor in the experiment. Although the precision improves by increasing the

incident dose, it has to be taken into account that every incident electron has

a ﬁ nite probability of damaging the structure. The structure can be damaged

either by displacing an atom from its position or by changing chemical bonds

due to ionization. Therefore, a compromise between precision and radiation

damage has to be made, which turns out to be optimal when the accelerating

voltage is reduced. However, at low accelerating voltages, the instrumental

aberrations become important. For this reason, correction of both the spherical

aberration and the chromatic aberration by either a chromatic aberration cor-

rector or a monochromator will be essential. By means of statistical experi-

mental design, it has been shown that a substantial improvement of the

precision may be obtained if both the spherical and chromatic aberration are

corrected (Van Dyck et al., 2003).

6 Ultimate Model-Based Resolution

In the preceding section, the CRLB on the precision of the estimated

distance of two peaks was used as a resolution criterion. Such a crite-

rion makes sense only if an estimator exists that actually achieves this

bound. Such an estimator exists in the form of the so-called ML estima-

tor. This estimator produces estimates that are normally distributed

about the true values of the parameters with a variance equal to the

CRLB. However, this is true only if the number of observations used

by the ML estimator is sufﬁ ciently large. For small numbers of observa-

tions, the estimates have statistical properties that are very different.

A detailed description is given in van den Bos (1992) and van den Bos

and den Dekker (1995, 2001).

6.1 Two Types of Observations

Central in this section is the space of observations, which has been

introduced in Section 5.1. This is the Euclidean space in which every

coordinate direction corresponds to a particular observation. Therefore,

the dimension of the space of observations is equal to the number of

observations and a particular set of observations is represented by a

single point in the space. For example, the observations shown in Figure

20–7 correspond to a single point in the space of observations. It has

been shown in the references mentioned that two types of sets of two-

peak observations may occur. If the two-peak model is ﬁ tted to the ﬁ rst

type of observations, distinct values for the locations are obtained. Then

the distance of the peaks is different from zero. If, on the other hand,

the two-peak model is ﬁ tted to observations of the second type, the

estimates for the locations exactly coincide. Then the distance of the peaks

is equal to zero. This means that the peaks exactly coincide. It is clear

that from the ﬁ rst type of observations both peaks are resolved in the

sense that they occur at distinct locations. On the other hand, for the

second type of observations, the peaks occur at the same location.

Therefore, the peaks add up to form a single peak and cannot be

resolved. From these considerations, it follows that the (hyper)surface

Chapter 20 The Notion of Resolution 1259

separating both types of observations in the space of observations may

be used as a resolution criterion. If the point representing a particular

set of observations is on the one side of the hypersurface, the peaks are

resolved, but if it is on the other side, they are not.

This perhaps unexpected behavior of the solutions for the locations

of the peaks has to do with the structure of criteria of goodness of ﬁ t

for the estimation of parameters such as the locations. The structure is

the pattern of the stationary points of the criterion, where a stationary

point is a point at which the gradient is equal to zero, such as a

minimum or maximum. The structure is important since the solution

of the model ﬁ tting problem is represented by a stationary point: the

absolute optimum of the criterion of goodness of ﬁ t. It has been found

that for observations made on overlapping peaks, two different struc-

tures may occur and that these correspond to distinct or coinciding

solutions for the locations, respectively. Which of these structures

occurs depends on the particular realization of the observations.

6.2 Consequences for Resolution

If the observations are statistical, they are, in the space of the observa-

tions, distributed about the point representing their expectations. This

expectation point, therefore, represents ideal exact observations. If the

model ﬁ tted is correctly speciﬁ ed, it perfectly ﬁ ts these exact observa-

tions. Then the solutions found for the locations are distinct and equal

the hypothetical true locations, no matter how closely located the

peaks. Therefore, the expectation point will be located on the “two-

peak side” of the separating hypersurface and not on the “one-peak

side,” since the solutions are distinct. Observations distributed about

the expectation point will, with a certain probability, only be on the

two-peak side of the separating hyperplane. It is concluded that resolu-

tion should be described in terms of probability of resolution. The dis-

tance of the expectation point and the separating hypersurface becomes

smaller and smaller as the hypothetical true locations are closer. This

means that the probability that a set of observations is located on the

one-peak side of the separating hyperplane increases and the probabil-

ity of resolution decreases correspondingly.

The separating hypersurface is computed from and, therefore, spe-

ciﬁ c to the peak model ﬁ tted. In this chapter, this is often the Gaussian

peak. The expectation point, however, is speciﬁ c to the model actually

present in the observations. If the model ﬁ tted and the model present

differ, the expectation point may even be located on the one-peak side

of the separating hyperplane. This implies that the peaks are not resolved

even if the statistical errors in the observations are very small.

It is clear that, ultimately, resolution is limited by errors, both system-

atic (modeling) errors and nonsystematic (statistical) errors. This limit is

fundamental: neither an increase in computer power nor improved soft-

ware can remove it. It would be present even if initial conditions would

be available for the locations of all peaks and sufﬁ cient computer power

would be available for ﬁ tting them. Removing these limits requires more

and preferably different observations of the same peaks.

1260 S. Van Aert et al.

7 Discussion and Conclusions

In this chapter, resolution has been investigated in detail in successive

steps. Throughout this chapter, emphasis has been put on electron

microscopy.

First, classical resolution criteria, such as Rayleigh’s and Sparrow’s,

have been discussed. These criteria are expressed in terms of the width

of the point spread function. The narrower the point spread function,

the higher the resolution.

Next, the diffraction limit to resolution and its relation to Rayleigh

and Sparrow resolution have been considered. The diffraction limit is

given by the cutoff frequency, which is the highest spatial frequency

that is transferred by the imaging system. It has been shown that the

diffraction limit is inversely proportional to the Rayleigh resolution. For

electron microscopy applications, it has been shown that the “width” of

the electrostatic potential of the atoms as well as the effect of thermal

vibrations of the atoms, the environment, and the detection have to be

taken into account. As a result of these extensions, it should be con-

cluded that the atoms themselves limit the diffraction limit and classical

resolution criteria. Also the notion of superresolution has been intro-

duced. Using superresolution algorithms, frequency components lying

beyond the diffraction limit can be reconstructed. Such algorithms use

prior knowledge of the object imaged by the imaging system.

Such prior knowledge has been incorporated in the form of a para-

metric model. Indeed, classical resolution criteria are in fact concerned

with calculated images, that is, noise-free images exactly describable

by a known parameterized mathematical model. Then, the unknown

parameters, such as the positions of projected atoms, can be measured

by means of model ﬁ tting. The practical limits to this so-called deter-

ministic model-based resolution are of a computational kind. As an

example, amorphous structures studied by atomic resolution TEM

have been discussed. For these structures, computational problems can

be overcome only if the thickness is very small. An upper bound to the

thickness is derived, which for amorphous silicon is of the order of 10

Å. For realistic thicknesses, electron tomography is needed. Moreover,

fundamental limits to deterministic model-based resolution exist if the

model is inaccurate, since then, systematic errors are introduced.

Then, unavoidable noise in the observations has been incorporated

by considering detected images instead of calculated images. The fun-

damental limit to this so-called statistical model-based resolution is

determined by the CRLB, which is a lower bound on the variance with

which the positions of, or the distance between, projected atoms or

atom columns can be measured using parameter estimation methods.

In this discussion, the ML estimator is very important since it achieves

the CRLB asymptotically. The effectiveness of the ML method has been

shown in an example, in which it has been applied to experimental

HRTEM images of an aluminium crystal. Moreover, it has been shown

that if the model is inaccurate, systematic errors determine fundamen-

tal limits to statistical model-based resolution as well. Apart from these

fundamental limits, computational limits exist, just as for deterministic

Chapter 20 The Notion of Resolution 1261

model-based resolution. It has also been shown how to improve statis-

tical model-based resolution using statistical experimental design.

Finally, ultimate model-based resolution is considered. Depending

on the observations, the solutions of the position estimates may coin-

cide exactly. It is shown that ultimately, resolution is limited by errors,

both systematic and nonsystematic. This limit is fundamental: no

increase in computer power or improved software can remove it. It

would be present even if the structure is resolved and sufﬁ cient com-

puter power would be available for ﬁ tting them. Removing these limits

requires more and preferably different observations of the same

structure.

Acknowledgments. Dr. S. Van Aert gratefully acknowledges the ﬁ nan-

cial support of the Fund for Scientiﬁ c Research—Flanders (FWO). The

research of Dr. A.J. den Dekker has been made possible by a fellowship

of the Royal Netherlands Academy of Arts and Sciences.

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