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This expression for the probability of error gives insight into the performance to make a correct decision and the sensitivity of this detection performance to the experimental settings Gonnissen et al. In the following sections, applications of statistical parameter estimation theory in the field of quantitative S TEM imaging are outlined. Aberration-corrected TEM, exit wave reconstruction methods or combinations of both are often used to measure shifts in atomic positions.

Whereas aberration correction has an immediate impact on the resolution of the experimental images, the purpose of exit wave reconstruction is to retrieve the complex electron wavefunction which is formed at the exit plane of the sample under study. Ideally, the exit wave is free from any imaging artifacts, thus enhancing the visual interpretability of the atomic structure.

Because of its potential to visualize light atomic columns, such as oxygen or nitrogen, with atomic resolution, exit wave reconstruction has become a powerful tool in high-resolution TEM Coene et al. Numerical calculations have shown that domain boundaries in CaTiO 3 are mainly ferrielectric with maximum dipole moments at the wall. The phase of the reconstructed exit wave is shown in Fig. This phase is directly proportional to the projected electrostatic potential of the structure.

In order to obtain quantitative numbers for the atomic column positions, statistical parameter estimation is needed den Dekker et al. This allows position measurements of all atomic columns with a precision of a few picometres without being restricted by the information limit of the microscope. Therefore, the phase of the reconstructed exit wave is considered as a data plane from which the atomic column positions are estimated in a statistical way. Nowadays, the physics behind the electron—object interaction is sufficiently well understood to have such a parameterized expression.

The parameters of this function, including the atomic column positions, can then be determined using the LS estimator. Therefore, this analysis is focused on the off-centring of the Ti atomic positions with respect to the centre of the neighbouring four Ca atomic positions. First, we average all displacements in planes parallel to the twin wall. Next, we average the results in the planes above with the corresponding planes below the twin wall. This second operation identifies the overall symmetry of the sample, with the twin wall representing a mirror plane.

The resulting displacements along and perpendicular to the twin wall are shown in Figs. In the direction perpendicular to the wall, systematic deviations for Ti of 3. A larger displacement is measured in the direction parallel to the wall in the layers adjacent to the twin wall. The average displacement in these layers is 6. In layers further away from the twin wall, no systematic deviations are observed.

These experimental results confirm the theoretical predictions Goncalves-Ferreira et al. The thickness of the domain wall is about two octahedra. The ferroelectric component is the smaller one and has an effect both parallel and perpendicular to the wall.

## On Advances in Statistical Modeling of Natural Images

We can also calculate the magnitude of the spontaneous polarization of the wall. In the model calculations it was found that the wall polarization is between 0. This value is comparable with the bulk spontaneous polarization of BaTiO 3 0. Another efficient technique to measure shifts in atomic positions is so-called negative spherical aberration imaging, in which the spherical aberration constant C s is tuned to negative values by employing an aberration corrector Jia et al.

Compared with traditional positive C s imaging, this imaging mode yields a negative phase contrast of the atomic structure, with atomic columns appearing bright against a darker background. For thin objects, this leads to a substantially higher contrast than for the dark-atom images formed under positive C s imaging.

This enhanced contrast has the effect of improving the measurement precision of the atomic positions and explains the use of this technique to measure atomic shifts of the order of a few picometres. Examples are measurements of the width of ferroelectric domain walls in PbZr 0. Depending on the shape and size of the STEM detector, different signals can be recorded Cowley et al. The high-angle scattering thus detected is dominated by Rutherford and thermal diffuse scattering. One of the advantages is thus the possibility of distinguishing visually between chemically different atomic column types.

Because of the incoherent imaging nature, the resolution observed in an HAADF STEM image is, to a large extent, determined by the intensity distribution of the illuminating probe. The combination of a high spatial resolution with a high chemical sensitivity makes HAADF STEM a very attractive tool for structure characterization at the atomic level. This is particularly the case when the difference in atomic number of distinct atomic column types is small, or if the signal-to-noise ratio becomes poor.

A performance measure which is sensitive to the chemical composition is the so-called scattering cross-section Retsky, ; Isaacson et al. Using statistical parameter estimation theory, the total intensity of scattered electrons can be quantified atomic column by atomic column using an empirical parameterized incoherent imaging model. The estimated scattering cross-sections allow us to differentiate between atomic columns with different compositions.

As such, differences in average atomic number of only 3 can clearly be distinguished in an experimental image, which is impossible by means of visual interpretation alone. This is an important advantage when studying interfaces, as illustrated in the following example. Even though the probe has been corrected for spherical aberration, no visual conclusions could be drawn concerning the sequence of the atomic planes at the interfaces.

The refined parameterized model is shown in Fig. The composition of the columns away from the interfaces is assumed to be in agreement with the composition in the bulk compounds, whereas the composition of the columns in the planes close to the interface shown in purple is unknown.

Histograms of the estimated scattering cross-sections of the known columns are presented in Fig. It is important to note that these tolerance intervals do not overlap, meaning that columns, for which the difference in average atomic number is only 3 TiO and MnO in this example, can clearly be distinguished. Based on this histogram, the composition of the unknown columns can be identified, as shown on the right-hand side of Fig.

Single-coloured dots are used to indicate columns whose estimated scattering cross-section falls inside a tolerance interval, whereas pie charts, indicating the presence of intermixing or diffusion, are used otherwise. The previous example shows how statistical parameter estimation theory can help to quantify the chemical composition in a relative manner. When aiming for an absolute quantification, intensity measurements relative to the intensity of the incoming electron beam are required LeBeau et al.

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In this manner, experimental scattering cross-sections can be compared directly with simulated scattering cross-sections Rosenauer et al. Reference cross-section values are then simulated by carefully matching experimental imaging conditions for a range of sample conditions, including thickness and composition. To illustrate this, Fig. By comparing experimental scattering cross-sections for each atomic column with simulated values, the thickness values for the PbO columns and the composition of the SrPbO columns have been determined, as shown in Fig.

The overall match between the simulated and experimental image intensities further confirms the results that have been obtained when using the scattering cross-sections approach. However, it should be noted that small deviations between simulated and experimental image intensities can not be avoided because of, for example, remaining uncertainties in the microscope settings such as defocus, source size or astigmatism. The high sensitivity of scattering cross-sections to composition is also an advantage when counting the number of atoms in an atomic column with single-atom precision.

Using the model-based approach explained above, the parameters of an empirical physics-based model have been estimated in the LS sense. For the cluster in the white boxed region, the refined model is shown in Fig. Based on the estimated parameters, scattering cross-sections have been computed for each atomic column and these are shown in the histogram in Fig. Since the thickness of the sample can be assumed to be constant over the particle area, substitution of an Al atom by an Ag atom leads to an increase in the estimated intensity.

Owing to a combination of experimental detection noise and residual instabilities, broadened — rather than discrete — peaks are observed. Therefore, these results cannot be interpreted directly in terms of the number of atoms in a column. From the estimated peak positions, the number of Ag atoms in each atomic column can be quantified, leading to the result shown in Fig. This counting procedure has also been applied to the same Ag cluster viewed along the [] direction, as shown in Figs.

For example, the atom counts presented in Figs. The most direct method for counting atoms is through comparison with image simulations LeBeau et al. Indeed, the assignment of numbers of atoms will always find a match by comparing experimental scattering cross-section values or peak intensities with simulated values. The reliability then depends solely on the accuracy with which, for example, the detector inner and outer angles have been determined and the accuracy with which the simulations have been carried out.

In comparison, the statistics-based method used to count the number of atoms shown in Fig. This approach is robust against systematic errors when two conditions are met: the number of experimental scattering cross-sections per unique thickness should be large enough and the spread of scattering cross-sections should be small enough compared with the difference between those of differing thicknesses De Backer et al. Ultimately, the simulations-based method and statistics-based method are combined into a hybrid approach.

An example analysis is presented in Fig. This validation step is required since more local minima are present in the ICL criterion shown in Fig. The precision of the atom counts is limited by the unavoidable presence of noise in the experimental images, resulting in overlap of the Gaussian components as shown in Fig. When the overlap increases, the probability of assigning an incorrect number of atoms will increase. The combination of a simulation-based and a statistics-based method thus allows for reliable atom counting with single-atom sensitivity.

As described in the previous sections of this feature article, new developments within the field of TEM enable the investigation of nanostructures at the atomic scale. Structural as well as chemical information can be extracted in a quantitative manner. However, such images are mostly two-dimensional projections of a three-dimensional object. To overcome this limitation, three-dimensional imaging by TEM or electron tomography can be used. Atomic resolution in three-dimensions has been the ultimate goal in the field of electron tomography during the past few years.

The underlying theory for atomic-resolution tomography has been well understood Saghi et al. In a next step, such atom-counting results can be used as input for discrete tomography. The discreteness that is exploited here is the fact that crystals can be thought of as discrete assemblies of atoms Jinschek et al. In this manner, a very limited number of two-dimensional images is sufficient to obtain a three-dimensional reconstruction with atomic resolution. This approach was applied to Ag clusters embedded in an Al matrix, as illustrated in Fig. An excellent match was found when comparing the three-dimensional reconstruction with additional projection images that were acquired along different zone axes.

In a similar manner, the core of a free-standing PbSe—CdSe core—shell nanorod could be reconstructed in three-dimensions Bals et al. The discrete approach that was used in these studies assumes that the atoms are situated on a fixed face-centred cubic lattice and that the particle contains no holes.

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These assumptions provide a decent start for the investigations described above, but deviations from a fixed grid, caused by defects, strain or lattice relaxation, are exactly the parameters that determine the physical properties of nanomaterials. Compressive sensing-based tomography has shown its ability to provide an adequate solution for problems that can be represented in the form of a sparse representation Donoho, a , b ; Saghi et al.

At the atomic scale, the approach exploits the sparsity of the object, since most of the voxels that need to be reconstructed correspond to vacuum and only a limited number of voxels are occupied by atoms Goris, Bals et al. An important advantage of this approach is that the actual positions of the atoms can be revealed without using assumptions concerning the crystal structure. This approach has been applied to reconstruct the structure of Au nanorods Goris, Bals et al.

Such bimetallic particles often provide novel properties compared with their monometallic counterparts Henglein, ; Hodak et al. To understand these properties, a complete three-dimensional characterization is often required where the exact positioning of the different chemical elements is crucial, especially at the interfaces. Owing to the atomic number Z -dependence of HAADF STEM intensities, the position and atom type of each atom have been determined from five high-resolution images acquired along different major zone axes.

Using statistical parameter estimation theory, the parameters of an incoherent imaging model have been estimated and the resulting models used as input for the compressive sensing-based algorithm. A detailed analysis of the position and atom type in a core—shell bimetallic nanorod was performed using orthogonal slices through the three-dimensional reconstruction, as shown in Fig. An intensity profile was acquired along the direction indicated by the white rectangular box in Fig.

In this manner, each atom in the cross-sections shown in Figs. The results are shown in Figs. In particular, the characterization of their structure is far from straightforward. At the same time, however, there is a clear need for a complete characterization in three-dimensions since these materials can no longer be considered as periodic objects. One of the main bottlenecks is that these clusters may rotate or show structural changes during investigation by TEM Li et al. Obviously, conventional electron tomography methods, even those that are based on a limited number of projections, can no longer be applied.

On the other hand, the intrinsic energy transfer from the electron beam to the cluster can be considered as a unique possibility to investigate the transformation between energetically excited configurations of the same cluster. This idea was exploited to study the dynamic behaviour of ultra-small Ge clusters consisting of less than 25 atoms Bals et al. In this manner, the number of atoms at each position could be determined, as illustrated in Fig. In order to extract three-dimensional structural information from these images without using prior knowledge of the structure, ab initio calculations were carried out.

Several starting configurations were constructed that are all in agreement with the experimental two-dimensional projection images. Although all of the cluster configurations stay relatively close to their starting structure after full relaxation, only those configurations in which a planar base structure was assumed were found to be still compatible with the two-dimensional experimental images. In this manner, reliable three-dimensional structural models are obtained for these small clusters and the transformation of a predominantly two-dimensional configuration into a compact three-dimensional configuration can also be characterized.

The use of statistical parameter estimation techniques in the field of electron microscopy is becoming increasingly important since it allows one to determine unknown structures quantitatively on a local scale. The theory of parameter estimation is well established and applications are becoming routine, partly through improvements in the underlying algorithms to estimate unknown structure parameters, and partly through the increase in computational power that allows fast processing and analysis.

Applications in the field of high-resolution S TEM show how statistical parameter estimation techniques can be used to overcome the traditional limits set by modern electron microscopy. The precision that can be achieved in this quantitative manner far exceeds the resolution performance of the instrument. The characterization limits are therefore no longer imposed by the quality of the lenses but are determined by the underlying physical principles. Structural, chemical, electronic and magnetic information can be obtained at the atomic scale.

As demonstrated in this feature article, not only can quantitative structure determination be carried out in two-dimensions, but also three-dimensional analyses are currently becoming standard. The examples discussed in this feature article demonstrate that statistical parameter estimation methods have been applied successfully to nanostructures which are relatively stable under the incoming electron beam, and therefore the atomic structure under investigation can be assumed to remain unchanged under illumination with high electron doses.

However, radiation damage becomes increasingly relevant not only in biological studies but also in the study of nanostructures Meyer et al. An important challenge that remains is therefore to push the development of quantitative methods toward its fundamental limits. Ultimately, the goal is to measure the atom positions of beam-sensitive nanostructures with picometre precision and to discern between adjacent atom types. This is very challenging and to reach this goal the allowable electron dose needs to be used in the most optimal way.

Indeed, every incoming electron counts and therefore needs to carry as much quantitative structural information as possible. For that purpose, the microscope and detector settings will be optimized using the principles of statistical experimental design den Dekker et al. This becomes increasingly important in an era where new data collection geometries are emerging Shibata et al. Therefore, the use of expressions representing the attainable precision, as discussed in this feature article, can be used to optimize the experimental design.

Statistical experimental design can be defined as the selection of free variables in an experiment to improve the precision of the measured parameters. By calculating the attainable precision, the experimenter is able to verify whether, for a given experimental design, the precision is sufficient for the purpose at hand. If not, the experiment design has to be optimized so as to attain maximum precision.

This will allow a significant reduction in the incoming electron dose to achieve maximum attainable precision or detectability. Finally, when lowering the incoming electron dose, it is expected that the use of a complete maximum likelihood method to estimate unknown structure parameters will be of great help den Dekker et al. Furthermore, an accurate description of the detector and noise properties, taking correlations between neighbouring pixel values into account, would then be required Niermann et al.

However, if successful, we will be able to measure unknown structure parameters as accurately and as precisely as possible using a given electron dose. In conclusion, the possibilities of statistical parameter estimation theory in the field of electron microscopy have been shown to be successful in the study of many materials problems so far.

Recent developments to explore the capabilities of new detector geometries will certainly open up a whole new range of possibilities to understand and characterize beam-sensitive nanostructures in particular. The authors thank the colleagues who have contributed to this work over the years, including K. Batenburg, R. Erni, B.

Goris, B. Partoens, A. Rosenauer, M. Rossell, B. Schoeters, D. Schryvers, J.

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Sijbers, S. Turner and J. Ayache, J. Nano Lett. Ultramicroscopy , 77 , 37— Status Solidi. B , , — Ultramicroscopy , 58 , 18— Mathematical Methods of Statistics. Science , , — Ultramicroscopy , , 46— Ultramicroscopy , , 56— Ultramicroscopy , , 23— Ultramicroscopy , , — Ultramicroscopy , , 34— Ultramicroscopy , , 83— Ultramicroscopy , 89 , — IEEE Trans. Theory , 52 , — Pure Appl. Ultramicroscopy , , 15— Nature , , — A , , — Image Recording in Microscopy.

Weinheim: VCH. Ultramicroscopy , , 59— Ultramicroscopy , 4 , — B , 79 , Fundamentals of Statistical Signal Processing , Vol.

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