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Electron Tomography in Materials Science

  • Rowan K. Leary
  • Paul A. MidgleyEmail author
Chapter
Part of the Springer Handbooks book series (SHB)

Abstract

This chapter illustrates how electron tomography has become a technique of primary importance in the three-dimensional () microscopic analysis of materials. The foundations of tomography are set out with descriptions of the Radon transform and its inverse and its relationship to the Fourier transform and the Fourier slice theorem. The acquisition of a tilt series of images is described and how the angular sampling in the series affects the overall 3-D resolution in the tomogram. The imaging modes available in the (scanning) transmission electron microscope are explored with reference to their application in electron tomography and how each mode can provide complementary information on the structural, chemical, electronic, and magnetic properties of the material studied. The chapter also sets out in detail methods for tomographic reconstruction from backprojection and iterative methods, such as simultaneous iterative reconstruction technique ( ) and algebraic reconstruction technique ( ), through to more recent compressed sensing approaches that aim to build in prior knowledge about the specimen into the reconstruction process. The chapter concludes with a look to the future.

Electron tomography STEM HAADF imaging compressed sensing reconstructions Radon transform analytical electron tomography algebraic iterative reconstruction 3D imaging 

By recording images of an object at different orientations, electron tomography ( ) provides a means to reconstruct that object in three dimensions, leading to a greater understanding of its internal structure, composition, and physico-chemical properties. Whilst ET was developed initially for the life sciences [26.1] it has, in the past 20 years or so, become an invaluable technique for the study of a vast range of materials across the physical sciences [26.2].

Important length scales in materials systems extend over several orders of magnitude (Fig. 26.1) and complementary tomographic techniques may be used for visualization. In this chapter, we focus on ET in the (scanning) transmission electron microscope, (S)TEM, which provides insight into the 3-D structure of materials with sub-nm resolution across sub-\(\mathrm{{\upmu}m}\) length scales.

There has been a rapid expansion in the number of imaging modes that may be used for tomographic acquisition. High-resolution image series can now reveal atomic structure in 3-D and a combination of spectroscopic methods (e. g., energy-dispersive x-ray ( ) and electron energy-loss spectroscopy ( )) and tomography leads to 3-D compositional, chemical, and electronic information. Moreover, a combination of diffraction and tomography provides 3-D crystallographic information and holotomography reconstructions furnish a 3-D visualization of electro-magnetic potentials. In all cases, the data acquisition is limited, and much of the recent progress in ET has been facilitated by improvements in the reconstruction routines, and these are explained below.

Fig. 26.1

3-D imaging methods shown as a function of the resolution possible and the volume of material that can be analyzed

26.1 Foundations of Tomography

Although the literal meaning of the term tomography refers to the visualization of slices, transmission tomography, such as (S)TEM tomography (referred to here as ET) or x-ray tomography, can be considered as a method of reconstructing the interior of an object from a set of projections through its structure. In essence, this form of tomography is conventionally achieved by recording a tilt series of projections, often about a single tilt axis. The ensemble of images is then used to form a reconstruction, or tomogram, via some operation that can essentially be seen as an inversion of the original projection process Fig. 26.2.

Fig. 26.2

(a) Acquisition of an angular series of two-dimensional () projections of an object, and (b) backprojection of these images into a 3-D space to obtain a reconstruction of the object

The mathematical framework for tomography originates from a seminal paper by Radon in 1917 [26.3], in which the projection of an \(N\)-dimensional object into a space of dimensions \(N-1\) was considered. That projection, or transform, known now as the Radon transform and its inverse will be discussed in more detail below. In 1956, Bracewell [26.4] showed how tomography can be considered in terms of the more widely known Fourier transform, and the relationship between the Fourier and Radon transforms was determined. The foundational principles of tomography stemming from these transforms are now well established, and further coverage beyond that given here can be found in [26.10, 26.5, 26.6, 26.7, 26.8, 26.9].

Although tomography is typically referred to as a 3-D reconstruction method, the most common single axis geometry permits the reconstruction process to be addressed in terms of a series of, in principle, independent 2-D reconstructions. In general, this is both algorithmically and computationally easier, and is used for the descriptions provided here. A possible disadvantage of slice-by-slice 2-D reconstruction of a 3-D volume is that it may be difficult to fully exploit 3-D prior knowledge during the reconstruction process, in which case fully 3-D reconstructions may be desirable.

26.1.1 The Radon Transform

The Radon transform \(\mathcal{R}\) describes directly the projection process, mapping a function \(f\) by line integrals along all possible lines \(L\). With increment \(\mathrm{d}s\) along \(L\), the transformation may be defined as
$$\breve{f}\left(l,\theta\right)=\mathcal{R}f=\int_{L}f\left(x,z\right)\mathrm{d}s$$
(26.1)
the geometry of which is illustrated in Fig. 26.3 [26.6]. The function \(f\) is defined here on the 2-D real space coordinates (\(x,z\)) and the Radon transform converts the data into Radon space (\(l,\theta\)), frequently referred to as a sinogram , where \(l\) is the line perpendicular to the projection direction and \({\theta}\) is the angle of projection (Fig. 26.4a-d). Using polar coordinates (\(r,\phi\)), related to Cartesian coordinates by
$$r=\sqrt{x^{2}+z^{2}}\quad\text{and}\quad{\phi}=\tan^{-1}\left(\frac{z}{x}\right),$$
a point object in real space (\(x=r\cos{\phi}\), \(z=r\sin{\phi}\)) is a line in Radon space (\(l,\theta\)) linked by \(l=r\cos({\theta}-{\phi})\).

In principle, the real space structure of the object \(f(x,z)\), can be recovered from the Radon domain \(\breve{f}\left(l,\theta\right)\) by an inversion of the Radon transform. Since the empirical sampling of an object by a projection is equivalent to a discrete sampling of the Radon integral, the goal in tomography is, then, to acquire a sufficient number of projections such that an inverse Radon transform, or some other means of reconstruction, can yield an adequate approximation of the object.

Fig. 26.3

The geometry of the Radon transform

Fig. 26.4a-d

The relationship between (a,c) real space and (b,d) Radon space. (a) Point object, (bsinogram of point object, (c) phantom, (dsinogram of phantom

26.1.2 The Fourier Transform and Fourier Slice Theorem

Closely related to the Radon transform is the Fourier transform, which for the general case of an \(N\)-dimensional function \(f\), may be written as
$$\tilde{f}\left(\boldsymbol{k}\right)=\mathcal{F}_{N}f=\int_{-\infty}^{\infty}f(\boldsymbol{x})\mathrm{e}^{-2\uppi\mathrm{i}\boldsymbol{k}\cdot\boldsymbol{x}}\mathrm{d}\boldsymbol{x}\;,$$
(26.2)
where \(\boldsymbol{x}=x_{1}\), \(x_{2}\), …, \(x_{N}\) are the coordinates in real space and \(k=k_{1}\), \(k_{2}\), …, \(k_{N}\) the corresponding coordinates in Fourier space. In words, the Fourier transform provides an alternative means of representing a function by decomposing it into (spatial) frequency components. Although perhaps most well known in the context of analyzing the characteristic periodicity in images of crystalline lattices, there is significance and practical application in the tomographic imaging context too, which is embodied in the Fourier slice theorem, as described below.
The work of Bracewell [26.4] highlighted the important relationship between projections formed via the Radon transform and the Fourier representation of an object, which has become known as the Fourier slice theorem [26.7]. The theorem states that

the Fourier transform of the projection of a function, at a given angle, is equivalent to a central section, at that angle, through the Fourier transform of the function.

Figure 26.5 illustrates the theorem for projections of a 2-D object; an analogous relationship applies to 2-D projections of an object and central sections in its 3-D Fourier space.
Fig. 26.5

The Fourier slice theorem

The Fourier slice theorem further aids the understanding of tomographic reconstruction and the consequences of finite sampling: acquiring projections at different angles is equivalent to sampling sections of the object's Fourier space over the range of frequencies in each central section. However, most objects will not be fully described by the frequencies in one section, or even a few sections, meaning that many projections at different angles are required to sample Fourier space sufficiently such that, in principle, it would be possible to obtain a satisfactory description of the object in real space by direct inversion of the sampled Fourier space.

The relationships between real, Radon, and Fourier space are shown explicitly in Fig. 26.6. From Fig. 26.6 and the foregoing discussion it is apparent that the tomographic reconstruction process can be approached via either a real space-based backprojection route, or a Fourier inversion route. The practicalities of ET however, reviewed hereinafter, mean that either approach is rarely straightforward to achieve.

Fig. 26.6

Relationship between Fourier space and Radon space

26.2 ET Acquisition

The first example of 3-D reconstruction using TEM came from DeRosier and Klug [26.11] who, harnessing prior knowledge of the helical symmetry of the object, obtained a 3-D reconstruction of the tail of the T4 bacteriophage from a single 2-D image. They also outlined the principles of reconstructing arbitrarily-shaped 3-D objects from a series of 2-D TEM images; or more specifically, TEM projections. Along with two other seminal papers from the same year [26.12, 26.13], this is often seen as the starting point of ET.

An ET investigation consists of a number of distinct but inter-related stages, summarized in Fig. 26.7. With the exception of sample preparation, which is referred to inter alia, these are reviewed sequentially in the following sections.

Fig. 26.7

Principal stages in an ET investigation

26.2.1 Finite and Limited Angular Sampling

The sampling theory outlined above suggests that the best tomographic reconstructions will be achieved by acquiring as many projections over as large an angular range as possible. For conventional reconstructions, without the application of prior knowledge (see later), regular tilt increments of \(1{-}2^{\circ}\) are used. Other tilting schemes are detailed in [26.14, 26.5]. However, several factors in ET always restrict the actual number of projections obtained to being fewer than the ideal. Firstly, in contrast to many other tomographic techniques, ET is performed in a highly restricted working space. In practice, the sample is located between the polepieces of an immersion objective lens, as illustrated in Fig. 26.8a-ea. In order to achieve high spatial resolution in each image, the polepiece gap must be kept as small as possible (ca. \(2{-}5\,{\mathrm{mm}}\)) minimizing the effects of spherical and chromatic aberration ([26.15] and references therein). This gap may limit the angular range over which the specimen can be tilted.

Fig. 26.8a-e

Reasons for limited angular sampling in ET: (a) restricted polepiece gap; (b) support grid or (c) holder shadowing; (d) specimen thickness; (e) limited depth-of-field

ET is thus a limited-angle tomographic problem, with a large unsampled region in Fourier space, known as the missing wedge (Fig. 26.9). The design of specialized narrow profile tomography holders has enabled the maximum tilt angle \({\theta}_{\mathrm{max}}\) to approach \(80^{\circ}\). However, additional limitations may come into play at high tilt angles; this may be due to shadowing from the specimen support grid or holder (in Fig. 26.8a-eb,c), because the increase in projected specimen thickness (Fig. 26.8a-ed) may render the projection unusable through blurring due to chromatic aberration in TEM or beam broadening, and thus loss of resolution, in STEM (Fig. 26.8a-ee).

Fig. 26.9

(a) Illustration of finite and limited angular sampling of Fourier space in ET; (b,c) its manifestation in practice. In (a), each projection, of an object of diameter \(D\), is a central section in Fourier space of thickness \(1/D\). The dashed line indicates the limit at which the information from adjacent projections just overlaps, corresponding to the Crowther criterion [26.16] defined in (26.4)

Where needed, any sample support grid used in an ET experiment should be judiciously chosen, striking a balance between sturdy support and occlusion of the sample during tilting. Selecting a wide grid bar spacing, to minimize shadowing, must be considered in conjunction with the rigidity of the support film, determined by its type and thickness. Typically, for many nanoparticulate specimens, a 200 mesh grid spacing (where smaller mesh numbers correspond to wider bar spacing) provides a reasonable compromise. Many manufacturers also offer slot grids comprising elongated sample viewing windows. Placing the long axis of these windows perpendicular to the tilt axis then provides an extended field of view without grid bar occlusion.

To further minimize the missing wedge problem, dual-axis tomography may be used [26.17, 26.18, 26.19, 26.20] which, by combining two mutually perpendicular tilt series, reduces the missing wedge to a missing pyramid. In single-axis ET, the effects of information loss due to the missing wedge can be minimized by ensuring that the sample, or important features, are oriented along the tilt axis. Dual-axis tomography may be crucial in cases where multiple feature orientations would mean that information loss due to the missing wedge would be intolerable [26.20, 26.21]. Disadvantages of dual-axis ET include additional electron beam exposure, and operator time, required to record two tilt series, as well as possible challenges in combining the two data sets accurately.

In the physical sciences, dual-axis ET is used far less than the single-axis geometry. As an alternative, there has been a growing trend in the preparation of needle-shaped samples, which, using specialist holders, can be rotated through the full \({\pm}90^{\circ}\) range, and avoid the aforementioned problems of shadowing or thickness increases [26.22, 26.23, 26.24, 26.25]. A prerequisite for this approach is that the sample must be amenable to fabrication into a needle shape using a focused ion beam or be attachable by some means to a needle-shaped support. This is ideal for analysis of targeted features extracted from bulk specimens.

Even for samples that can be tilted over the full angular range, undersampling still occurs because of the finite angular increment \({\Updelta}{\theta}\) over which the sample is tilted between each projection. The chosen angular sampling is ultimately set by the electron dose that the specimen can withstand, as faithful tomographic reconstruction is reliant on the premise that the specimen does not change (significantly) during collection of the image series. (An exception to this rule is dynamic tomography, which seeks to track continuously occurring changes, but is reliant on prior knowledge to be able to predict the nature of changes; such methods have not yet been developed in ET.) While pertinent in all (S)TEM studies, the repeated imaging of a region in ET means that consideration of electron beam induced changes and carbonaceous contamination are paramount. Every effort should be made to assess the potential for and measures taken to mitigate beam damage and contamination, including sample preparation and strategy for the tilt series acquisition.

Automated or semi-automated tilt series acquisition and low-dose procedures have facilitated the application of ET to beam-sensitive specimens that damage by inelastic processes (viz. heating and radiolysis), and have been critical in the biological sciences [26.1]. Likewise, operation below the threshold for knock-on damage can open the door to analysis of specimens that damage predominantly by elastic scattering [26.26]. Although less critical for most specimens in the physical sciences, the idea of dose fractionation [26.5] is one that should always be borne in mind when recording a tilt series, optimizing the information acquired about the whole specimen across the tilt series.

26.2.2 Limited Sampling: Artefacts and Reconstruction Resolution

The finite and limited angular sampling in ET can lead to serious artefacts in the tomographic reconstructions, which are well documented in the literature [26.2, 26.27, 26.28, 26.29]. In fact, structures may be partially or entirely absent from the tomogram if it is affected strongly by the missing wedge [26.30]. In general, the missing wedge causes an apparent elongation \(e\) in the beam direction, determined by \({\theta}_{\mathrm{max}}\) [26.31]
$$e=\sqrt{\frac{\theta_{max}+\sin\theta_{\text{max}}\cos\theta_{\text{max}}}{\theta_{\text{max}}-\sin\theta_{\text{max}}\cos\theta_{\text{max}}}}$$
(26.3)
as exemplified in Fig. 26.10. Experimentally, it has been shown that deviations away from the ideal tomographic reconstruction fall to acceptably small values when \({\theta}_{\mathrm{max}}={\pm}80^{\circ}\) [26.21, 26.28, 26.32]. Qualitatively, the effect of finite angular sampling is to cause streaking artefacts, the severity of which increases with tilt increment (Fig. 26.10).
Fig. 26.10

The effect of finite tilt increment and limited angular tilt range on tomographic reconstruction of a spherical phantom (weighted backprojection reconstructions). The maximum tilt angle and tilt increment used for each reconstruction is denoted by the column and row headings, respectively

A consequence of the limited angular sampling is that the reconstruction resolution will vary with direction [26.33], the determining variables for which in the context of ET were first considered by Crowther [26.16]. The resolution parallel to the tilt axis \(d_{y}\) should, in principle, be equal to that of the input projections. Perpendicular to the tilt axis, the Crowther criterion (which assumes \({\theta}_{\mathrm{max}}={\pm}90^{\circ}\)) gives the resolution \(d_{x}\) as being determined by the number of projections \(N_{\text{p}}\) and the diameter of the region to be reconstructed \(D\)
$$d_{x}=\frac{\uppi D}{N_{\text{p}}},$$
(26.4)
the geometry of which is illustrated in Fig. 26.9a. Due to the missing wedge, the reconstruction in the direction of the electron beam \(d_{z}\) is further degraded by the elongation factor \(e_{xz}\)
$$d_{z}=d_{x}\cdot e_{xz}\;. $$
(26.5)
Although these expressions are often cited as useful guides, the reconstruction resolution is often seen to be considerably better than predicted by this criterion when constrained reconstruction techniques are used ([26.5, Chap. 10] or [26.32, 26.34] for methods). As summarized by Midgley and Weyland [26.2], the actual reconstruction resolution is likely to depend on a combination of the effects of the sampling regime, noise characteristics, shape of the object to be reconstructed, and the nature of the reconstruction routine. A rough rule of thumb is that the achievable resolution in an electron tomographic reconstruction using conventional techniques will typically be ca. \(1/100\) of the object diameter.

26.3 ET Imaging Modes

In the most basic sense, the transmission of the electron beam through the specimen allows the (S)TEM to be described as a structure projector. However, as exemplified by Hawkes [26.5, Chap. 3], and Midgley and Weyland [26.2], it is critically important to consider the extent to which the signals obtained in (S)TEM constitute a projection that is valid, or at least useful, for tomographic reconstruction.

While the ideal projection involves a sum integral of some physical property, as defined by the Radon transform, this is rarely achieved in (S)TEM. Instead, it is generally regarded as sufficient that a (S)TEM signal is a monotonic function of a projected physical quantity. It is this more relaxed stipulation that is generally referred to as the projection requirement in ET. It is clear from ET studies to date that directly interpretable signals satisfying the projection requirement are significantly easier to process, and the ET reconstructions are more readily interpreted than those that do not. Nonetheless, ET is often performed using signals that do not fully satisfy the projection requirement. In those cases especially, careful interpretation of the resulting tomograms may be required to differentiate real structure from artefacts, or special reconstruction approaches may need to be adopted to achieve 3-D reconstruction.

26.3.1 TEM or STEM for Tomography in Materials Science

Here it is worth contrasting specimen characteristics and consequent ET practices in the physical and biological sciences. Although the two fields share many principles and practices, the specimen demands and the dominant imaging modes are, in the majority, quite distinct. ET in the biological sciences has been practiced for many years using bright-field ( ) TEM, and this is still dominantly the case. BF-TEM is highly suitable for biological specimens because they are often noncrystalline and/or thin, weakly scattering objects. For these, mass-thickness or phase contrast are the main determinants in image formation, and the images obtained, perhaps after correcting for lens aberrations, may be considered true projections of the underlying structure.

For strongly scattering crystalline specimens on the other hand, as are common in the physical sciences, several factors (detailed below) can readily lead to violation of the projection requirement in BF-TEM. On the other hand, specimens in the physical sciences are often much more electron beam tolerant, permitting use of a range of imaging modes that would, in general, be far too damaging for biological structures. Alternative imaging modes can both satisfy better the projection requirement for specimens in the physical sciences and enable measurement of not just morphological characteristics in 3-D, but also chemical, magnetic, electronic, and crystallographic properties.

While many insights have been made, and continue to be made, using TEM for ET in the physical sciences, and certain niche techniques inherently require TEM-based techniques, STEM has become increasingly popular. Overwhelmingly, it is clear that major reasons for the popularity of STEM are (i) the annular dark-field ( ) imaging mode, which can often provide both intuitive and high-fidelity analysis due to its direct interpretability; and (ii) the ability to acquire simultaneously a range of signals, opening the door to extended multidimensional and multimodal analyses.

Table 26.1 summarizes the main signal modes that are used in ET. The most important of these are reviewed in the following sections.

Table 26.1

Principal (S)TEM imaging modes for ET of solid catalysts

Signal mode

Contrast mechanism

Early/selected studies

Status\({}^{\mathrm{a}}\)

Suitable studies

Morphological imaging modes

BF-CTEM

Phase, amplitude

DeRosier and Klug (1968) [26.11]; Spontak et al (1988) [26.35]; Koster et al (2000) [26.36]

E

WPOs, biological specimens, amorphous materials

ADF-STEM

Atomic number (\(Z\))

Midgley and Weyland (2001) [26.35]

E

Crystalline specimens, \(Z\)-contrast

Cs-CTEM

 

Bar Sadan et al (2008) [26.37]

A

Atomic-scale, WPOs

Cs-STEM

Atomic number (\(Z\))

Van Aert et al (2011) [26.38]; Goris et al (2012) [26.39]

A

Atomic-scale; heavy metal nanoparticles

Cc-TEM

 

Baudoin et al (2013) [26.40]

A

Thick biological specimens

DF-TEM

Angularly selective scattering

Barnard et al (2006) [26.41];Bals et al (2006) [26.42]

A

Lattice defects, low-contrast soft matter

Precession

BF-CTEM

 

Rebled et al (2011) [26.43]

A

Crystalline specimens

BF-STEM

 

Sousa et al (2011) [26.44]

A

Thick specimens, polymers, biological sections

IBF-STEM

 

Ercius et al (2006) [26.45]

A

Thick specimens

MAADF-STEM

 

Sharp et al (2008) [26.46]

A

Dislocations

STEM in ESEM

 

Jornsanoh et al (2011) [26.47]

A

Nonconductive or hydrated specimens

Multidimensional imaging modes

EFTEM

 

Weyland and Midgley (2001) [26.48];Möbus and Inkson (2001) [26.49]

E

Chemical segregation, optical properties, bonding variations

EELS

Inelastic scattering

Jarausch et al (2009) [26.22]

A

Chemical environment

Elemental distribution (core-loss)

Optical properties (low-loss)

EDXS

Secondary x-ray emission

Möbus et al (2003) [26.50];Lepinay et al (2013) [26.51]

A

Elemental distribution

Diffraction

 

Kolb et al (2007) [26.52]

A

Crystalline materials

Holography

Reconstructed phase and amplitude

Twitchett-Harrison et al (2007) [26.53]

A

Mean inner potential, electrostatic and magnetic fields

Time resolved

 

Kwon and Zewail (2010) [26.54]

A

New commercial TEMs

\({}^{\mathrm{a}}\)E \(=\) established, A \(=\) advanced

26.3.2 BF-TEM Tomography

Although BF-TEM has been used for several decades, the images are not always straightforward to interpret, for several reasons [26.55, 26.56]. In BF-TEM, the image intensity, generally, does not show a monotonic dependence on the specimen thickness, depending strongly and in an involved manner on defocus. As such, contrast reversals in the image can occur through the specimen thickness or as a result of small changes in the electron optical conditions. By their nature, BF-TEM images typically yield only weak chemical sensitivity—a considerable drawback when seeking to investigate complex multi-element samples and/or when seeking to resolve fine-scale features against contrast generated from any specimen support. For strongly scattering crystalline specimens, further complications in BF-TEM can be introduced due to strong Fresnel contrast and domination of the image by diffraction contrast (Bragg scattering). These signals carry a wealth of information that is of interest in certain contexts, such as diffraction contrast imaging of planar defects and strain fields [26.55, Chaps. 25 and 26], but in other contexts they can preclude a general facile interpretation that a monotonically varying signal endows and cause marked problems for ET. Significant image complications, so-called delocalization, may also arise at high resolution due to lens aberrations. Aberration-corrected (AC) optics and the use of ‘‘negative spherical aberration imaging'' (Urban et al in [26.57]) can permit compensation of aberrations and allow more readily interpretable TEM images to be formed. However, direct information on chemical composition is still lacking.

Considering the effects outlined above, the suitability with regards to the tomographic projection requirement has been extensively discussed in foundational [26.35, 26.58], [26.10, Chap. 3] and review literature [26.2, 26.20, 26.27, 26.28, 26.59, 26.60, 26.61, 26.62, 26.63, 26.64, 26.65, 26.66]. There seems to be general consensus in the literature that BF-TEM is capable of approximately reconstructing the exterior shape of convex homogeneous crystalline objects, while the intensity of the interior may be subject to erroneous nonlinearity due to diffraction effects [26.65, 26.67, 26.68].

Another imaging mode, ADF-TEM, can be implemented using an annular aperture in the back focal plane. This can yield chemically sensitive tomograms in a manner similar to ADF-STEM, and may have particular merits for fast acquisition and low-contrast soft matter [26.42, 26.69]. However, for most samples dark-field imaging in STEM can often provide a more powerful approach. Interestingly, there are significant new opportunities for TEM ET studies exploiting recent advances with direct electron detectors.

Despite the technical challenges, there have been cases where important information has been revealed using BF-TEM tomography, e. g., [26.70] where due consideration is given to possible violation of the projection requirement, or the effects are insignificant at the level of interest in the tomogram. The morphology of polymer systems has been studied with BF tomography [26.71], though STEM is increasingly being adopted for these too [26.72]. Thin carbonaceous or similar materials, i. e., those that are weakly scattering, have also been profitably studied with (AC) BF-TEM [26.26, 26.37]. Figure 26.11a-e illustrates the local nanoporosity obtained using BF-TEM ET [26.73] of SBA-15 mesoporous silica. Whilst Fig. 26.11a-ea shows a single BF image, Fig. 26.11a-eb–e show slices from the tomographic reconstruction revealing, in far greater detail than is possible with a single BF image, the presence of locally disordered and merged pores (Fig. 26.11a-eb–e). The reconstructions were used to prove that, in samples subjected to higher temperature hydrothermal treatments, pore fraction increases detected by nitrogen sorption were because of the increase in the number and volume fraction of disordered merged pores. A similar BF-TEM ET study [26.74] of hydrothermally treated zeolitic catalyst showed how image analysis techniques may be used to reveal a hierarchical porosity [26.75].

Fig. 26.11a-e

BF-TEM image of an ordered mesoporous silica (SBA-15). (b) Slice through an ET reconstruction, showing local disorder caused by merged pores, examples of which are highlighted in color in (ce), which are slices taken at different heights from the boxed region in (a). Reproduced from [26.73] with permission of The Royal Society of Chemistry (RSC) on behalf of the European Society for Photobiology, the European Photochemistry Association, and the RSC

26.3.3 STEM Tomography

STEM, using the ADF imaging mode, has become the most widely utilized technique for ET in the physical sciences [26.10, 26.2, 26.27, 26.59, 26.60, 26.61, 26.75, 26.76, 26.77], [26.10, Chap. 12], [26.78, Chap. 8]. The motivation for collecting an ADF signal is that at high detection angles, and with a large angular integration range, coherent contributions to the image from Bragg-scattered beams are minimized. With the detected signal then dominated by Rutherford-like and thermal diffuse scattering, the scattering detected from each atom can be considered as transversely incoherent. The signal intensity should then vary monotonically with the thickness of the specimen and the atomic number \(Z\) of the constituent atoms, approaching a \(Z^{2}\) relationship. The actual \(Z\) exponent lies somewhere in the region of \(Z^{1.3-2}\), depending (primarily) on the inner detection angle [26.79, 26.80]. Unlike the phase contrast transfer function of BF-TEM, the optical transfer function of ADF-STEM does not oscillate rapidly with changing spatial frequency or defocus; it is these characteristics that endow direct interpretability and high contrast.

It was realized in the early development of STEM in the 1970s (Pennycook's historical review in Chap. 1 of [26.78]) that high-contrast chemically sensitive atomic resolution images can be obtained of heavy metal nanoparticles, clusters, or even single atoms on low-\(Z\) support materials. These are characteristics fulfilled by many supported nanoparticulate catalyst systems and also, in general, by heavy metal nanoparticles deposited on low-\(Z\) TEM sample support grids, and ADF-STEM tomography has been applied to many catalyst and nanoparticle systems over the past decade or so ([26.2, 26.27, 26.59, 26.77], [26.78, Chap. 8], [26.81, 26.82]). Illustrative examples are shown in Fig. 26.12a-c.

Fig. 26.12a-c

ET reconstructions of supported heavy metal nanoparticles (a) (Pt,Ru) nanocatalysts supported on a disordered mesoporous silica. The surface of the silica has been color-coded according to the local Gaussian curvature. The nanocatalysts (red) appear to prefer to anchor themselves at the (blue) saddle-points (for details, [26.83]). (b) Surface-rendered visualization of Au nanocatalysts (red) supported on titania (blue). The nanocatalysts are located in the crevices between titania crystallites; confirmed by the aberration-corrected STEM image in panel (c). (b,c) reprinted from [26.84], with permission from Elsevier

In 2001, it was shown [26.69, 26.70] that the characteristics of ADF-STEM also make it a particularly successful imaging mode for 3-D imaging of strongly scattering crystalline specimens via ET. It is widely agreed that for many specimens in the physical sciences, ADF-STEM can satisfy the projection requirement to a sufficient approximation. Thus, it is has often been concluded that ADF-STEM is the most suitable technique for ET nanometrology [26.65, 26.85, 26.86]. A clear example can be found in a study by Lu et al [26.86], who found that BF-TEM substantially overestimated the constituent volume fraction of carbon-black in polymer composites compared to ADF-STEM, which provided acceptable accuracy.

However, ADF-STEM is not immune to potential difficulties that can lead to complications in interpretation. It is often pointed out that for crystalline specimens, the intensity of the image may be modified due to strong Bloch wave channeling when the crystal is near zone-axis orientation, which tends to concentrate the beam intensity onto atomic columns [26.87]. Primarily, this can increase the high angle scattering, and thereby the intensity in the image. In general though, this effect tends to occur only at a small number of crystal orientations across an ET tilt series, is more uniform across a crystal, and is less pronounced relative to diffraction contrast in BF-TEM. Nevertheless, a decision may need to be made as to whether to discard images strongly affected by channeling, or to proceed with using them in the reconstruction. For relatively minor occurrences, the effects may be sufficiently negated by virtue of the combination of many tilt series images during the reconstruction process.

Caution should also be noted in that very large differences in atomic number may lead to signals that could readily exceed the dynamic range of the ADF detector, and consequently impose restriction to low contrast of the low-\(Z\) component(s), or lead to contrast saturation of those of high \(Z\) that would violate the projection requirement. Detector saturation or contrast reversal can also result from very thick samples, the latter due to scattering to high angles beyond the outer radius of the detector. These can lead to artefacts such as voids or erroneous core-shell structures in ET reconstructions [26.45, 26.82], the nature of which are examined in detail in [26.67]. Good practice is to assess the potential for contrast saturation at different tilt angles before embarking on acquisition of the tilt series. Detector gain and offset (contrast and brightness settings) should not be altered during the tilt series, as doing so would violate the projection requirement.

While ADF-STEM using high angles is currently the imaging mode that, arguably, is likely to best satisfy the projection requirement for the widest range of specimens, there may be certain scenarios when other variants of STEM become more suitable, for example, the use of a type of BF-STEM imaging for particularly thick specimens that would produce contrast reversals in ADF-STEM. Ercius et al [26.45] have shown that coherence artefacts can be minimized by using a large bright field detector, whose broad integration area effectively suppresses diffraction contrast, providing an incoherent bright-field ( ) image that satisfies the projection requirement. In particular, while ADF-STEM provides some compositional contrast through the \(Z\) dependence of the signal, where a more direct measurement of composition is needed, the use of analytical signals capable of measuring composition (and other properties) directly is required.

26.3.4 Aberration-Corrected and Atomic-Scale TEM and STEM Tomography

AC TEM and STEM, in tandem with new advanced reconstruction schemes, have opened up opportunities for atomic-scale ET, which have seen considerable development over the last 5 years or so. ET has pushed beyond the long standing \({\mathrm{1}}\,{\mathrm{nm^{3}}}\) gold standard [26.69, 26.88], well into the atomic regime.

Although early pioneering studies showed the possibility to achieve 3-D atomic-level detail [26.37, 26.89], the seminal study of Van Aert et al using discrete constraints and a regular atomic lattice was the first to achieve 3-D atomic-scale reconstruction of an Ag nanocrystal [26.38]. More recently, the development of new reconstruction schemes (see later) has enabled 3-D study of atomic-scale defects and subtle changes in atomic-scale morphology. These include crystal domain (grain) structure and atomic packing [26.90, 26.91, 26.92, 26.93, 26.94, 26.95], crystallographic defects including dislocations, stacking faults and vacancies [26.91, 26.92, 26.95, 26.96], atom-by-atom chemical distributions [26.93, 26.96, 26.97, 26.98], and atomic-scale strain fields [26.92], offering exciting insights and opportunities for materials science (Fig. 26.13a-c).

Similarly, the crystallographic or noncrystallographic structure of decahedral and icosahedral nanoparticles has been a perplexing issue for decades, with competing theories as to how strain is accommodated in fivefold twinned geometry e. g., [26.99] and structural complexities hidden in projection images. 3-D studies by atomic-scale ET provide new means to address long-standing issues, opening the door, for example, to mapping the 3-D strain state as shown in Fig. 26.14. Here, a systematic lattice expansion is measured along both the \(x\) and \(z\) directions, but the expansion along \(z\) is limited to only the outer few atomic layers and shows asymmetry likely due to the decahedron resting on an amorphous carbon support [26.92].

Fig. 26.13a-c

3-D determination of atomic coordinates, chemical species and grain structure of an FePt nanoparticle. (a) Overview of the 3-D positions of individual atomic species with Fe atoms in red and Pt atoms in blue. (b) Multislice images obtained from the experimental 3-D atomic model along the [100], [010], and [001] directions. Scale bar is \({\mathrm{2}}\,{\mathrm{nm}}\). (c) The nanoparticle consists of two large L1\({}_{2}\) grains, three small L1\({}_{2}\) grains, three small L1\({}_{0}\) grains, and a Pt-rich A1 grain. From [26.98]

Fig. 26.14

(a) 3-D visualization of the tomographic reconstruction of a gold nanodecahedron; the arrows indicate planar defects inside the decahedron. (b,c) 3-D strain analysis. Slices through (b) the \({\upvarepsilon}_{xx}\) volume and (c) the \({\upvarepsilon}_{zz}\) volume. Reprinted with permission from [26.92], published under ACS AuthorChoice License, permissions requests should be directed to the ACS

26.3.5 Analytical Electron Tomography

Nano-analytical techniques undoubtedly play a significant role in many (S)TEM investigations, enabling mapping of physical properties, such as local chemistry. These may be extended to 3-D by utilizing the signal in ET, providing a tomogram with one or more additional signal dimensions beyond the spatial domain (Fig. 26.15). Multidimensional or analytical electron tomography ( ) has grown significantly in recent years, with EELS and energy-dispersive x-ray spectroscopy ( ) being the mostly widely implemented forms of AET to date. They can provide element selective imaging for 3-D mapping of composition [26.100, 26.101, 26.102] and, under suitable circumstances, chemistry (e. g., local valency [26.22]), electronic properties [26.103], and optical properties [26.104]. Under favorable circumstances, analytical tomograms can be interrogated quantitatively, for example, to determine local elemental concentration [26.100]. Although more electron dose intensive than conventional structural imaging techniques, recent developments in hardware coupled with data handling capabilities and advanced reconstruction algorithms have bought these signal modes into feasibility for ET and generated significant activity to further develop rich new opportunities. While most AET studies to date have been on beam-resistant specimens, optimized and novel methodologies can and should increasingly enable application in a wider range of contexts. Further techniques including electron holographic and crystallographic tomography in (S)TEM add to a growing scope for multidimensional AET investigations.

Fig. 26.15

Principle of AET, in which each voxel in 3-D space contains additional signal dimensionality, in the form of, for example, an energy spectrum, or diffraction pattern

EFTEM and STEM-EELS Tomography

(S)TEMs equipped with a post-column (or occasionally in-column) electron energy spectrometer offer the opportunity to pursue EELS tomography, recording characteristic losses of the electron beam on interaction with the specimen in the form of energy-loss images. This may be performed using energy-filtered TEM ( ), where an energy selecting window is placed over a characteristic energy-loss feature, and an image formed using only electrons inside that window; or by STEM-EELS spectrum-imaging, where a spectrum covering a chosen energy range is recorded pixel-by-pixel.

Depending on the particular energy-loss range, the images may characterize different properties, requiring specific consideration of the projection requirement. The signal from core-loss ionization edges (whose onset is determined by the characteristic energy required to promote an inner-shell electron of a particular type of atom) can be obtained in EFTEM by using additional (usually two pre-edge) EFTEM images to enable subtraction of the background under the edge. This then yields an elemental map, and by acquiring a tilt series of such maps for tomographic reconstruction, an element-sensitive tomogram can be obtained.

EFTEM ET has been a recognized method for a number of years (it was first demonstrated in 2001 [26.48, 26.49]) and has proven to be of value in many contexts [26.105, 26.106]. Figure 26.16a, for example, shows a 3-D iron elemental map revealing the morphology of an iron-based catalyst nanoparticle at the top of a multiwall carbon nanotube ( ). Determining the 3-D form of each chemical constituent in the iron-filled CNTs is one of the key factors for understanding the growth mechanism and potential applications [26.105].

However, the size of energy windows required for mapping (typically \(10{-}20\,{\mathrm{eV}}\)) limits analysis of fine spectral information. With the development of fast and efficient spectrometers enabling acquisition in acceptable times and electron doses, there has been rising development of ET based on STEM-EELS. Advances in spectrometers and in the design and wide availability of monochromators for the incident electron beam have also brought improved energy resolution.

The spectrum at each pixel in a STEM-EELS spectrum image can be analyzed post-facto, enabling maps to be obtained from any energy-loss channel in a versatile manner. Similar to EFTEM, 3-D elemental maps can be obtained using core-loss ionization edges across a tilt series of STEM-EELS spectrum images, and it is also possible to utilize the fine structure at these edges to map specific phases, and electronic properties and bonding across 3-D space [26.107, 26.108, 26.22]. Figure 26.16b shows the bonding states of silicon in a semiconductor device in 3-D; the silicon is identified either in its elemental form or as an oxide, or as part of a metal silicide or silicon nitride [26.22]. In favorable cases, it may be possible to directly map the valency of certain elements. Materials with empty 3d and 4f shells are amenable to this because of a pronounced and intense EELS fine structure that can be used as a fingerprint to determine the valency. The example in Fig. 26.17a,ba shows a ceria nanoparticle in which the particle surface is shown to be predominantly \(\mathrm{Ce^{3+}}\) in character and the core \(\mathrm{Ce^{4+}}\) [26.103], and Fig. 26.17a,bb shows the iron distribution in an iron oxide core-shell particle, distinguishing between the Fe(II) and mixed Fe(II) and Fe(III) contributions of the two oxides [26.107].

Fig. 26.16

(a) EFTEM ET elemental mapping of an iron-filled multiwalled carbon nanotube. Reprinted from [26.105]. (b) STEM-EELS ET chemical state mapping of silicon in a W-to-Si contact from a semiconductor device. Reprinted from [26.22], with permission from Elsevier

Fig. 26.17a,b

STEM-EELS ET valence state mapping revealing different surface states of (a) a ceria nanoparticle. Reprinted with permission from [26.103]. Copyright 2014 The American Chemical Society. (b) Changes in Fe valency in a FeO/\(\mathrm{Fe_{3}O_{4}}\) nanocube. Reprinted with permission from [26.107]. Copyright 2016 The American Chemical Society

STEM-EELS and EFTEM in the low-loss region may also encode valuable electronic information such as plasmonic behavior. The bulk, or volume, plasmon energy depends on the local electron density, and plasmon EFTEM images, utilizing narrow energy windows ca. \(1{-}2\,{\mathrm{eV}}\) in width, may be used as input for tomographic reconstructions. Figure 26.18a shows an early example in which islands of silicon can be distinguished from the silicon oxide matrix by the shift in the plasmon energy [26.109]. More recently, the surface plasmon modes of a Ag nanoparticle were reconstructed in 3-D (Fig. 26.18b) [26.104] and a similar reconstruction undertaken using cathodoluminescence to study the modes of a gold nanosphere [26.110].

Fig. 26.18

(a) EFTEM tomography plasmon mapping, revealing the morphology of silicon nanoparticles embedded in silicon oxide. Reprinted from [26.109], with the permission of AIP Publishing. (b) 3-D rendering of surface plasmon modes of a silver nanocube, reconstructed using STEM-EELS tomography [26.104]. (c) Cathodoluminescence tomography maps of a gold-polystyrene nanocrescent [26.110]

Mapping elemental concentration directly is ideally suited to meeting the projection requirement. However, several factors can complicate STEM-EELS or EFTEM, restricting the range of amenable specimens or requiring careful protocols to mitigate their potential impact. At specimen thicknesses above a characteristic inelastic mean free path for the material (typically \(\approx{\mathrm{100}}\,{\mathrm{nm}}\) [26.111]), the influence of multiple inelastic and plural scattering becomes significant, and the apparent elemental signal may actually begin to fall. Thus the core-loss signal from thick specimens will no longer satisfy the projection requirement. Methods to remove the effects of plural scattering, while available, are not straightforward and are complicated by changes in thickness with tilt, and have not been significantly developed for ET to date. Diffraction contrast in crystalline specimens can similarly complicate STEM-EELS, and further complications may arise in anisotropic materials whose response will change markedly with tilt.

The introduction of new spectrometer technology enabling near simultaneous acquisition of both low-loss and core-loss spectra (dual EELS) [26.112] is a significant development, enabling the development of STEM-EELS ET in which many of the traditional challenges can be overcome [26.108, 26.113]. It also opens the door to enhanced analyses and new possibilities, including absolute (as opposed to relative) quantification of elemental concentration [26.108].

EDXS Tomography

ET utilizing energy dispersive x-ray spectroscopy ( ) has recently advanced significantly with the availability of x-ray detectors for (S)TEM with significantly higher solid angles, detection efficiency, and processing capabilities. Early attempts at EDXS ET [26.114, 26.50] were limited by the poor efficiency of conventional detectors. These subtend solid angles of only \(0.1{-}0.3\,{\mathrm{sr}}\) and are positioned on one side of the specimen, which creates considerable problems from holder shadowing when tilting in ET. Modern silicon drift detectors with much higher solid angles (\(\approx{\mathrm{1}}\,{\mathrm{sr}}\)), faster processing capabilities, and multiple detector chips positioned symmetrically around the optic axis (such that there are always chips in the sight of the specimen) have stimulated new scope, and EDXS ET studies are rapidly growing in number, fidelity, and value (Fig. 26.19).

Fig. 26.19

(a) STEM-EDXS elemental tomograms of Au-Ag bimetallic nanorings, showing (left) irregular Au surface segregation and (right) more uniform Ag segregation. Reprinted with permission from [26.100] published under ACS AuthorChoice License, permissions requests should be directed to the ACS. (b) STEM-EDXS tomograms revealing 3-D elemental distributions in an organic/inorganic core/multishell nanowire. Reproduced from [26.101], published under CC-BY 4.0 license

In essence, EDXS provides a signal well suited to tomographic reconstruction, providing a direct measure of elemental concentration. With characteristic peaks that lie on a relatively low background and few of the plural scattering problems that can complicate EELS of thick specimens, EDXS can in many regards provide a much simpler means of compositional mapping in both 2-D and 3-D. Nevertheless, 3-D mapping with EDXS ET is not without its challenges, which may compromise the projection requirement and require correction for high-fidelity ET and especially for quantitative analysis in 3-D. Even with new detectors providing vastly improved collection efficiency and positioned symmetrically around the specimen, detector shadowing can be a major consideration. Recent efforts have sought to characterize experimental configurations and parameterize the most important determining factors, including holder and detector geometry as a function of tilt angle, to enable corrections to be applied for detector shadowing [26.115, 26.116]. A second potential challenge that may require correction is x-ray absorption. Absorption correction procedures for ET have been proposed using the Cliff–Lorimer  [26.115] and \({\zeta}\)-factor [26.117] methods, which can also incorporate shadowing. Another approach has been to progressively refine absorption correction in an iterative reconstruction process [26.101]. In thick specimens, additional effects such as x-ray fluorescence, may become significant. Samples fabricated into needles by focused ion beam milling overcome holder shadowing [26.102, 26.118]. By reducing the volume of material on the x-ray path to detectors relative to a slab geometry they also reduce, though do not eliminate, absorption or fluorescence effects.

In many modern instruments, it is possible to collect the EDXS and EELS signals simultaneously. The great advantage of simultaneous acquisition is that the EELS signal is especially sensitive to light elements, and the EDXS to heavier elements, so the combination can provide a more complete 3-D chemical picture. Figure 26.20 shows 3-D elemental maps of an Yb-doped Al-\({\mathrm{5}}\,{\mathrm{wt\%}}\) Si alloy using simultaneous (right) STEM-EELS and (left) STEM-EDXS tomography [26.102]. Though such studies have been limited to date, there is considerable scope for advanced multidimensional multimodal analysis.

Fig. 26.20

3-D elemental maps of Yb (green), Si (red), and Al (blue) from an Al-\({\mathrm{5}}\,{\mathrm{wt\%}}\) Si alloy (6100 ppm Yb), obtained from simultaneous (right) STEM-EELS and (left) STEM-EDXS tomography. Local spectra are shown from the Yb-rich precipitates. Reproduced from [26.102] with permission of the Royal Society of Chemistry

In both STEM-EELS and EDXS ET, signal acquisition within a reasonable electron dose and/or time is still not easy. Methods that enable robust reconstruction from fewer images may be the only way to open up these imaging modes to less beam resistant specimens.

As well as the spectroscopic signals, the past 5 years or so have seen the inception, or progression, of a number of other advanced signal modes for ET. While remaining primarily the practice of a select number of groups, holographic and diffraction techniques have now been confirmed as valuable ET signal modes (they are reviewed in [26.119, 26.59] and [26.120], respectively). Indeed, automated acquisition has been developed for both techniques.

Crystallographic ET

Diffraction contrast that arises in TEM can, under favorable circumstances, be profitably utilized in dark-field TEM ET, using the objective aperture in the back focal plane of the objective lens to select a particular Bragg reflection to contribute to the image. By ensuring that the diffraction conditions remain approximately constant at each tilt angle, it has been shown that the projection requirement can be satisfied sufficiently well. Although the acquisition is challenging, this technique can be highly sensitive to small changes in crystalline orientation and has been used for imaging of defects such as precipitates [26.121] and dislocations (using the weak-beam dark-field technique) [26.41] or buried structures such as quantum dots [26.122].

For polycrystalline samples, the 3-D distribution of grains within the specimen volume may be reconstructed by acquiring a large number of dark field ( ) images varying sample tilt and scattering angle. By acquiring an ensemble of ca. \({\mathrm{100}}\,{\mathrm{k}}\) DF images the 3-D grain distribution in a polycrystalline Al specimen was reconstructed [26.123].

An alternative approach involves raster-scanning a near-parallel beam and recording a full 2-D diffraction pattern at each point in the raster—a technique that is sometimes known as scanning electron diffraction ( ). By repeating such a scan at a series of specimen tilt angles crystallographic information from a volume of material may be recovered (Eggeman et al [26.124] and more recently Meng and Zuo [26.125]). Such tilt series data can be extremely information rich, allowing for versatile 3-D crystallographic analysis. The diffraction patterns can be analyzed computationally post facto, and virtual dark-field or component images may be formed (the latter using multivariate statistical analysis ( ) methods), which can then be used to reconstruct, in 3-D, both real and reciprocal spaces, as illustrated in Fig. 26.21b. By interrogating subvolumes to retrieve local 3-D crystallography, it is possible to determine, for example, the orientation relationships between grains or phases and across interfaces. The scanned diffraction ET data sets also provide a promising means for 3-D mapping of crystallographic strain at the nanoscale [26.126].

Fig. 26.21

3-D crystallographic reconstruction of an Ni-based superalloy from scanning precession electron diffraction tomography showing a faceted metal carbide (blue), \({\upeta}\)-phase (green), and surrounding matrix (orange). At each point in real space (left), reciprocal-space information is available, as shown (right) for the lath-like \({\upeta}\)-phase. The colored overlay of spots is the auto-correlation of the zero-order Laue zone reflections with reciprocal lattice basis vectors marked [26.124]

26.3.6 Holographic ET

As discussed elsewhere in this book, electron holography (both in-line and off-axis holography) is exquisitely sensitive to changes in the phase of the wave brought about by variations in the sample's electrostatic or magnetic potential. Combining holography and tomography should, then, offer a route to exploring that electro-magnetic potential in 3-D. In general, we can describe the reconstructed phase image as a projection of a 3-D potential through the equation
$$\varphi\left(x,y\right)=C_{\text{E}}\int V\left(x,y,z\right)\mathrm{d}z-\frac{e}{\hslash}\iint\boldsymbol{B}\left(x,y\right)\cdot\mathrm{d}\boldsymbol{S}\;,$$
(26.6)
where the first integral is made parallel to the beam direction, \(z\), \(V\) is the crystal potential, \(\boldsymbol{B}\) is the magnetic induction and related to the magnetic potential \(\boldsymbol{A}\) by \(\boldsymbol{B}=\text{rot}\,\boldsymbol{A}\), and \(\boldsymbol{S}\) is normal to the area mapped out by the trajectories of the electrons going from source to detector; \(C_{\text{E}}\) is a wavelength-dependent constant.
In the absence of magnetic fields and diffraction contrast, and in the absence of stray fields outside the specimen, the phase change can be related directly to the mean inner potential \(V_{0}\) of the specimen
$$\varphi(x,y)=C_{E}V_{0}\left(x,y\right)t(x,y)\;,$$
(26.7)
where \(t\) is the specimen thickness.

This has been used to investigate changes in the electrostatic potential in 3-D in semiconductor devices containing a p-n junction where variations in the depletion region near the junction were revealed [26.53].

Mapping magnetic fields (or the magnetic induction) in 3-D is possible in principle, but the vectorial nature of the magnetic induction \(\boldsymbol{B}\) makes this challenging, as three components of \(\boldsymbol{B}\) must be found at each reconstructed voxel. The theoretical basis for undertaking such vector tomography has been extensively set out [26.127, 26.128, 26.129, 26.130]. The reconstructed phase from the hologram will be sensitive to both electrostatic and magnetic components and, in order to separate the electrostatic and magnetic phase shifts, a tilt series over a full \(360^{\circ}\) tilt range is needed. (Alternatively, two tilt series may be acquired, one before and one after reversing the direction of magnetization in the specimen, e. g., using the TEM objective lens or flipping the sample up-side down.) If two tilt series over \(360^{\circ}\) are acquired with mutually perpendicular tilt axes, two independent components of the magnetic induction may be reconstructed. Application of the no monopole condition \(\nabla\boldsymbol{B}=0\) enables the third component of \(\boldsymbol{B}\) to be found from the other two.

The first successful reconstruction of the full 3-D magnetic induction and vector potential was achieved by Phatak et al [26.131], using the transport of intensity ( ) approach to study a magnetic permalloy plate (Fig. 26.22a-g). More recently, Wolf and coworkers [26.132, 26.133] used a simplified experimental scheme, with off-axis electron holography, to achieve a quantitative reconstruction of one component of \(\boldsymbol{B}\) for needle samples, with the component parallel to the needle (and tilt) axis, Fig. 26.23a-f.

Fig. 26.22a-g

ET reconstruction of the magnetic vector potential outside a multidomain permalloy island. (a) The magnetic vector potential \(\boldsymbol{A}\) (red arrows) and the magnetic induction (blue). (b) Acquisition scheme to achieve data for the vector field ET. (ce) Fresnel images at a single tilt angle (under, in, and over focus, respectively). The location of a central vortex is marked with an arrow. (f,g) 3-D visualization of the reconstructed magnetic induction and vector potential, respectively. Reprinted with permission from [26.131]. Copyright 2010 by The American Physical Society

Fig. 26.23a-f

3-D electron holographic reconstructions of cobalt and \(\mathrm{Co_{2}FeGa}\) magnetic nanowires (NW s). 3-D volume rendering of electric potential (in volt) (a,b) and axial (predominant) \(\boldsymbol{B}\)-field component (in tesla) (c,d) inside the NWs. (e) Axial \(\boldsymbol{B}\)-field component inside the Co NW obtained from micromagnetic simulation. The arrow plots visualize the out-of-plane components showing the twist of magnetic induction. (f) Line scans in the axial direction through the centre of the NW from the tip to the back. (a,c,e) reprinted with permission from [26.132] published under ACS AuthorChoice License, permissions requests should be directed to the ACS. (b,d,f) reprinted with permission from [26.133]. Copyright 2016 The American Chemical Society

26.3.7 Time-Resolved ET

Compared to x-ray tomography, there has been relatively little use made of time-resolved ET. One key leap forward was made in the ultrafast electron community in a paper by Kwon and Zewail [26.54], in which a stroboscopic pump-probe technique was able to capture the 3-D dynamic changes in the vibration of a nanowire. For nonstroboscopic measurements, there has been some progress in acquiring tilt series over ever smaller time periods, allowing the possibility of dynamic tomography, especially in the BF-TEM mode, where use can be made of the remarkable sensitivity and efficiency of new direct electron detectors. One demonstration of this was done by Migunov et al [26.134], who, via continuous tilting, recorded a rapid low-dose tilt series in just a few seconds.

It is also plausible to see how unique in situ changes could be analyzed in 3-D via ET, if the tilt series of images were recorded sufficiently rapidly with respect to any change. This is clearly another aspect where few-image reconstructions could help. Many sample holders used for in situ TEM (e. g., with heating elements or gaseous chambers) have a significantly restricted tilt range, but rising interest is leading to the development of high tilt in situ holders by a number of manufacturers.

26.4 Tilt Series Alignment

Automated feature tracking during acquisition, computer control of goniometers, and improvements in stage design have greatly facilitated the successful acquisition of ET tilt series (ensuring that the specimen remains close to the centre of the field of view in each image). Nonetheless, post-acquisition alignment of the projections to a common tilt axis is almost always required and should, ideally, be with sub-pixel accuracy.

Where specimens have distinctive features, common in the physical sciences, the alignment is usually carried out using cross-correlation [26.2, 26.62] and [26.5, Chap. 6]. An alternative is to place high-contrast markers on the specimen or support film, usually gold nanoparticles, and to track these in each image to determine the required shifts [26.5, Chap. 5]. This approach is more common in biological applications, where the specimens often show lower contrast and, where features are more sporadically distributed throughout the reconstruction volume, may suffer less from obstruction by the markers. A number of software packages also provide facilities for manual adjustments to be made.

Alignment by cross-correlation is illustrated in Fig. 26.24a-f. The cross-correlation determines the match between two images across all lateral and vertical displacements and provides an output image (Fig. 26.24a-fc) whose intensity peak indicates the shift required to bring the features from the two images into coincidence. Often the sharpness of the cross-correlation peak, and, therefore, the accuracy of the determined shifts, can be improved by applying one or more filtering processes to emphasize or reduce the influence of certain features in the images. In Fig. 26.24a-fd–f, for example, a much sharper cross-correlation peak has been obtained by use of a Sobel filter to highlight edges. Since the projected view of the specimen is similar but not identical at successive tilts, the cross-correlation match will never be exact, and this may be particularly so for slab-like specimens and/or where additional objects enter into the field of view. Foreshortening of features in projection at successively higher tilts can be significant in extended slab-like specimens (Fig. 26.25). This can be alleviated by applying a linear stretch of \(1/\cos{\theta}\) to the projections, perpendicular to the tilt axis [26.135], restoring the spatial correspondence between successive projections.

Fig. 26.24a-f

Determining the relative shift between two tilt series images by cross-correlation. (a,b) Successive tilt series images and (c) the corresponding cross-correlation indicating their relative shift. (d,e) Sobel filtering of the images to yield a sharper cross-correlation peak (f)

Fig. 26.25

The spatial relationship between sample features at successive tilt angles, leading to foreshortening in projection. The correspondence between these projections can be restored by a stretch of \(1/\cos{\theta}\), as indicated by the red arrows

An advantage of marker-based alignment is that, as well as determining the required shifts, it also enables determination of the position and angle of the tilt axis, whereas with standard cross-correlation approaches this has to be performed by other means. Often, projecting a tilt series along the \(z\)-direction by summation, or other means such as maximum intensity projection, can provide an initial coarse estimate of the angle and lateral position of the tilt axis. Features of an object located at some distance from a fixed tilt axis of rotation appear, in projection, to move perpendicularly to the axis. Assuming that a tilt series is well aligned in \(x\) and \(y\), the tracks of distinctive features in a \(z\)-projection can, therefore, reveal the angle of the tilt axis, as exemplified in Fig. 26.26a,b. Features lying directly on the tilt axis appear stationary in location through a tilt series, and where such trackless features are seen, and from where tracks appear to emanate, reveals the lateral position of the tilt axis.

Fig. 26.26a,b

Tilt axis identification from \(z\)-direction projection. (a) A single HAADF-STEM image, at \(0^{\circ}\) tilt, of gold nanoparticles on a carbon support. (b) Maximum intensity \(z\)-projection of the full tilt series and its Fourier transform (inset), where the particle tracks reveal the tilt axis angle, \(15^{\circ}\) from the vertical, and location

Alignment of the tilt axis is also critical for high-fidelity reconstructions. Figure 26.27a-d illustrates how this can be achieved through minimizing the tarcing of features if the axis is misaligned. It is important to emphasize that accurate alignment is fundamental for high-fidelity and high-resolution ET reconstructions to be obtained. Indeed, critical to extending the achievable resolution and fidelity in recent years has been the development of specialized alignment procedures such as centre-of-mass-based approaches and refinement during iterative reconstruction [26.136, 26.137, 26.138]. These have been especially pertinent in atomic-scale ET and are also of importance to enable a growing trend for ET reconstruction from very few tilt series images. There is also ready opportunity for alignment procedures and software implementations developed in the biological sciences, such as feature or local patch tracking [26.5, Chap. 6], [26.139] to be applied in physical science contexts [26.140, 26.141].

Fig. 26.27a-d

Tilt axis alignment by minimization of arcing. (a) HAADF-STEM image of gold dog-bone nanoparticles, showing the location of a central and top/bottom slices chosen for preliminary reconstruction and tilt axis adjustment. (b) WBP reconstructions of each slice when the tilt axis is correctly positioned. (c) Incorrect tilt axis angle manifests as arcing artefacts in opposite directions in each of the top/bottom slices (as indicated schematically in the lower right-hand corner). (d) Incorrect lateral position of the tilt axis results in arcing artefacts in a common direction in all three reconstructed slices

26.5 ET Reconstruction

Various classifications have been used to differentiate or to group tomographic reconstruction algorithms. Considering the established algorithms in contemporary ET, they are classed here as falling into two groups:
  1. 1.

    Direct transform methods, including backprojection and Fourier techniques

     
  2. 2.

    Algebraic iterative methods, including the ART and SIRT-type classes.

     
Comprehensive mathematical description of the methods discussed can be found in [26.5, 26.6, 26.7, 26.8, 26.9], [26.10, Chap. 2]. A notable concise summary of the many different forms of ART and SIRT algorithms is given in [26.142].

26.5.1 Backprojection

The reconstruction method favored by the ET community has for many years been the weighted backprojection ( ) algorithm, owing in large part to its speed of execution and because the algorithm is well understood. In the most basic description [26.62], [26.78, Chap. 8], backprojection consists of smearing each projection from a tilt series back into space at the angle at which it was originally formed. By backprojecting a sufficient number of projections, the summation of the backprojected rays in the space will generate the original object; such direct backprojection was illustrated schematically in Fig. 26.2.

However, ET reconstructions from simple backprojection appear blurred because the radial sampling regime of ET (Fig. 26.9) leads to relative undersampling of higher spatial frequencies (Fig. 26.28a-cb). This can be corrected using a ramp-like weighting filter, usually applied to the projections in Fourier space. The result is a WBP [26.143], [26.5, Chap. 8], as shown in Fig. 26.28a-cc. While this filtering process has the benefit of enhancing edges, it can complicate any quantitative analysis of the voxel intensities in the tomogram.

Fig. 26.28a-c

The application of a weighting filter in backprojection reduces the blurring effects brought about through relative under-sampling of high spatial frequencies

26.5.2 Direct Fourier Inversion

Fourier-based reconstruction methods exploit the Fourier slice theorem outlined in Sect. 26.1.2, Fourier-Based Methods. Essentially, for an \(N\)-dimensional reconstruction, Fourier reconstruction entails application of an (\(N-1\))-dimensional discrete Fourier transform across the spatial dimension of the projections to obtain radial Fourier data. An \(N\)-dimensional inverse Fourier transform is then applied to this data set to recover the function in real space. However, as was indicated in Fig. 26.6 and is shown explicitly in Fig. 26.29, the data in the Fourier domain of the function lies on Cartesian coordinates, whereas the radial Fourier data is on a polar grid. To convert the data between the two coordinate systems requires some form of interpolation or gridding process. This step is challenging and can result in poor quality reconstructions if simple interpolation (such as bilinear) is used. As such, direct Fourier inversion methods have generally been disregarded in ET.

Fig. 26.29

(a) Radial and (b) Cartesian data points; and (c) an interpolation (gridding) process to convert between the two

Nonetheless, several Fourier-based reconstruction methods using sophisticated nonuniform Fourier transform, or gridding, procedures have been proposed recently in the biological ET and single particle microscopy context, with potential performance enhancements; reviewed by, for example, Penczek [26.9]. A more recent approach, in both biological [26.144, 26.145] and physical [26.104, 26.146, 26.147, 26.148, 26.149, 26.91, 26.94] sciences ET has been to combine sophisticated Fourier-based operators with iterative reconstruction.

26.5.3 Algebraic Iterative Reconstruction

In a qualitative description, algebraic iterative reconstruction ( ) techniques in ET operate by constraining the reconstruction to match the original projections, with the match being improved at successive iterations (Fig. 26.30). A difference reconstruction is obtained via a comparison of projections of the reconstruction with the original projections, either by division in multiplicative techniques or subtraction in additive techniques. The current reconstruction is then updated via multiplication or addition of the difference, respectively. This kind of iterative refinement by projection and reprojection can also be described mathematically in terms of projection onto convex sets [26.150].

Fig. 26.30

Principle of iterative tomographic reconstruction. (Shown specifically for a SIRT-type algorithm; based on the diagram of Weyland and coworkers [26.62])

Formally, AIR is based on the discretization of the projection process into a finite number of basis functions (\(n\) in total), as illustrated in Fig. 26.31. The projection system is represented by the matrix \(\boldsymbol{\Upphi}\), and the vector \(\boldsymbol{x}\) contains the function in \(n\) discrete points in space. The vector \(\boldsymbol{b}\) contains the ray sums, corresponding to a discretized sinogram, with \(m\) entries in total. The tomography reconstruction process can then be formulated as a system of linear equations
$$b_{i}=\sum_{j=1}^{n}\phi_{i,j}x_{j}\qquad\mathrm{for}i=1,\ldots,m\;,$$
(26.8)
where each \({\phi}_{i,j}\) is often calculated as the fraction of the \(j\)-th basis function intersected by the \(i\)-th projection ray, implying \(0\leq\phi_{i,j}\leq 1\). Equation (26.8) can be abbreviated as
$$\boldsymbol{b}=\boldsymbol{\Upphi}\boldsymbol{x}\;.$$
(26.9)
These equations represent an inverse problem, where the task is to estimate \(\boldsymbol{x}\) given the data \(\boldsymbol{b}\) and the projection matrix \(\boldsymbol{\Upphi}\). In ET, the limited number of tilt series projections means that there are far fewer equations than unknowns (i. e., \(m\ll n\)), and the system of equations is underdetermined, implying there is an infinite number of solutions consistent with the projection data. This is compounded by the ill-posedness arising from data imperfections, such as noise, projection misalignment, or diffraction contrast. AIR techniques such as ART [26.151] and SIRT [26.152] were proposed in the context of ET in the 1970s. With advances in computational power and efficient algorithmic implementations e. g., [26.153], they have been the mostly widely utilized reconstruction methods in materials science. Due primarily to greater stability when the projections are noisy, the SIRT algorithm has usually been preferred to ART in ET and, in the physical sciences, is generally seen as the established standard.
Fig. 26.31

Discrete representation of a tomographic experiment. The function \(f\) is represented by \(n\) discrete basis functions, which can be written as a vector \(\boldsymbol{x}\). Each ray sum is denoted by \(b_{i}\), with the set of ray sums forming the vector \(\boldsymbol{b}\). Linking \(\boldsymbol{b}\) and \(\boldsymbol{x}\) is the projection matrix \(\boldsymbol{\Upphi}\), where each element \({\phi}_{i,j}\) describes the contribution of the \(j\)-th basis function \(x_{j}\) to the \(i\)-th projection ray \(\boldsymbol{b}_{i}\)

26.5.4 Algebraic Reconstruction Technique (ART)

The classic AIR technique to solve (26.9) is that of Kaczmarz [26.154]. In the literature, this is often referred to simply as ART, but it is important to realize that ART also refers to a class of AIR techniques. Kaczmarz's method can be expressed with the additive update scheme
$$\hat{\boldsymbol{x}}^{k+1}={\hat{\boldsymbol{x}}^{k}}+\rho_{k}\cdot\frac{b_{i}-\left\langle\boldsymbol{\phi}_{i},\hat{\boldsymbol{x}}^{k}\right\rangle}{\left\|\boldsymbol{\phi}_{i}\right\|_{\ell_{2}}^{2}}\cdot\boldsymbol{\phi}_{i}\;,$$
(26.10)
where each \(\boldsymbol{\phi}_{i}\) represents a row of \(\boldsymbol{\Upphi}\) in (26.9), and hence
$$\left\langle\boldsymbol{\phi}_{i},\hat{\boldsymbol{x}}^{k}\right\rangle$$
denotes the standard inner product of the vectors \(\boldsymbol{\upvarphi}_{i}\) and \(\hat{\boldsymbol{x}}^{k}\). The index \(i\) addressed at the \(k\)-th iteration is given by \(i=(k\,\text{mod}\,m)+1\); \({\rho}\) is the relaxation parameter, which influences the sensitivity of the update to noise and may be fixed (\({\rho}\)) or vary at each iteration (\({\rho}_{\mathrm{k}}\)).
ART was introduced to the ET community by Gordon et al [26.151], who also presented a multiplicative form (sometimes referred to as MART). An update scheme can be written as
$$x_{j}^{k+1}=\left(\frac{b_{i}}{\left\langle\boldsymbol{\phi}_{i},\hat{\boldsymbol{x}}^{k}\right\rangle}\right)^{{\rho_{k}}\cdot{\varphi_{i,j}}}\cdot x_{j}^{k}\;,$$
(26.11)
where \(0{\leq}{\phi}_{i,j}{\leq}1\) and again \(i=(k\,\text{mod}\,m)+1\).
Another form of ART that has been extensively discussed in the ET literature is block-ART, first introduced by Eggermont et al [26.155], which for \(O\) blocks of \(P\) equations (where \(O\cdot P=m\)) may be written as [26.10, Chap. 2], [26.5, Chap. 7]
$$\hat{\boldsymbol{x}}^{k+1}={\hat{\boldsymbol{x}}^{k}}+\rho_{k}\cdot\sum_{i=o_{k}\cdot P+1}^{\left(o_{k}+1\right)P}\frac{b_{i}-\left\langle\boldsymbol{\phi}_{i},\hat{\boldsymbol{x}}^{k}\right\rangle}{\left\|\boldsymbol{\phi}_{i}\right\|_{\ell_{2}}^{2}}\cdot\boldsymbol{\phi}_{i}\;,$$
(26.12)
where \(o_{k}\) (\(0{\leq}o_{k}{\leq}O\)) is the index of the block to be used in the \(k\)-th iterative step.

As outlined by Kuba and Herman [26.10, Chap. 2], the essential difference between block-ART and the more conventional ART methods is that in the former the update proceeds by taking into account groups (blocks) of measurements that come from a particular projection, compared to the latter dealing with only one measurement at a time (i. e., one ray integral). If \(P=m\) (and, hence, \(O=1\)), then the method is said to be fully simultaneous [26.156, p. 100] and is closely related to SIRT-type methods. An intermediate case is when the blocks are formed by all the equations associated with a single projection, an example of which is the simultaneous algebraic reconstruction technique ( ) [26.157].

26.5.5 Simultaneous Iterative Reconstruction Technique (SIRT)

The other major class of AIR algorithms in ET are SIRT-type methods. As the name suggests, information from all the equations (projections) is used at the same time for the update process. This accounts for SIRT often being less sensitive to noise than ART.

SIRT methods can be written in the general additive form
$$\hat{\boldsymbol{x}}^{k+1}={\hat{\boldsymbol{x}}^{k}}+\rho_{k}\cdot\boldsymbol{\Upupsilon}\boldsymbol{\Upphi^{*}}\boldsymbol{\Upomega}(\boldsymbol{b}-\boldsymbol{\Upphi}\hat{\boldsymbol{x}}^{k})\;,$$
(26.13)
where \(\boldsymbol{\Upphi^{*}}\) is the (conjugate) transpose of \(\boldsymbol{\Upphi}\). The matrices \(\boldsymbol{\Upomega}\) and \(\boldsymbol{\Upupsilon}\) are symmetric and positive definite, and for most implementations \(\boldsymbol{\Upupsilon}\) is the identity transform ([26.142] for examples of functional roles played by these matrices). The most common variant is the Landweber method [26.158]
$$\hat{\boldsymbol{x}}^{k+1}={\hat{\boldsymbol{x}}^{k}}+\rho_{k}\cdot\boldsymbol{\Upphi^{*}}(\boldsymbol{b}-\boldsymbol{\Upphi}\hat{\boldsymbol{x}}^{k})$$
(26.14)
which corresponds to setting \(\boldsymbol{\Upomega}=\boldsymbol{\Upupsilon}\), the identity transform in (26.13). It is well known that in overdetermined cases (i. e., \(m> n\)) for which there is a unique solution, SIRT effectively solves a (weighted) least-squares problem of the form [26.159]
$$\hat{\boldsymbol{x}}=\arg\min_{\hat{\boldsymbol{x}}}\left\|\boldsymbol{\Upphi}\hat{\boldsymbol{x}}-\boldsymbol{b}\right\|_{\ell_{2}}^{2}\;.$$
(26.15)
However, in underdetermined problems (typical of ET) an infinite number of solutions \(\hat{\boldsymbol{x}}\) may exist that yield minimal discrepancy in (26.15) (or other AIR algorithms). As will be discussed later, in such cases, it can be distinctly advantageous to apply additional constraints during the iterative reconstruction that help to select from the possible solutions.

Further, in highly ill-posed scenarios, the standard AIR algorithms can exhibit marked semi-convergence, whereby initial iterations tend towards better approximations of the solution, but at some point may start to deteriorate to a poorer approximation (Fig. 26.32) [26.142, 26.160] and references therein]. This can be particularly problematic in ET, where semi-convergent type behaviors may occur when there is a high noise level in the projections or other significant inconsistencies such as projection misalignment, which become exacerbated at large iteration numbers. Important aspects that remain to be adequately addressed in this regard are optimal choice of the variables in the basic AIR algorithms, namely the relaxation parameter \({\rho}\) and the total number of iterations \(k_{\text{max}}\), both of which are important parameters influencing the outcome of the algorithm. Determination of the optimal number of iterations in ET often requires reconstruction for different iteration numbers and some form of qualitative or quantitative (e. g., [26.34]) assessment. Some ET software packages do not even allow \({\rho}\) to be altered. These parameters have been discussed in the past, but robust automated (i. e., nonempirical) methods for choosing or intelligently varying them (e. g., [26.142, 26.159, 26.160]) are yet to find marked endorsement in ET.

Fig. 26.32

Principle of semi-convergence in tomographic reconstruction: the initial iterates tend to better approximation of the exact solution, but above a certain number of iterations they begin to deteriorate

Slight variants of the conventional ET reconstruction algorithms include dual-axis SIRT [26.102, 26.161],  [26.162], and  [26.163, 26.164]. Dual axis SIRT [26.161] has been less popular, possibly due to the inherent difficulties in aligning dual axis tilt series and the added computational demands but has more recently been advocated as yielding potentially valuable resolution enhancement compared to single-axis ET, even for structures not affected by missing wedge artefacts [26.102]. WSIRT, proposed by Wolf et al [26.162], combines WBP and SIRT, showing improved convergence, resolution, and reconstruction error compared to SIRT alone, including a reduced point spread in the missing wedge direction. The DIRECTT algorithm of Lange et al [26.163, 26.164] resembles SIRT, but at each iteration only a selected portion of voxels in the reconstruction is updated, based on either their gray level error or local contrast. This favors high-density/contrast features, and the gradual introduction of voxel updates acts as a regularizing mechanism.

26.5.6 Advanced Reconstruction in ET

Compressed Sensing Electron Tomography

The relative paucity of data in ET experiments means that to achieve higher-fidelity reconstructions requires advanced methods that make best use of that data during tomographic reconstruction. The highly underdetermined and ill-posed nature of the ET reconstruction process implies that seeking data fidelity alone will be insufficient. In this case, it is well-known, from the field of inverse problems, that to improve the fidelity or quality of a tomographic reconstruction, some form of prior knowledge constraints (often called regularization) can be introduced during the reconstruction process. The regularization selects out of the possible solutions to the underdetermined system of equations those which additionally satisfy the prior knowledge characteristics, and therefore, in principle, should reduce the number of projections required for reconstruction.

In general, as the level of undersampling increases, so must the strength or efficacy of the prior knowledge constraints, if reconstruction fidelity is to be maintained. Caution should be noted in this regard though, as the fidelity of the outcome depends on validity of any prior knowledge constraints imposed. Stronger constraints can be introduced to bias the results towards a particular outcome, but this outcome will only be of high fidelity if the constraints are valid. The ideal scenario is one in which the prior knowledge constraints are relatively liberal but effective during the optimization process and accurately describe the object. In some cases though, it may be necessary to sacrifice some degree of reconstruction fidelity to obtain a reconstruction that possesses other desirable characteristics. For example, a reconstruction that has been biased so that each of its constituent objects possess homogeneous density and sharp boundaries—whether this is true or an approximation of the object—may make it easier to identify and analyze those objects.

With advances in computational power and mathematical methods there have been considerable recent developments in bringing reconstruction methods incorporating prior-knowledge constraints to ET. One method of signal recovery from undersampled data that has seen huge growth in interest and application recently is compressed sensing ( , also referred to as compressive sensing or compressive sampling) [26.165, 26.166]. By exploiting the sparsity implicit in many signals, CS methods are able to recover signals with remarkably high fidelity from far fewer measurements than traditionally would have been necessary. CS or more generally sparse regularization and related approaches have now gained significant attention in the context of ET and have provided high-fidelity tomographic reconstructions even from very few projections [26.146, 26.147].

The application of recovery methods exploiting sparsity is growing rapidly, including in areas such as x-ray computed tomography [26.167], magnetic resonance imaging ( ) [26.168, 26.169] and single particle microscopy [26.170, 26.171]. CS harnesses principles of transform coding and sparse approximation that are well established from their use in image compression algorithms. For example, for the ubiquitous JPEG and JPEG-2000 image compression standards, sparse representation is provided by the discrete cosine transform (DCT ) and the discrete wavelet transform ( ), respectively [26.172].

Formally, the representation of a signal \(\boldsymbol{x}\) (such as an ET reconstruction) in a basis \({\Uppsi}\) is said to be sparse if there are few (\(s\) in total) nonzero coefficients in that representation, i. e., \(s\ll n\), where \(n\) is the full dimension of the signal in its native domain. In this case, only \(s\) coefficients in the basis contain all the information about \(\boldsymbol{x}\). If \(\boldsymbol{x}\) can be well approximated by \(s\ll n\) nonzero coefficients, \(\boldsymbol{x}\) is said to be compressible in \({\Uppsi}\); here, there may be many small negligible coefficients, which can be set to zero, and only \(s\) significant coefficients. A compressible representation of \(\boldsymbol{x}\) in the basis captures only the most important information about \(\boldsymbol{x}\) in \(s\) coefficients. A wide variety of transforms are available for this task, offering scope for sparse representation approaches to be wide reaching. A simple, illustrative, example of sparse representation is shown in Fig. 26.33a-f.

Fig. 26.33a-f

The key operators, domains, and prerequisites in CS-ET. The image in (a) can be represented sparsely in the gradient domain (ai), via a spatial finite differences operator. In ET a limited number of images (b) are acquired in the transmission electron microscope over a finite tilt range (leaving a missing wedge of unsampled information (c)). CS recovery proceeds (ce) by minimizing the number of nonzero coefficients in the (sparse) transform domain (e), whilst ensuring consistency with the measured data (c) yielding a reconstruction (f) that is sparse in the transform domain (fi)

Fig. 26.34a,b

Comparison between the reconstructions of an ensemble of nanoparticles using (a) CS and (b) SIRT methods. The reduction in fan artefacts in particular is clearly seen in both orthoslice orientations

Consider the approach first for image compression. An image is first fully sampled and then transformed into a chosen domain (e. g., a wavelet domain). If the transform has been chosen correctly, the number of significant transform coefficients will be relatively small with many less important ones being discarded. Thus, the amount of stored information representing the image is reduced or compressed. However, such data reduction, if carried out correctly, should not lead to any significant loss of fidelity in the recovered image.

Within the CS framework, however, we keep in mind the possibility of using transform sparsity and compressibility during the initial acquisition, with the aim to record a relatively small number of samples but that are sufficient to capture the important information in the signal. In other words, we aim to record the signal directly in compressed form.

Unlike image compression methods, where the sparse coefficients are known, with a CS approach these need to be recovered by searching for the sparsest signal in the transform domain that is consistent with the measured data. This can be performed very effectively using the \(\ell_{1}\)-norm, defined as the sum of the absolute values. A popular formulation is
$$\hat{\boldsymbol{x}}_{\lambda}=\arg\underset{\hat{\boldsymbol{x}}}{\min}\left(\left\|\Upphi\hat{\boldsymbol{x}}-\boldsymbol{b}\right\|_{\ell_{2}}^{2}+\lambda\left\|\Uppsi\hat{\boldsymbol{x}}\right\|_{{\ell_{1}}}\right)$$
(26.16)
where \({\lambda}\) is a parameter that weights the relative importance of the transform domain sparsity versus the data fidelity in the reconstruction. As further illustrated in Fig. 26.33a-f, CS recovery seeks the sparsest signal in the transform domain that is also consistent with the measured data.

The ability to use many different imaging modes for (S)TEM-based tomography leads to a range of image contrast and texture. For each mode, we need to consider the most appropriate sparsifying transforms for a CS-ET reconstruction. Many effective transforms have now been developed [26.173] for CS-ET, and the most important are outlined below.

A strong focus of ET is in the 3-D reconstruction of nanoscale objects, which are often restricted in one, two, or even three dimensions. As such, many (S)TEM images and ET reconstructions may be considered sparse in the image domain itself, and thus the sparsifying transform \({\Uppsi}\) is simply the identity transform. The finite and limited angular sampling in ET can lead to prominent streaking artefacts in the reconstruction, especially in the missing wedge direction. By imposing sparsity in the image domain such artefacts may be reduced [26.146] (Fig. 26.34a,b). This has proven to be effective even in atomic resolution STEM tomography of gold nanorods [26.39], where each atomic potential may be considered as sufficiently localized in space such that an atomic scale ET reconstruction should be inherently sparse. Sparsity constraints in the image domain work well only if the background is zero, and so any background intensity should be excluded from reconstructions. If that is not possible, or if the object of interest occupies a large portion of the field of view, then an image domain sparsity constraint is less applicable and other sparsifying transforms should be considered.

An alternative, and increasingly popular, sparsifying transform is spatial finite differences. In this transform, a constraint is imposed on the number of discontinuities in the image and the homogeneity of objects. The \(\ell_{1}\)-norm of the spatial gradients of the image, often referred to as the total variation ( )-norm [26.174], penalizes many small variations in the image intensity, but allows a limited (i. e., sparsely distributed) number of large gradients. A TV constraint is especially suitable for images that consist of homogeneous regions with sharp boundaries, often referred to as piecewise constant, and in the physical sciences is ideal for reconstructing small numbers of homogeneous phases, such as nanoparticle systems [26.39]. In an early application of CS-ET [26.146], TV-minimization was applied simultaneously with image domain sparsity to reconstruct with high-fidelity concave iron oxide nanoparticles using just nine projections.

Another useful sparsifying transform is that of DWT [26.175], in which wavelet coefficients capture both spatial position and spatial frequency information. Wavelets are, therefore, able to represent smooth, and piecewise smooth, signal content, including nonperiodic features such as jumps and spikes. There are now a number of studies using DWTs, including denoising of biological ET reconstructions [26.176] and single particle images [26.177], orientation determination in single particle microscopy [26.178], and single particle 3-D reconstruction [26.171].

Finally, a discrete Fourier transform may be used to provide a sparse representation of an image containing periodic features (such as a crystal lattice). In real systems, however, defect structures or finite periodicity will decrease the sparsity of any Fourier representation [26.179]. DCT, a variant of the discrete Fourier transform, may be applied locally and provide sparse representation of locally oscillating textures in natural images [26.180]. However, although providing sparse representations, the Fourier and DCT domains are not incoherent with the signal domain used for ET (the Radon/sinogram or Fourier domain). As such, these transforms are generally not suited for ET reconstructions, but they may be of use for pre/postprocessing in ET, for example, CS-based in-painting of fiducial markers [26.179].

Discrete Tomography

Another way of incorporating prior knowledge during tomographic reconstruction that has been quite extensively developed in the context of ET is the method of discrete tomography [26.181, 26.8], which can be used to provide high-quality and high-fidelity reconstructions if features of the specimen can be considered in discrete terms. A specimen could, for instance, be considered to consist of a discrete number of constituents of uniform density [26.182] or to consist of discrete elements that lie on a regular grid, such as atomic positions in a (perfect) nanocrystal [26.183, 26.38]. The application of such techniques in ET have been advanced by Batenburg and coworkers in particular, using a class of algorithm known as the discrete algebraic reconstruction technique (  [26.182]; a more mathematical description is given in [26.184]). These have shown profitable results, including reduction of missing wedge artefacts [26.182, 26.25] and reconstruction from few projections [26.146] in nanoscale ET, as well as enabling atomistic ET studies [26.185, 26.38].

DART has received quite wide recognition in the physical sciences, and a number of variants have been proposed by the Batenburg group and others [26.186, 26.187, 26.188]. At their core, these algorithms harness SIRT, but additionally introduce thresholding and gray level assignment during the iterative refinement. A considerable advantage of discrete approaches is that objects are segmented during the reconstruction process, as they are assigned to a particular discrete group. In many regards, this partitioning into homogeneous regions with sharp boundaries is very similar to a total variation constraint in CS-ET, but is stricter in forcing regions to a specific gray level, rather than still permitting small variations. Figure 26.35a-f shows conventional SIRT and DART reconstructions of a bamboo-like carbon nanotube containing an iron catalyst.

Fig. 26.35a-f

Comparison of (a,b,c) SIRT reconstruction and (d,e,f) DART reconstruction of part of a bamboo-like carbon nanotube. Reprinted from [26.182], with permission from Elsevier

However, the ET practitioner must consider that many real samples may not fully satisfy discrete constraints. Even if a high-quality discrete reconstruction of such samples can be obtained, it may not be of high fidelity. Moreover, often such strong prior knowledge is not available, although methods for automatic gray level selection may help in this regard [26.187]. To relax the strict constraints and increase the level of automation, further modifications of DART that have been developed have included partial discreteness [26.189], adaptivity [26.186], and combination with TV regularization [26.190]. Other variants of discrete tomography advocated for ET include the binary algebraic reconstruction technique ( , [26.191]) and the Bayesian approach of Wollgarten and Habeck [26.192]. Figure 26.36a-i shows a comparison of SIRT, total variation, and discrete tomography reconstructions of facetted nanoparticles, where it can be seen that elongation in the missing wedge direction present in the SIRT reconstruction is largely negated with total variation or discrete tomography. The total variation regularization promotes broadly homogeneous intensity in the nanoparticles and sharp boundaries, compared to the binary discrete tomography reconstruction. One slight problem noted in this study is that diffraction contrast was present in some of the tilt series images. As noted previously this, in general, is bad for the tomographic reconstruction, as it breaks the purely thickness dependence of the signal. The discrete tomography reconstruction struggled to deal with this, hindering correct estimation of the particle boundaries and leading to the small artefacts indicated by the arrows [26.97].

Fig. 26.36a-i

Comparison of (a,d,g) SIRT reconstruction, (b,e,h) TV-based reconstruction and (c,f,i) a DART reconstruction of a series of facetted nanoparticles. The double-headed arrow in (g) shows the elongating effect of the missing wedge, and (i) the arrow shows a possible artefact at the surface in the DART-based reconstruction. Reprinted from [26.97], with permission from Elsevier

Geometric Tomography

Another class of advanced algorithms comes from the field of geometric tomography [26.193]. These are chiefly concerned with recovering the shape of objects and mainly incorporate prior knowledge regarding convexity and homogeneity. As shown by Saghi et al [26.67], these approaches can be valuable when nonlinearities, such as diffraction contrast or detector saturation, are prevalent in the tilt series projections. In addition to a geometric surface tangent algorithm proposed by Petersen and Ringer [26.194], a selection of geometric algorithms from the mathematical literature has been explored by Alpers et al [26.191], including reconstruction from very few projections when strong geometric prior knowledge is available. Limitations of geometric algorithms are that the mass-density distribution is neglected, i. e., they assume homogeneity and/or recover only external or internal shape or edges, and in a number of cases, the object to be reconstructed must be convex.

Fourier-Based Methods

The past few years have seen a resurgence of Fourier-based ET reconstruction, with major advances being made through combination with iterative refinement and implementation of constraints. This includes Fourier-based implementations of CS-ET (which can also be performed using a real space projection operator) and the development and application of an approach known as equally sloped tomography ( ) [26.148, 26.94]. A distinct feature of EST is preference for acquisition of projections at equal slope increments, as opposed to conventional equal angular increments (or other schemes such as the Saxton scheme [26.195]). Equally sloped sampling, in principle, enables high-accuracy implementation of a pseudopolar fast Fourier transform ( ) to convert between the pseudopolar coordinates of the projections and Cartesian coordinates. However, significant efficacy of the algorithm arises because of the combination of a PPFFT with oversampling and iterative refinement, during which constraints such as positivity and finite spatial support can be imposed. As outlined by Miao et al [26.148], the iterative process in EST can result in filling in some information in the missing wedge due to correlation among Fourier components.

EST has been used in a series of studies at the atomic scale, combining the reconstruction with methods for identification of atom positions to show defect structures in small nanoparticles [26.149, 26.91, 26.94]. More recently, a generalized Fourier iterative reconstruction ( ) algorithm has been developed to incorporate various physical constraints and refinements, such as tilt angles, and was used in the aforementioned reconstruction of a bimetallic FePt nanoparticle to reveal chemical order/disorder (Fig. 26.13a-c) [26.98].

Single-Particle Reconstructions

Another notable development has been the application of single-particle microscopy techniques in the physical sciences. Well established in the biological sciences e. g., [26.196, 26.197, 26.5], single-particle methods involve recording many thousands of images of identical particles (e. g., viruses) at different orientations and using the ensemble of images to reconstruct the particle in 3-D. Single-particle methods in materials science have been more limited by virtue of the fact that significant populations of identical or near identical specimens (usually nanoparticles) are less common. However, there do exist certain magic number atomic clusters and nanoparticles whose configurations can be particularly stable, and for which single-particle approaches can be used. While the identification of magic nanoparticle morphologies from comparison of 2-D (S)TEM images to model structures has long been performed, Azubel et al [26.90] recently adapted low-dose aberration-corrected TEM and SPM approaches to determine the 3-D atomic structure of \(\mathrm{Au_{68}}\) nanoparticles.

Park et al [26.95] determined the structure of few-nm Pt nanoparticles at near atomic resolution using single particle methods by exploiting free rotation of the nanoparticles in a graphene liquid cell to obtain multiple viewing angles. This approach not only enabled the study of unique nanoparticles and their defects, but is also significant in extending 3-D electron microscopy to address in-situ contexts. The use of direct electron detection to obtain high-quality, low-dose, and rapidly acquired images of particles (that are undergoing motion) mirrors the successful exploitation of this new technology in biological contexts [26.198, 26.199].

Machine Learning

An increasingly pertinent need is to extract information content efficiently from potentially large multidimensional ET data sets. Often, this involves reducing the data down to a more manageable size. Here multivariate statistical analysis or machine learning methods can be of particular value. These can be used for improvement of signal to noise, but also provide powerful separation of significant components in data in a more objective manner. Examples of use in ET include principal component analysis and independent component analysis [26.107, 26.200], and nonnegative matrix factorization [26.104, 26.124, 26.201]. As an example, Fig. 26.37 shows how component analysis can be carried out successfully on a tilt series of EDX spectrum images of an Ni base superalloy to discover that there are six components of interest in the sample with corresponding loading maps at each tilt. Those loading maps may be used as input to reconstructions so as to form a 3-D loading map indicating where that EDX component dominates in the specimen volume.

Fig. 26.37

(a) A HAADF STEM image of a region close to the tip of the needle and 2-D element maps of the major alloying elements extracted from EDX spectra. (b) A scree plot of the 50 principal components. (c) Independent component weightings (IC#0–IC#4) at \({-}\)30\({}^{\circ}\) tilt and (d) their corresponding component spectra. (e) Orthoslices of the reconstructions of two independent components. (f) Segmentation and visualization of the \({\upgamma}^{\prime}\) strengthening phase after ellipsoid fitting. From [26.200], published under CC-BY license

Model-Based Reconstructions

Another way in which electron tomographic reconstruction may be improved is through better modeling of the electron–specimen interactions. Typically, as alluded to already, the signal used for conventional tomographic reconstruction should be a monotonic function of a projected physical quantity, such as composition or thickness—the projection requirement, However, as ET expands into increasing use of analytical signals, probing perhaps more complex properties or with more complex electron–specimen interactions, deviations from the projection requirement will need to be accounted for. Thus, the recent expansion of ET reconstruction methods incorporating constraints is being accompanied by more comprehensive modeling of signal formation processes as part of the reconstruction scheme [26.202]. Model-based reconstruction approaches offer the ability to account for and utilize signals that are not simple projections. As well as improving the fidelity of established imaging modes, where violation of the projection requirement has been tolerated, they offer scope for significantly broadening the range of properties and phenomena that can be studied by ET. Figure 26.38, for example, summarizes a model-fitting approach developed by Collins et al [26.201], matching simulated and experimental EEL spectra, for refinement of the underlying charge density of the individual localized surface plasmon eigenmodes.

Fig. 26.38

(a) Summary of key processing steps in eigenmode tomography. (b) Electron tomography and tilt series EELS of the surface plasmon modes of a silver right bipyramid on a \(\mathrm{MoO_{3}}\) substrate. Surface representation of a CS-ET reconstruction with the zero tilt HAADF micrograph and a plan-view perspective of the right bipyramid. The inset scale bar is 25 nm. Surface mesh used for simulations. Spectral NMF decomposition factors corresponding to surface plasmon mode excitations. HAADF images, experimental NMF component maps, and simulated loss probability maps at the corresponding tilt angles. (c) Calculated corner (dipole) eigenmode and corresponding surface charge reconstructions for a silver right bipyramid on a \(\mathrm{MoO_{3}}\) dielectric substrate. Reprinted with permission from [26.201] published under ACS AuthorChoice License, permissions requests should be directed to the ACS

26.6 Segmentation, Visualization, and Quantitative Analysis

In order to analyze a tomogram quantitatively, for example to determine surface area, volume fraction, crystallography, or porosity, it must first be segmented. This involves assigning each voxel in the tomogram to a feature of interest, for instance a nanoparticle, the vacuum, or the substrate. Segmentation is based upon classifying the voxels based on some similarity metric, such as their intensity, spatial location, or local characteristics such as the image gradient or textural patterns. The difficulty in achieving segmentation of ET reconstructions has meant that, often, they have been treated only in a qualitative manner. Alternatively, in a number of cases where segmentation has been achieved, it has only been through labor intensive manual procedures, in which the identification of features and delineation of their boundaries is open to individual interpretation. Indeed, segmentation can be the most time-consuming part of the ET work flow. Image processing techniques may facilitate automated or semi-automated segmentation, and recognition of their important role in ET has gradually grown over recent years in both the biological and the physical sciences.

Segmentation is one of the most difficult aspects of ET to summarize because of the wide variety of different methods employed. The segmentation requirements will depend on both the nature and the quality of the ET reconstruction, and therefore often need to be developed on a case-by-case basis. However, the key aspects of image processing-based segmentation can be generally applicable to many similar data sets with small adjustments.

Several advanced segmentation methods have received attention [26.203, 26.204, 26.205], [26.10, Chap. 11], [26.78, Chap. 8], [26.5, Chaps. 11–15], mainly originating from the biological ET community. These include denoising by anisotropic nonlinear diffusion [26.206], watershed transformation [26.198], and gradient vector flows [26.207]. Examples of more recently proposed advanced segmentation techniques are noise reduction utilizing Beltrami flow [26.208], application of fuzzy set theory [26.209], and segmentation of thin structures using orientation fields [26.210].

Segmentation procedures used in physical science ET investigations are sometimes well described in particular studies [26.211, 26.212, 26.213, 26.214]. Fernandez [26.203] recently reviewed computational methods for ET, including segmentation techniques in the physical sciences, and a number of the aspects covered are recounted here. Many of the image processing operations that are readily applicable to materials science ET reconstructions can also be found in other tomographic contexts or general reviews of 3-D tomographic data analysis [26.10], as well as in standard image processing texts [26.215]. Often, effective segmentation schemes consist of a number of standard image processing operations strung together, as exemplified in Fig. 26.39a-m [26.211].

Fig. 26.39a-m

Semi-automated segmentation routine applied to an ET reconstruction of unsupported nanoparticulate Ga-Pd catalysts, to enable identification of individual and agglomerated nanoparticles constituting the densely populated cluster. (al) The principal stages of the routine on a 2-D slice from the \(x\)\(y\) plane of the tomogram. (m) A 3-D voxel projection visualization. (l,m) The final segmented tomogram, where individual voxels have been given a color according the nanoparticle or group of strongly agglomerated nanoparticles to which they belong. Reprinted with permission from [26.211]. Copyright 2012 The American Chemical Society

Typically, for most ET reconstructions to date, segmentation begins with a procedure for denoising and/or enhancing the features of interest. This could involve basic regional averaging such as low-pass [26.212] or median filtering [26.70], histogram equalization [26.213], edge enhancement such as Sobel filtering [26.216], difference of Gaussians [26.211], or unsharp masking [26.146, 26.147], or more sophisticated processes such as anisotropic nonlinear diffusion [26.206]. Denoising or feature enhancement is typically followed by feature extraction based on a similarity metric. One of the most widely applicable similarity metrics in materials science, where many samples consist of regions of homogeneous density (e. g., nanoparticles), is simply the voxel intensity [26.146, 26.214, 26.25]. In this case, features can be differentiated by global thresholding on the image gray level histogram or by local spatially aware thresholding, for which a variety of threshold selection methods exist [26.217].

Optimal threshold selection in tomography is an active area of research [26.218], and the success will still be dependent on the structural complexity of the system under consideration and the quality of the reconstruction. Applicable to sufficiently high-quality data where there is a clear intensity difference between features and background, Otsu's method [26.219] is one of the most well-known automated threshold selection techniques and seeks the optimal separation based on minimizing the intraclass variance in the image histogram. Accordingly, a number of ET studies have used the Otsu or multilevel Otsu method [26.146, 26.214, 26.220]. More sophisticated threshold selection procedures proposed specifically for tomography have involved analysis of edge profiles [26.221] or projection data error minimization [26.222].

Alternative procedures for feature extraction might include detection of specific shapes. Such methods have primarily arisen in the biological field (e. g., for extracting membranes and filaments [26.223]), but can be equally powerful in materials contexts too, for example sphere extraction [26.212]. Discrete or partially discrete reconstruction algorithms [26.182, 26.184, 26.186, 26.187, 26.188] that incorporate gray level assignment a priori as part of the reconstruction process could be classed as a distinct approach to segmentation.

Subsequent to initial feature identification, additional procedures may be used to better delineate or differentiate identified features. These might include, for example, morphological operations to denoise or regularize the boundaries of objects [26.146, 26.147] or the Watershed transform [26.224] to separate mildly touching objects [26.211, 26.212, 26.214] and/or to locate their centroids [26.212, 26.83].

With segmented components of a reconstruction there are a variety of quantitative measures that can be obtained, which can be of high value in the catalytic context. Thus, in spite of the challenges, examples of quantitative catalytically relevant data obtained from ET are growing in number. Examples include size, shape, and local distribution of nanoparticulate catalysts [26.211, 26.214, 26.70], determination of porosity, surface area, local curvature, and fractal dimension of catalyst supports [26.83]. Crystallographic analysis can also be undertaken on segmented or unsegmented tomograms, and there are numerous examples revealing the crystallography of catalytically relevant nanoparticles that was unobtainable from 2-D projections [26.141, 26.225].

It is important to emphasize that success of segmentation routines and the accuracy of quantitative analysis is often dominated by the quality of the input data. In this regard, the appropriate choice of signal mode, acquisition geometry and reconstruction technique can greatly facilitate the segmentation/quantification process. A high-quality reconstruction will facilitate simpler segmentation procedures. Indeed, in a follow-up study to the one shown in Fig. 26.39a-m [26.76], which obtained a higher fidelity tomogram using CS-ET (Fig. 26.34a,b) [26.147], the segmentation procedure could begin by thresholding directly on the raw tomogram without need for postreconstruction edge enhancement. Despite indications to the contrary in some particular cases [26.21, 26.28, 26.32], the effects of the missing wedge and finite sampling often cannot be ignored in quantitative analysis. Methods that overcome or negate these problems, such as the use of needle samples [26.22, 26.23, 26.24, 26.25] or advanced reconstruction algorithms, may provide the only routes to truly reliable quantitative ET. The development of more widely applicable methods to tackle these issues is one of the most worthy areas of development in ET.

Moving beyond 2-D images to a data volume inherently adds additional complexity to visualization, requiring an extended set of methods. Different visualization techniques will convey the data in the volume in different ways and must be selected carefully to show the desired information. Often, more than one technique is needed to enable a complete interpretation to be established. Visualization can be performed before and/or after segmentation.

Volume rending, also known as voxel projection, provides a versatile means of visualizing the intensity distribution through a reconstruction volume. As the name suggests, the rendering is in essence achieved by computing a projection through the volume, analogous to the original projection process on the microscope. Forming a 2-D projection once again may at first seem counterintuitive, but with the reconstructed volume on the computer, it is possible to control various properties to enhance information of interest. These include manipulating the view to give a sense of depth or perspective, changing the type of projection (such as maximum intensity projection), using color, or altering the level of transparency. It is also possible to view the reconstruction from any chosen direction and to restrict the visualization to just a subvolume. Techniques for volume rendering can reach a high level of sophistication, enabled by modern computer graphics capabilities. While offering significant versatility, care should also be taken that significant information is not lost or overlooked when generating a volume rendering. For reconstructions suffering from tomographic artefacts, some compromise may need to be made between artefact and object visibility.

Surface rendering can often provide a more distinctive visualization of the 3-D form of objects. Here, a surface is rendered around features in a reconstruction providing an intuitive visualization of their morphology. A common approach is to generate a surface around voxels at a given intensity, forming an isosurface. For a high-fidelity reconstruction where the object of interest has a well-defined intensity distribution, such an isosurface should reveal its structure in a meaningful way. In practice, as with volume rendering, some careful choices may be needed to visualize genuine structure and not artefacts. Color, transparency, orientation, and lighting effects can all be used to give a sense of 3-D structure and to display multiple components of the reconstruction simultaneously.

In many regards, the most objective and definitive manner in which to interrogate a tomographic reconstruction is to display 2-D slices from the volume. These show a plane cut at a chosen position in the 3-D volume and should not be confused with projections through it. Figure 26.40 illustrates this distinction. Often, the slices are taken orthogonally to the primary reconstruction axes and are known as orthoslices, but a slice can be computed (using some form of 3-D interpolation) at any arbitrary cutting plane.

Fig. 26.40

A series of orthoslices from the top to the bottom surface illustrating the shape and dispersion of Pt nanoparticles on the surfaces of an Au nanotriangle

To illustrate further, Fig. 26.41a-c shows different aspects of an ET reconstruction of a hierarchical macro and mesoporous SBA-15 silica selectively loaded with small (\(\approx{\mathrm{2}}\,{\mathrm{nm}}\)) Pt nanoparticles in the mesopores and larger (\(\approx{\mathrm{6}}\,{\mathrm{nm}}\)) Pd nanoparticles in the macropores [26.226]. The volume rendering (Fig. 26.41a-ca) gives an overall feel of the structure, showing simultaneously all of the Pd and Pt nanoparticles and the general form of the macropore and mesopore structure. However, similar to the original projections from the microscope, the overlap of the complex pore structures in projection makes them difficult to discern. Here, surface renderings provide a more suitable approach. Figure 26.41a-cb reveals the macropore structure with a surface rendering of the outer morphology of the SBA-15. Figure 26.41a-cb also displays the volume rendering at the same time, showing that the larger Pd nanoparticles are located in the macropores. Figure 26.41a-cc reveals the internal mesopore structure by combining an opaque surface rendering of the mesopores along with the surface from Fig. 26.41a-cb shown with semitransparency. Still, to determine the location of the smaller Pt nanoparticles definitively requires (ortho)slices. Figure 26.42a,ba uses a surface rendering to show the position of an orthoslice in the ET reconstruction of another part of this hierarchical catalyst. As shown in Fig. 26.42a,bb, the orthoslice clearly reveals the hexagonal arrangement of internal mesopores, as well as the Pt nanoparticles within those mesopores. The multidirectional form of the mesopores, however, requires slices to be computed in various directions in order to understand fully the 3-D structure.

Fig. 26.41a-c

Different visualizations of an ET reconstruction of a hierarchical macro and mesoporous SBA-15 silica selectively loaded with small (\(\approx{\mathrm{2}}\,{\mathrm{nm}}\)) Pt nanoparticles in the mesopores and larger (\(\approx{\mathrm{6}}\,{\mathrm{nm}}\)) Pd nanoparticles in the macropores. (a) Volume rendering, (b) volume rendering combined with surface rendering of the outer morphology of the SBA-15, (c) semitransparent rendering of the surface in (b) along with surface rendering of the internal mesopore structure

Fig. 26.42a,b

2-D slice in the \(x\)\(y\) plane of an ET reconstruction of a small fragment of the hierarchical system described in Fig. 26.41a-c. (a) indicates the position of the slice in the fragment, where a surface rendering shows the outer morphology of the SBA-15. (b) The extracted slice, revealing the internal mesopore structure as well as Pt nanoparticles within the mesopores

26.7 Conclusions

Over the past 20 years or so, ET has changed from being a rather niche technique practised by very few in the physical sciences to one that is now a routine and a widely-used method to determine the nanoscale structure of materials in three dimensions. In parallel, there has been a remarkable growth in the number of imaging modes that have been used in a tomographic way to reveal not only the morphology of the region of interest but also composition, chemistry, electro-magnetic properties, optical properties, and local crystallography.

The growth of tomography in materials science, using not only electrons but also x-rays and to a lesser extent other forms of radiation, has been aided also by the extraordinary rise in computational power and the advent of new reconstruction algorithms with which to take advantage of it. In particular, the use of CS and related methods holds great promise to improve not only the fidelity of the tomogram but also to enable a model-based approach to reconstruction, which allows materials properties to be recovered that are not immediately accessible through conventional backprojection routes.

The introduction of more efficient and faster spectrometers and cameras enables analytical tomography to be performed more quickly and with greater use of what signal is provided by the electron–specimen interaction. Even with materials science specimens, which are generally more robust than those in the life sciences, care needs to be taken to ensure beam damage does not accumulate to unacceptable levels over the time to acquire a tilt series.

There is now the technical capability to automate much of the image acquisition process and with machine learning and AI, and the possibility of analyzing the data as it is acquired. Modern cameras and spectrometers enable vast quantities of data to be acquired in short timescales and on-the-fly processing will undoubtedly become a common feature in years to come. Such processing also allows judgement as to whether just enough data has been acquired to move to a new region or to tackle a new problem and thus enable more efficient use of the microscope, a feature that is perhaps more critical with tomography than other TEM techniques, given the necessarily long acquisition times/large doses required.

The complexity of modern materials and devices will only increase in the future and much of that complexity will be three-dimensional and chemically heterogeneous. ET is a technique that will continue to progress rapidly, with improved spatial resolution and reconstruction fidelity and with the application of new analytical methods.

Notes

Acknowledgements

The research leading to these results was possible through funding from the European Union Seventh Framework Program under Grant Agreement 312483-ESTEEM2 (Integrated Infrastructure Initiative–I3), from the European Research Council under the European Union's Seventh Framework Program (FP/2007–2013)/ERC Grant Agreement 291522–3-DIMAGE, and funding from the EPSRC, grant number EP/R008779/1. R.K.L. acknowledges a Junior Research Fellowship at Clare College. The authors acknowledge the many people with whom they have worked, including most recently Sir John Meurig Thomas, Francisco de la Pena, Sean Collins, Adam Lee, Emilie Ringe, Alex Eggeman, Jon Barnard, Duncan Johnstone, and David Rossouw.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Dept. of Materials Science & MetallurgyUniversity of CambridgeCambridgeUK

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