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# Electron Tomography in Materials Science

## 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.

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.

## 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.

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

*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

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.

### 26.1.2 The Fourier Transform and Fourier Slice Theorem

*Fourier slice theorem*, as described below.

*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

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.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.

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.

## 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.

### 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.

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).

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

*streaking*artefacts, the severity of which increases with tilt increment (Fig. 26.10).

*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 \(D\)

*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.

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 |
| E | WPOs, biological specimens, amorphous materials |

ADF-STEM | Atomic number (\(Z\)) |
| E | Crystalline specimens, \(Z\)-contrast |

Cs-CTEM |
| A | Atomic-scale, WPOs | |

Cs-STEM | Atomic number (\(Z\)) | A | Atomic-scale; heavy metal nanoparticles | |

Cc-TEM |
| A | Thick biological specimens | |

DF-TEM | Angularly selective scattering | A | Lattice defects, low-contrast soft matter | |

Precession BF-CTEM |
| A | Crystalline specimens | |

BF-STEM |
| A | Thick specimens, polymers, biological sections | |

IBF-STEM |
| A | Thick specimens | |

MAADF-STEM |
| A | Dislocations | |

STEM in ESEM |
| A | Nonconductive or hydrated specimens |

Multidimensional imaging modes | ||||
---|---|---|---|---|

EFTEM |
| E | Chemical segregation, optical properties, bonding variations | |

EELS | Inelastic scattering |
| A | Chemical environment Elemental distribution (core-loss) Optical properties (low-loss) |

EDXS | Secondary x-ray emission | A | Elemental distribution | |

Diffraction |
| A | Crystalline materials | |

Holography | Reconstructed phase and amplitude |
| A | Mean inner potential, electrostatic and magnetic fields |

Time resolved |
| A | New commercial TEMs |

### 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].

### 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.

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].

### 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.

#### 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].

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].

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).

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.

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].

### 26.3.6 Holographic ET

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.

### 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.

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.

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].

## 26.5 ET Reconstruction

- 1.
Direct transform methods, including backprojection and Fourier techniques

- 2.
Algebraic iterative methods, including the ART and SIRT-type classes.

### 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.

### 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.

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].

*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.

### 26.5.4 Algebraic Reconstruction Technique (ART)

*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

*Gordon*et al [26.151], who also presented a multiplicative form (sometimes referred to as MART). An update scheme can be written as

*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]

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.

*Landweber*method [26.158]

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.

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.

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.

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.

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].

#### 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.

#### 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.

## 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].

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.

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.

## 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.

## References

- A.J. Koster, R. Grimm, D. Typke, R. Hegerl, A. Stoschek, J. Walz, W. Baumeister: Perspectives of molecular and cellular electron tomography, J. Struct. Biol.
**120**(3), 276–308 (1997)CrossRefGoogle Scholar - P.A. Midgley, M. Weyland: 3D electron microscopy in the physical sciences: the development of Z-contrast and EFTEM tomography, Ultramicroscopy
**96**(3/4), 413–431 (2003)CrossRefGoogle Scholar - J. Radon: Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Ber. Verh. K. Sächs. Ges. Wiss. Leipz. Math.-Phys. Kl.
**69**, 262–277 (1917)Google Scholar - R.N. Bracewell: Two-dimensional aerial smoothing in radio astronomy, Aust. J. Phys.
**9**, 297–314 (1956)CrossRefGoogle Scholar - J. Frank:
*Three-Dimensional Electron Microscopy of Macromolecular Assemblies: Visualization of Biological Molecules in Their Native State*(Oxford Univ. Press, New York 2006)CrossRefGoogle Scholar - S.R. Deans:
*The Radon Transform and Some of its Applications*(Wiley, New York, Chichester 1983)Google Scholar - A.C. Kak, M. Slaney:
*Principles of Computerized Tomographic Imaging*(IEEE, New York 1988)Google Scholar - G.T. Herman, A. Kuba:
*Advances in Discrete Tomography and Its Applications*(Birkhauser, Boston 2007)CrossRefGoogle Scholar - P.A. Penczek, J.J. Grant: Fundamentals of three-dimensional reconstruction from projections, Methods Enzymol.
**482**, 1–33 (2010)CrossRefGoogle Scholar - J. Banhart:
*Advanced Tomographic Methods in Materials Research and Engineering*(Oxford Univ. Press, Oxford 2008)CrossRefGoogle Scholar - D.J. De Rosier, A. Klug: Reconstruction of three dimensional structures from electron micrographs, Nature
**217**(5124), 130–134 (1968)CrossRefGoogle Scholar - W. Hoppe, R. Langer, G. Knesch, C. Poppe: Proteinkristallstrukturanalyse mit Elektronenstrahlen, Naturwissenschaften
**55**(7), 333–336 (1968)CrossRefGoogle Scholar - R.G. Hart: Electron microscopy of unstained biological material: The polytropic montage, Science
**159**(3822), 1464–1467 (1968)CrossRefGoogle Scholar - T. Dahmen, P. Trampert, N. De Jonge, P. Slusallek: Advanced recording schemes for electron tomography, MRS Bulletin
**41**(7), 537–541 (2016)CrossRefGoogle Scholar - R. Leary, R. Brydson: Chromatic aberration correction: The next step in electron microscopy, Adv. Imaging Electron Phys.
**165**, 73–130 (2011)CrossRefGoogle Scholar - R.A. Crowther, D.J. DeRosier, A. Klug: The reconstruction of a three-dimensional structure from projections and its application to electron microscopy, Proc. Royal Soc. A
**317**(1530), 319–340 (1970)CrossRefGoogle Scholar - D.N. Mastronarde: Dual-axis tomography: An approach with alignment methods that preserve resolution, J. Struct. Biol.
**120**(3), 343–352 (1997)CrossRefGoogle Scholar - P. Penczek, M. Marko, K. Buttle, J. Frank: Double-tilt electron tomography, Ultramicroscopy
**60**(3), 393–410 (1995)CrossRefGoogle Scholar - J. Tong, I. Arslan, P. Midgley: A novel dual-axis iterative algorithm for electron tomography, J. Struct. Biol.
**153**(1), 55–63 (2006)CrossRefGoogle Scholar - I. Arslan, J.R. Tong, P.A. Midgley: Reducing the missing wedge: High-resolution dual axis tomography of inorganic materials, Ultramicroscopy
**106**(11/12), 994–1000 (2006)CrossRefGoogle Scholar - N. Kawase, M. Kato, H. Nishioka, H. Jinnai: Transmission electron microtomography without the ‘‘missing wedge'' for quantitative structural analysis, Ultramicroscopy
**107**(1), 8–15 (2007)CrossRefGoogle Scholar - K. Jarausch, P. Thomas, D.N. Leonard, R. Twesten, C.R. Booth: Four-dimensional STEM-EELS: Enabling nano-scale chemical tomography, Ultramicroscopy
**109**(4), 326–337 (2009)CrossRefGoogle Scholar - M. Koguchi, H. Kakibayashi, R. Tsuneta, M. Yamaoka, T. Niino, N. Tanaka, K. Kase, M. Iwaki: Three-dimensional STEM for observing nanostructures, J. Electron Microsc.
**50**(3), 235–241 (2001)Google Scholar - M. Kato, N. Kawase, T. Kaneko, S. Toh, S. Matsumura, H. Jinnai: Maximum diameter of the rod-shaped specimen for transmission electron microtomography without the ‘‘missing wedge'', Ultramicroscopy
**108**(3), 221–229 (2008)CrossRefGoogle Scholar - E. Biermans, L. Molina, K.J. Batenburg, S. Bals, G. Van Tendeloo: Measuring porosity at the nanoscale by quantitative electron tomography, Nano Lett.
**10**(12), 5014–5019 (2010)CrossRefGoogle Scholar - J. Leschner, J. Biskupek, A. Chuvilin, U. Kaiser: Accessing the local three-dimensional structure of carbon materials sensitive to an electron beam, Carbon
**48**(14), 4042–4048 (2010)CrossRefGoogle Scholar - H. Friedrich, P.E. de Jongh, A.J. Verkleij, K.P. de Jong: Electron tomography for heterogeneous catalysts and related nanostructured materials, Chem. Rev.
**109**(5), 1613–1629 (2009)CrossRefGoogle Scholar - S.S. van Bavel, J. Loos: Volume organization of polymer and hybrid solar cells as revealed by electron tomography, Adv. Funct. Mater.
**20**, 3217–3234 (2010)CrossRefGoogle Scholar - X.Y. Wang, R. Lockwood, M. Malac, H. Furukawa, P. Li, A. Meldrum: Reconstruction and visualization of nanoparticle composites by transmission electron tomography, Ultramicroscopy
**113**, 96–105 (2012)CrossRefGoogle Scholar - H. Sugimori, T. Nishi, H. Jinnai: Dual-axis electron tomography for three-dimensional observations of polymeric nanostructures, Macromolecules
**38**(24), 10226–10233 (2005)CrossRefGoogle Scholar - M. Radermacher, W. Hoppe: Properties of 3-D reconstruction from projections by conical tilting compared to single-axis tilting. In:
*7th Eur. Congr. Electron Microsc., Den Haag, Leiden, The Netherlands*, ed. by P. Brederoo, G. Boom (1980) pp. 132–133Google Scholar - D. Chen, H. Friedrich, G. de With: On resolution in electron tomography of beam sensitive materials, J. Phys. Chem. C
**118**(2), 1248–1257 (2013)CrossRefGoogle Scholar - R.N. Bracewell, A.C. Riddle: Inversion of fan-beam scans in radio astronomy, Astrophys. J.
**150**, 427–434 (1967)CrossRefGoogle Scholar - H. Heidari Mezerji, W. Van den Broek, S. Bals: A practical method to determine the effective resolution in incoherent experimental electron tomography, Ultramicroscopy
**111**(5), 330–336 (2011)CrossRefGoogle Scholar - P.A. Midgley, M. Weyland, J.M. Thomas, B.F.G. Johnson: Z-contrast tomography: A technique in three-dimensional nanostructural analysis based on Rutherford scattering, Chem. Commun.
**18**(10), 907–908 (2001)CrossRefGoogle Scholar - A.J. Koster, U. Ziese, A.J. Verkleij, A.H. Janssen, K.P. de Jong: Three-dimensional transmission electron microscopy: A novel imaging and characterization technique with nanometer scale resolution for materials science, J. Phys. Chem. B
**104**(40), 9368–9370 (2000)CrossRefGoogle Scholar - M. Bar Sadan, L. Houben, S.G. Wolf, A. Enyashin, G. Seifert, R. Tenne, K. Urban: Toward atomic-scale bright-field electron tomography for the study of fullerene-like nanostructures, Nano Lett.
**8**(3), 891–896 (2008)CrossRefGoogle Scholar - S. Van Aert, K.J. Batenburg, M.D. Rossell, R. Erni, G. Van Tendeloo: Three-dimensional atomic imaging of crystalline nanoparticles, Nature
**470**(7334), 374–377 (2011)CrossRefGoogle Scholar - B. Goris, S. Bals, W. Van den Broek, E. Carbó-Argibay, S. Gómez-Graña, L.M. Liz-Marzán, G. Van Tendeloo: Atomic-scale determination of surface facets in gold nanorods, Nat. Mater.
**11**, 930–935 (2012)CrossRefGoogle Scholar - J.-P. Baudoin, J.R. Jinschek, C.B. Boothroyd, R.E. Dunin-Borkowski, N. de Jonge: Chromatic aberration-corrected tilt series transmission electron microscopy of nanoparticles in a whole mount macrophage cell, Microsc. Microanal.
**19**(04), 814–820 (2013)CrossRefGoogle Scholar - J.S. Barnard, J. Sharp, J.R. Tong, P.A. Midgley: High-resolution three-dimensional imaging of dislocations, Science
**313**(5785), 319 (2006)CrossRefGoogle Scholar - S. Bals, G. Van Tendeloo, C. Kisielowski: A new approach for electron tomography: Annular dark-field transmission electron microscopy, Adv. Mater.
**18**(7), 892–895 (2006)CrossRefGoogle Scholar - J.M. Rebled, L. Yedra, S. Estradé, J. Portillo, F. Peiró: A new approach for 3D reconstruction from bright field TEM imaging: Beam precession assisted electron tomography, Ultramicroscopy
**111**(9/10), 1504–1511 (2011)CrossRefGoogle Scholar - A.A. Sousa, A.A. Azari, G. Zhang, R.D. Leapman: Dual-axis electron tomography of biological specimens: extending the limits of specimen thickness with bright-field STEM imaging, J. Struct. Biol.
**174**(1), 107–114 (2011)CrossRefGoogle Scholar - P. Ercius, M. Weyland, D.A. Muller, L.M. Gignac: Three-dimensional imaging of nanovoids in copper interconnects using incoherent bright field tomography, Appl. Phys. Lett.
**88**(24), 243116 (2006)CrossRefGoogle Scholar - J.H. Sharp, J.S. Barnard, K. Kaneko, K. Higashida, P.A. Midgley: Dislocation tomography made easy: a reconstruction from ADF STEM images obtained using automated image shift correction, J. Phys. Conf. Ser.
**126**(1), 012013 (2008)CrossRefGoogle Scholar - P. Jornsanoh, G. Thollet, J. Ferreira, K. Masenelli-Varlot, C. Gauthier, A. Bogner: Electron tomography combining ESEM and STEM: a new 3D imaging technique, Ultramicroscopy
**111**(8), 1247–1254 (2011)CrossRefGoogle Scholar - M. Weyland, P.A. Midgley: 3D-EFTEM: Tomographic reconstruction from tilt series of energy loss images. In:
*Proc. Inst. Phys. EMAG Conf. Ser. 2001*, Vol. 161 (2001) p. 239Google Scholar - G. Möbus, B.J. Inkson: Three-dimensional reconstruction of buried nanoparticles by element-sensitive tomography based on inelastically scattered electrons, Appl. Phys. Lett.
**79**, 1369 (2001)CrossRefGoogle Scholar - G. Möbus, R.C. Doole, B.J. Inkson: Spectroscopic electron tomography, Ultramicroscopy
**96**, 433 (2003)CrossRefGoogle Scholar - K. Lepinay, F. Lorut, R. Pantel, T. Epicier: Chemical 3D tomography of 28 nm high K metal gate transistor: STEM XEDS experimental method and results, Micron
**47**, 43–49 (2013)CrossRefGoogle Scholar - U. Kolb, T. Gorelik, C. Kübel, M.T. Otten, D. Hubert: Towards automated diffraction tomography: Part I—Data acquisition, Ultramicroscopy
**107**(6/7), 507–513 (2007)CrossRefGoogle Scholar - A.C. Twitchett-Harrison, T.J.V. Yates, S.B. Newcomb, R.E. Dunin-Borkowski, P.A. Midgley: High-resolution three-dimensional mapping of semiconductor dopant potentials, Nano Lett.
**7**(7), 2020–2023 (2007)CrossRefGoogle Scholar - O.-H. Kwon, A.H. Zewail: 4D electron tomography, Science
**328**(5986), 1668–1673 (2010)CrossRefGoogle Scholar - D.B. Williams, C.B. Carter:
*Transmission Electron Microscopy: A Textbook for Materials Science*(Springer, New York 2009)CrossRefGoogle Scholar - J.C.H. Spence:
*High-Resolution Electron Microscopy*, 4th edn. (Oxford Univ. Press, Oxford 2013)CrossRefGoogle Scholar - D. Cockayne, A.I. Kirkland, P.D. Nellist, A. Bleloch: New possibilities with aberration-corrected electron microscopy, Philos. Trans. Royal Soc. A
**367**(1903), 3633–3870 (2009)CrossRefGoogle Scholar - M. Weyland, P.A. Midgley, J.M. Thomas: Electron tomography of nanoparticle catalysts on porous supports: A new technique based on Rutherford scattering, J. Phys. Chem. B
**105**(33), 7882–7886 (2001)CrossRefGoogle Scholar - P.A. Midgley, R.E. Dunin-Borkowski: Electron tomography and holography in materials science, Nat. Mater.
**8**(4), 271–280 (2009)CrossRefGoogle Scholar - P.A. Midgley, M. Weyland, T.J.V. Yates, I. Arslan, R.E. Dunin-Borkowski, J.M. Thomas: Nanoscale scanning transmission electron tomography, J. Microsc.
**223**(3), 185–190 (2006)CrossRefGoogle Scholar - P.A. Midgley, E.P.W. Ward, A.B. Hungria, J.M. Thomas: Nanotomography in the chemical, biological and materials sciences, Chem. Soc. Rev.
**36**, 1477–1494 (2007)CrossRefGoogle Scholar - M. Weyland, P.A. Midgley: Electron tomography. In:
*Nanocharacterisation*, 2nd edn., ed. by A.I. Kirkland, S.J. Haigh (Royal Society of Chemistry, Cambridge 2007)Google Scholar - H. Jinnai, R.J. Spontak: Transmission electron microtomography in polymer research, Polymer
**50**(5), 1067–1087 (2009)CrossRefGoogle Scholar - H. Jinnai, R.J. Spontak, T. Nishi: Transmission electron microtomography and polymer nanostructures, Macromolecules
**43**(4), 1675–1688 (2010)CrossRefGoogle Scholar - C. Kübel, A. Voigt, R. Schoenmakers, M. Otten, D. Su, T.-C. Lee, A. Carlsson, J. Bradley: Recent advances in electron tomography: TEM and HAADF-STEM tomography for materials science and semiconductor applications, Microsc. Microanal.
**11**, 378–400 (2005)CrossRefGoogle Scholar - G. Möbus, B.J. Inkson: Nanoscale tomography in materials science, Mater. Today
**10**(12), 18–25 (2007)CrossRefGoogle Scholar - Z. Saghi, X. Xu, G. Möbus: Electron tomography of regularly shaped nanostructures under non-linear image acquisition, J. Microsc.
**232**(1), 186–195 (2008)CrossRefGoogle Scholar - H. Friedrich, M.R. McCartney, P.R. Buseck: Comparison of intensity distributions in tomograms from BF TEM, ADF STEM, HAADF STEM, and calculated tilt series, Ultramicroscopy
**106**(1), 18–27 (2005)CrossRefGoogle Scholar - F. Leroux, E. Bladt, J.-P. Timmermans, G. Van Tendeloo, S. Bals: Annular dark-field transmission electron microscopy for low contrast materials, Microsc. Microanal.
**19**(03), 629–634 (2013)CrossRefGoogle Scholar - G. Prieto, J. Zečević, H. Friedrich, K.P. de Jong, P.E. de Jongh: Towards stable catalysts by controlling collective properties of supported metal nanoparticles, Nat. Mater.
**12**(1), 34–39 (2013)CrossRefGoogle Scholar - R.J. Spontak, M.C. Williams, D.A. Agard: Three-dimensional study of cylindrical morphology in a styrene-butadiene-styrene block copolymer, Polymer
**29**(3), 387–395 (1988)CrossRefGoogle Scholar - H. Jinnai, X. Jiang: Electron tomography in soft materials, Curr. Opin. Solid State Mater. Sci.
**17**(3), 135–142 (2013)CrossRefGoogle Scholar - P. Yuan, L. Tan, D. Pan, Y. Guo, L. Zhou, J. Yang, J. Zou, C. Yu: A systematic study of long-range ordered 3D-SBA-15 materials by electron tomography, New J. Chem.
**35**, 2456–2461 (2011)CrossRefGoogle Scholar - K.P. de Jong, J. Zečević, H. Friedrich, P.E. de Jongh, M. Bulut, S. van Donk, R. Kenmogne, A. Finiels, V. Hulea, F. Fajula: Zeolite Y crystals with trimodal porosity as ideal hydrocracking catalysts, Angew. Chem. Int. Ed.
**49**, 10074–10078 (2010)CrossRefGoogle Scholar - J. Zečević, K.P. de Jong, P.E. de Jongh: Progress in electron tomography to assess the 3D nanostructure of catalysts, Curr. Opin. Solid State Mater. Sci.
**17**(3), 115–125 (2013)CrossRefGoogle Scholar - R. Leary, P.A. Midgley, J.M. Thomas: Recent advances in the application of electron tomography to materials chemistry, Acc. Chem. Res.
**45**(10), 1782–1791 (2012)CrossRefGoogle Scholar - M. Weyland: Electron tomography of catalysts, Top. Catal.
**21**(4), 175–183 (2002)CrossRefGoogle Scholar - S.J. Pennycook, P.D. Nellist:
*Scanning Transmission Electron Microscopy*(Springer, New York 2011)CrossRefGoogle Scholar - O.L. Krivanek, M.F. Chisholm, V. Nicolosi, T.J. Pennycook, G.J. Corbin, N. Dellby, M.F. Murfitt, C.S. Own, Z.S. Szilagyi, M.P. Oxley, S.T. Pantelides, S.J. Pennycook: Atom-by-atom structural and chemical analysis by annular dark-field electron microscopy, Nature
**464**(7288), 571–574 (2010)CrossRefGoogle Scholar - M.M.J. Treacy: Z dependence of electron scattering by single atoms into annular dark-field detectors, Microsc. Microanal.
**17**(06), 847–858 (2011)CrossRefGoogle Scholar - P. Ercius, O. Alaidi, M.J. Rames, G. Ren: Electron tomography: A three-dimensional analytic tool for hard and soft materials research, Adv. Mater.
**27**, 5638–5663 (2015)CrossRefGoogle Scholar - M. Weyland, P.A. Midgley: Electron tomography, Mater. Today
**7**(12), 32–40 (2004)CrossRefGoogle Scholar - E.P.W. Ward, T.J.V. Yates, J.-J. Fernandez, D.E.W. Vaughan, P.A. Midgley: Three-dimensional nanoparticle distribution and local curvature of heterogeneous catalysts revealed by electron tomography, J. Phys. Chem. C
**111**(31), 11501–11505 (2007)CrossRefGoogle Scholar - J.C. Hernández-Garrido, K. Yoshida, P.L. Gai, E.D. Boyes, C.H. Christensen, P.A. Midgley, N.C. Greenham: The location of gold nanoparticles on titania: A study by high resolution aberration-corrected electron microscopy and 3D electron tomography, Catal. Today
**160**, 165–169 (2011)CrossRefGoogle Scholar - J.M. Thomas, P.A. Midgley, T.J.V. Yates, J.S. Barnard, R. Raja, I. Arslan, M. Weyland: The chemical application of high-resolution electron tomography: Bright field or dark field?, Angew. Chem. Int. Ed.
**43**(48), 6745–6747 (2004)CrossRefGoogle Scholar - K. Lu, E. Sourty, R. Guerra, G. Bar, J. Loos: Critical comparisonof volume data obtained by different electron tomography techniques, Macromolecules
**43**(3), 1444–1448 (2010)CrossRefGoogle Scholar - P.B. Hirsch, A. Howie, P.B. Nicholson, D.W. Pashley, W.J. Whelan:
*Electron Microscopy of Thin Crystals*(Krieger, New York 1977)Google Scholar - I. Arslan, T.J.V. Yates, N.D. Browning, P.A. Midgley: Embedded nanostructures revealed in three dimensions, Science
**309**(5744), 2195–2198 (2005)CrossRefGoogle Scholar - Z.Y. Li, N.P. Young, M.D. Vece, S. Palomba, R.E. Palmer, A.L. Bleloch, B.C. Curley, R.L. Johnston, J. Jiang, J. Yuan: Three-dimensional atomic-scale structure of size-selected gold nanoclusters, Nature
**451**, 46 (2008)CrossRefGoogle Scholar - M. Azubel, J. Koivisto, S. Malola, D. Bushnell, G.L. Hura, A. Koh, H. Tsunoyama, T. Tsukuda, M. Pettersson, H. Häkkinen, R.D. Kornberg: Electron microscopy of gold nanoparticles at atomic resolution, Science
**345**(6199), 909–912 (2014)CrossRefGoogle Scholar - C.-C. Chen, C. Zhu, E.R. White, C.-Y. Chiu, M.C. Scott, B.C. Regan, L.D. Marks, Y. Huang, J. Miao: Three-dimensional imaging of dislocations in a nanoparticle at atomic resolution, Nature
**496**(7443), 74–77 (2013)CrossRefGoogle Scholar - B. Goris, J. De Beenhouwer, A. De Backer, D. Zanaga, K.J. Batenburg, A. Sánchez-Iglesias, L.M. Liz-Marzán, S. Van Aert, S. Bals, J. Sijbers, G. Van Tendeloo: Measuring lattice strain in three dimensions through electron microscopy, Nano Lett.
**15**, 6996–7001 (2015)CrossRefGoogle Scholar - G. Haberfehlner, P. Thaler, D. Knez, A. Volk, F. Hofer, W.E. Ernst, G. Kothleitner: Formation of bimetallic clusters in superfluid helium nanodroplets analysed by atomic resolution electron tomography, Nat. Commun.
**6**, 8779 (2015)CrossRefGoogle Scholar - M.C. Scott, C.-C. Chen, M. Mecklenburg, C. Zhu, R. Xu, P. Ercius, U. Dahmen, B.C. Regan, J. Miao: Electron tomography at 2.4-ångström resolution, Nature
**483**(7390), 444–447 (2012)CrossRefGoogle Scholar - J. Park, H. Elmlund, P. Ercius, J.M. Yuk, D.T. Limmer, Q. Chen, K. Kim, S.H. Han, D.A. Weitz, A. Zettl, A.P. Alivisatos: 3D structure of individual nanocrystals in solution by electron microscopy, Science
**349**, 290–295 (2015)CrossRefGoogle Scholar - T. Willhammar, K. Sentosun, S. Mourdikoudis, B. Goris, M. Kurttepeli, M. Bercx, D. Lamoen, B. Partoens, I. Pastoriza-Santos, J. Perez-Juste, L.M. Liz-Marzan, S. Bals, G. Van Tendeloo: Structure and vacancy distribution in copper telluride nanoparticles influence plasmonic activity in the near-infrared, Nat. Commun.
**8**, 14925–14932 (2017)CrossRefGoogle Scholar - B. Goris, T. Roelandts, K.J. Batenburg, H. Heidari Mezerji, S. Bals: Advanced reconstruction algorithms for electron tomography: From comparison to combination, Ultramicroscopy
**127**, 40–47 (2013)CrossRefGoogle Scholar - Y. Yang, C.-C. Chen, M.C. Scott, C. Ophus, R. Xu, A. Pryor, L. Wu, F. Sun, W. Theis, J. Zhou, M. Eisenbach, P.R.C. Kent, R.F. Sabirianov, H. Zeng, P. Ercius, J. Miao: Deciphering chemical order/disorder and material properties at the single-atom level, Nature
**542**(7639), 75–79 (2017)CrossRefGoogle Scholar - L.D. Marks: Experimental studies of small particle structures, Rep. Prog. Phys.
**57**(6), 603–649 (1994)CrossRefGoogle Scholar - T.J.A. Slater, A. Macedo, S.L.M. Schroeder, M.G. Burke, P. O'Brien, P.H.C. Camargo, S.J. Haigh: Correlating catalytic activity of Ag-Au nanoparticles with 3D compositional variations, Nano Lett.
**14**, 1921 (2014)CrossRefGoogle Scholar - P. Burdet, Z. Saghi, A.N. Filippin, A. Borrás, P.A. Midgley: A novel 3D absorption correction method for quantitative EDX-STEM tomography, Ultramicroscopy
**160**, 118 (2016)CrossRefGoogle Scholar - G. Haberfehlner, A. Orthacker, M. Albu, J. Li, G. Kothleitner: Nanoscale voxel spectroscopy by simultaneous EELS and EDS tomography, Nanoscale
**6**, 14563 (2014)CrossRefGoogle Scholar - B. Goris, S. Turner, S. Bals, G. Van Tendeloo: Three-dimensional valency mapping in ceria nanocrystals, ACS Nano
**8**, 10878 (2014)CrossRefGoogle Scholar - O. Nicoletti, F. de la Pena, R.K. Leary, D.J. Holland, C. Ducati, P.A. Midgley: Three-dimensional imaging of localized surface plasmon resonances of metal nanoparticles, Nature
**502**, 80 (2013)CrossRefGoogle Scholar - N.Y. Jin-Phillipp, C.T. Koch, P.A. van Aken: Toward quantitative core-loss EFTEM tomography, Ultramicroscopy
**111**, 1255 (2011)CrossRefGoogle Scholar - M.H. Gass, K.K.K. Koziol, A.H. Windle, P.A. Midgley: Four-dimensional spectral tomography of carbonaceous nanocomposites, Nano Lett.
**6**(3), 376–379 (2006)CrossRefGoogle Scholar - P. Torruella, R. Arenal, F. de la Peña, Z. Saghi, L. Yedra, A. Eljarrat, L. López-Conesa, M. Estrader, A. López-Ortega, G. Salazar-Alvarez, J. Nogués, C. Ducati, P.A. Midgley, F. Peiró, S. Estradé: 3D visualization of the iron oxidation state in FeO/Fe
_{3}O_{4}core–shell nanocubes from electron energy loss tomography, Nano Lett.**16**, 5068–5073 (2016)CrossRefGoogle Scholar - F. de la Peña, T. Ostaševičius, R.K. Leary, C. Ducati, P.A. Midgley, R. Arenal: Quantitative elemental and bonding EELS tomography of a complex nanoparticle. In:
*Proc. Eur. Microsc. Congr.*(2016), https://doi.org/10.1002/9783527808465.EMC2016.6431CrossRefGoogle Scholar - A. Yurtsever, M. Weyland, D.A. Muller: Three-dimensional imaging of nonspherical silicon nanoparticles embedded in silicon oxide by plasmon tomography, Appl. Phys. Lett.
**89**, 151920 (2006)CrossRefGoogle Scholar - A.C. Atre, B.J.M. Brenny, T. Coenen, A. García-Etxarri, A. Polman, J.A. Dionne: Nanoscale optical tomography with cathodoluminescence spectroscopy, Nat. Nanotechnol.
**10**, 429 (2015)CrossRefGoogle Scholar - R.F. Egerton:
*Electron Energy-Loss Spectroscopy in the Electron Microscope*(Springer, New York 2011)CrossRefGoogle Scholar - J. Scott, P.J. Thomas, M. Mackenzie, S. McFadzean, J. Wilbrink, A.J. Craven, W.A. Nicholson: Near-simultaneous dual energy range EELS spectrum imaging, Ultramicroscopy
**108**(12), 1586–1594 (2008)CrossRefGoogle Scholar - A. Al-Afeef, W.P. Cockshott, I. MacLaren, S. McVitie: Electron tomography image reconstruction using data-driven adaptive compressed sensing, Scanning
**38**, 251–276 (2016)CrossRefGoogle Scholar - Z. Saghi, X. Xu, Y. Peng, B. Inkson, G. Mobus: Three-dimensional chemical analysis of tungsten probes by energy dispersive x-ray nanotomography, Appl. Phys. Lett.
**91**, 251906 (2007)CrossRefGoogle Scholar - T.J.A. Slater, A. Janssen, P.H.C. Camargo, M.G. Burke, N.J. Zaluzec, S.J. Haigh: STEM-EDX tomography of bimetallic nanoparticles: A methodological investigation, Ultramicroscopy
**162**, 61–73 (2016)CrossRefGoogle Scholar - C.S. Yeoh, D. Rossouw, Z. Saghi, P. Burdet, R.K. Leary, P.A. Midgley: The dark side of EDX tomography: Modeling detector shadowing to aid 3D elemental signal analysis, Microsc. Microanal.
**21**(3), 759–764 (2015)CrossRefGoogle Scholar - D. Zanaga, T. Altantzis, J. Sanctorum, B. Freitag, S. Bals: An alternative approach for ζ-factor measurement using pure element nanoparticles, Ultramicroscopy
**164**, 11–16 (2016)CrossRefGoogle Scholar - Z. Saghi, G. Divitini, B. Winter, R. Leary, E. Spiecker, C. Ducati, P.A. Midgley: Compressed sensing electron tomography of needle-shaped biological specimens – Potential for improved reconstruction fidelity with reduced dose, Ultramicroscopy
**160**, 230–238 (2016)CrossRefGoogle Scholar - D. Wolf, A. Lubk, F. Röder, H. Lichte: Electron holographic tomography, Curr. Opin. Solid State Mater. Sci.
**17**(3), 126–134 (2013)CrossRefGoogle Scholar - U. Kolb, E. Mugnaioli, T.E. Gorelik: Automated electron diffraction tomography—A new tool for nano crystal structure analysis, Cryst. Res. Technol.
**46**(6), 542–554 (2011)CrossRefGoogle Scholar - K. Kimura, S. Hata, S. Matsumura, T. Horiuchi: Dark-field transmission electron microscopy for a tilt series of ordering alloys: Toward electron tomography, J. Electron Microsc.
**54**(4), 373–377 (2005)Google Scholar - R. Beanland, A. Sánchez, J. Hernandez-Garrido, D. Wolf, P. Midgley: Electron tomography of III-V quantum dots using dark field 002 imaging conditions, J. Microsc.
**237**(2), 148–154 (2010)CrossRefGoogle Scholar - H.H. Liu, S. Schmidt, H.F. Poulsen, A. Godfrey, Z.Q. Liu, J.A. Sharon, X. Huang: Three-dimensional orientation mapping in the transmission electron microscope, Science
**332**, 833–834 (2011)CrossRefGoogle Scholar - A.S. Eggeman, R. Krakow, P.A. Midgley: Scanning precession electron tomography for three-dimensional nanoscale orientation imaging and crystallographic analysis, Nat. Commun.
**6**, 7267 (2015)CrossRefGoogle Scholar - Y. Meng, J.-M. Zuo: Three-dimensional nanostructure determination from a large diffraction data set recorded using scanning electron nanodiffraction, IUCr Journal
**3**, 300–308 (2016)CrossRefGoogle Scholar - D. Johnstone, A. Van Helvoort, P. Midgley: Nanoscale strain tomography by scanning precession electron diffraction, Microsc. Microanal.
**23**(S1), 1710–1711 (2017)CrossRefGoogle Scholar - S.J. Lade, D. Paganin, M.J. Morgan: Electron tomography of electromagnetic fields, potentials and sources, Opt. Commun.
**253**(4–6), 392–400 (2005)CrossRefGoogle Scholar - G. Lai, T. Hirayama, K. Ishizuka, T. Tanji, A. Tonomura: Three-dimensional reconstruction of electric-potential distribution in electron-holographic interferometry, Appl. Opt.
**33**(5), 829–833 (1994)CrossRefGoogle Scholar - C. Phatak, M. Beleggia, M. De Graef: Vector field electron tomography of magnetic materials: Theoretical development, Ultramicroscopy
**108**, 503–513 (2008)CrossRefGoogle Scholar - V. Stolojan, R.E. Dunin-Borkowski, M. Weyland, P.A. Midgley: Three-dimensional magnetic fields of nanoscale elements determined by electron-holographic tomography, Electron Microsc. Anal.
**2001**, 243–246 (2001)Google Scholar - C. Phatak, A.K. Petford-Long, M. De Graef: Three-dimensional study of the vector potential of magnetic structures, Phys. Rev. Lett.
**104**, 253901 (2010)CrossRefGoogle Scholar - D. Wolf, L.A. Rodriguez, A. Béché, E. Javon, L. Serrano, C. Magen, C. Gatel, A. Lubk, H. Lichte, S. Bals, G. Van Tendeloo, A. Fernández-Pacheco, J.M. De Teresa, E. Snoeck: 3D magnetic induction maps of nanoscale materials revealed by electron holographic tomography, Chem. Mater.
**27**(19), 6771–6778 (2015)CrossRefGoogle Scholar - P. Simon, D. Wolf, C. Wang, A.A. Levin, A. Lubk, S. Sturm, H. Lichte, G.H. Fecher, C. Felser: Synthesis and three-dimensional magnetic field mapping of Co
_{2}FeGa Heusler nanowires at 5 nm resolution, Nano Lett.**16**, 114 (2016)CrossRefGoogle Scholar - V. Migunov, H. Ryll, X. Zhuge, M. Simson, L. Strüder, K.J. Batenburg, L. Houben, R.E. Dunin-Borkowski: Rapid low dose electron tomography using a direct electron detection camera, Sci. Rep.
**5**, 14516 (2015)CrossRefGoogle Scholar - R. Guckenberger: Determination of a common origin in the micrographs of tilt series in three-dimensional electron microscopy, Ultramicroscopy
**9**, 167–173 (1982)CrossRefGoogle Scholar - L. Houben, M. Bar Sadan: Refinement procedure for the image alignment in high-resolution electron tomography, Ultramicroscopy
**111**(9/10), 1512–1520 (2011)CrossRefGoogle Scholar - T. Sanders, M. Prange, C. Akatay, P. Binev: Physically motivated global alignment method for electron tomography, Adv. Struct. Chem. Imaging
**1**, 1–11 (2015)CrossRefGoogle Scholar - D. Gürsoy, Y.P. Hong, K. He, K. Hujsak, S. Yoo, S. Chen, Y. Li, M. Ge, L.M. Miller, Y.S. Chu, V. De Andrade, K. He, O. Cossairt, A.K. Katsaggelos, C. Jacobsen: Rapid alignment of nanotomography data using joint iterative reconstruction and reprojection, Sci. Rep.
**7**, 11818 (2017)CrossRefGoogle Scholar - C.O. Sanchez Sorzano, C. Messaoudi, M. Eibauer, J.R. Bilbao-Castro, R. Hegerl, S. Nickell, S. Marco, J.M. Carazo: Marker-free image registration of electron tomography tilt-series, BMC Bioinformatics
**10**, 124 (2009)CrossRefGoogle Scholar - J. Kwon, J.E. Barrera, T.Y. Jung, S.P. Most: Measurements of orbital volume change using computed tomography in isolated orbital blowout fractures, Arch. Facial Plast. Surg.
**11**(6), 395–398 (2009)CrossRefGoogle Scholar - T. Furnival, R.K. Leary, P.A. Midgley: Denoising time-resolved microscopy image sequences with singular value thresholding, Ultramicroscopy
**178**, 112–124 (2017)CrossRefGoogle Scholar - P.C. Hansen, M. Saxild-Hansen: AIR Tools—a MATLAB package of algebraic iterative reconstruction methods, J. Comput. Appl. Math.
**236**(8), 2167–2178 (2012)CrossRefGoogle Scholar - P.F.C. Gilbert: The reconstruction of a three-dimensional structure from projections and its application to electron microscopy, II. Direct methods, Proc. Royal Soc. B
**182**(1066), 89–102 (1972)CrossRefGoogle Scholar - E. Lee, B.P. Fahimian, C.V. Iancu, C. Suloway, G.E. Murphy, E.R. Wright, D. Castaño Díez, G.J. Jensen, J. Miao: Radiation dose reduction and image enhancement in biological imaging through equally-sloped tomography, J. Struct. Biol.
**164**(2), 221–227 (2008)CrossRefGoogle Scholar - Y. Chen, F. Förster: Iterative reconstruction of cryo-electron tomograms using nonuniform fast Fourier transforms, J. Struct. Biol.
**185**(3), 309–316 (2014)CrossRefGoogle Scholar - Z. Saghi, D.J. Holland, R. Leary, A. Falqui, G. Bertoni, A.J. Sederman, L.F. Gladden, P.A. Midgley: Three-dimensional morphology of iron oxide nanoparticles with reactive concave surfaces, a compressed sensing-electron tomography (CS-ET) approach, Nano Lett.
**11**(11), 4666–4673 (2011)CrossRefGoogle Scholar - R. Leary, Z. Saghi, P.A. Midgley, D.J. Holland: Compressed sensing electron tomography, Ultramicroscopy
**131**, 70–91 (2013)CrossRefGoogle Scholar - J. Miao, F. Förster, O. Levi: Equally sloped tomography with oversampling reconstruction, Phys. Rev. B
**72**(5), 052103 (2005)CrossRefGoogle Scholar - C. Zhu, C.-C. Chen, J. Du, M.R. Sawaya, M.C. Scott, P. Ercius, J. Ciston, J. Miao: Towards three-dimensional structural determination of amorphous materials at atomic resolution, Phys. Rev. B
**88**, 100201 (2013)CrossRefGoogle Scholar - M.I. Sezan: An overview of convex projections theory and its application to image recovery problems, Ultramicroscopy
**40**(1), 55–67 (1992)CrossRefGoogle Scholar - R. Gordon, R. Bender, G.T. Herman: Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and X-ray photography, J. Theor. Biol.
**29**(3), 471–481 (1970)CrossRefGoogle Scholar - P. Gilbert: Iterative methods for the three-dimensional reconstruction of an object from projections, J. Theor. Biol.
**36**(1), 105–117 (1972)CrossRefGoogle Scholar - J.I. Agulleiro, J.J. Fernandez: Fast tomographic reconstruction on multicore computers, Bioinformatics
**27**(4), 582–583 (2011)CrossRefGoogle Scholar - S. Kaczmarz: Angenäherte Auflösung von Systemen linearer Gleichungen, Bull. Int. Acad. Pol. Sci. Lett. A
**35**, 355–357 (1937)Google Scholar - P.P.B. Eggermont, G.T. Herman, A. Lent: Iterative algorithms for large partitioned linear systems, with applications to image reconstruction, Linear Algebra Appl.
**40**, 37–67 (1981)CrossRefGoogle Scholar - Y. Censor, S.A. Zenios:
*Parallel Optimization: Theory and Algorithms*(Oxford Univ. Press, New York 1997)Google Scholar - A.H. Andersen, A.C. Kak: Simultaneous algebraic reconstruction technique (SART): A superior implementation of the ART algorithm, Ultrason. Imaging
**6**(1), 81–94 (1984)CrossRefGoogle Scholar - L. Landweber: An iteration formula for Fredholm integral equations of the first kind, Am. J. Math.
**73**(3), 615–624 (1951)CrossRefGoogle Scholar - J. Gregor, T. Benson: Computational analysis and improvement of SIRT, IEEE Trans. Med. Imaging
**27**(7), 918–924 (2008)CrossRefGoogle Scholar - E. Elfving, T. Nikazad, P.C. Hansen: Semi-convergence and relaxation parameters for a class of SIRT algorithms, Electron. Trans. Numer. Anal.
**37**, 321–336 (2010)Google Scholar - J. Tong, I. Arslan, P. Midgley: A novel dual-axis iterative algorithm for electron tomography, J. Struct. Biol.
**153**(1), 55–63 (2006)CrossRefGoogle Scholar - D. Wolf, A. Lubk, H. Lichte: Weighted simultaneous iterative reconstruction technique for single-axis tomography, Ultramicroscopy
**136**, 15–25 (2014)CrossRefGoogle Scholar - A. Lange, A. Kupsch, M.P. Hentschel, I. Manke, N. Kardjilov, T. Arlt, R. Grothausmann: Reconstruction of limited computed tomography data of fuel cell components using direct iterative reconstruction of computed tomography trajectories, J. Power Sources
**196**(12), 5293–5298 (2011)CrossRefGoogle Scholar - S. Lück, A. Kupsch, A. Lange, M.P. Hentschel, V. Schmidt: Statistical analysis of tomographic reconstruction algorithms by morphological image characteristics, Image Anal. Stereol.
**29**(2), 61–77 (2010)CrossRefGoogle Scholar - E.J. Candès, J. Romberg, T. Tao: Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inf. Theory
**52**(2), 489–509 (2006)CrossRefGoogle Scholar - D.L. Donoho: Compressed sensing, IEEE Trans. Inf. Theory
**52**(4), 1289–1306 (2006)CrossRefGoogle Scholar - E.Y. Sidky, X.C. Pan: Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization, Phys. Med. Biol.
**53**(17), 4777–4807 (2008)CrossRefGoogle Scholar - D.J. Holland, D.M. Malioutov, A. Blake, A.J. Sederman, L.F. Gladden: Reducing data acquisition times in phase-encoded velocity imaging using compressed sensing, J. Magn. Reson.
**203**(2), 236–246 (2010)CrossRefGoogle Scholar - M. Lustig, D. Donoho, J.M. Pauly: Sparse MRI: The application of compressed sensing for rapid MR imaging, Magn. Reson. Med.
**58**(6), 1182–1195 (2007)CrossRefGoogle Scholar - M.W. Kim, J. Choi, L. Yu, K.E. Lee, S.S. Han, J.C. Ye: Cryo-electron microscopy single particle reconstruction of virus particles using compressed sensing theory, Proceedings SPIE
**6498**, 64981G (2007)CrossRefGoogle Scholar - C. Vonesch, W. Lanhui, Y. Shkolnisky, A. Singer: Fast wavelet-based single-particle reconstruction in cryo-EM. In:
*IEEE Symp. Biomed. Imaging*(2011), https://doi.org/10.1109/ISBI.2011.5872791CrossRefGoogle Scholar - D.S. Taubman, M.W. Marcellin: JPEG2000: Standard for interactive imaging, Proceedings IEEE
**90**(8), 1336–1357 (2002)CrossRefGoogle Scholar - J.L. Starck, F. Murtagh, J.M. Fadili:
*Sparse Image and Signal Processing: Wavelets, Curvelets, Morphological Diversity*(Cambridge Univ. Press, Cambridge 2010)CrossRefGoogle Scholar - L. Rudin, S. Osher, E. Fatemi: Non-linear total variation noise removal algorithm, Physica D
**60**, 259–268 (1992)CrossRefGoogle Scholar - S. Mallat:
*A Wavelet Tour of Signal Processing*(Academic Press, Burlington 2008)Google Scholar - A. Stoschek, R. Hegerl: Denoising of electron tomographic reconstructions using multiscale transformations, J. Struct. Biol.
**120**(3), 257–265 (1997)CrossRefGoogle Scholar - C.O.S. Sorzano, E. Ortiz, M. López, J. Rodrigo: Improved Bayesian image denoising based on wavelets with applications to electron microscopy, Pattern Recognit.
**39**(6), 1205–1213 (2006)CrossRefGoogle Scholar - C.O.S. Sorzano, S. Jonić, C. El-Bez, J.M. Carazo, S. De Carlo, P. Thévenaz, M. Unser: A multiresolution approach to orientation assignment in 3D electron microscopy of single particles, J. Struct. Biol.
**146**(3), 381–392 (2004)CrossRefGoogle Scholar - K. Song, L.R. Comolli, M. Horowitz: Removing high contrast artifacts via digital inpainting in cryo-electron tomography: An application of compressed sensing, J. Struct. Biol.
**178**(2), 108–120 (2012)CrossRefGoogle Scholar - J.L. Starck, M. Elad, D.L. Donoho: Image decomposition via the combination of sparse representations and a variational approach, IEEE Trans. Image Process.
**14**(10), 1570–1582 (2005)CrossRefGoogle Scholar - G.T. Herman, A. Kuba:
*Discrete Tomography: Foundations, Algorithms and Applications*(Birkhauser, Boston 1999)CrossRefGoogle Scholar - K.J. Batenburg, S. Bals, J. Sijbers, C. Kübel, P.A. Midgley, J.C. Hernandez, U. Kaiser, E.R. Encina, E.A. Coronado, G. Van Tendeloo: 3D imaging of nanomaterials by discrete tomography, Ultramicroscopy
**109**(6), 730–740 (2009)CrossRefGoogle Scholar - J.R. Jinschek, K.J. Batenburg, H.A. Calderon, R. Kilaas, V. Radmilovic, C. Kisielowski: 3-D reconstruction of the atomic positions in a simulated gold nanocrystal based on discrete tomography: Prospects of atomic resolution electron tomography, Ultramicroscopy
**108**(6), 589–604 (2008)CrossRefGoogle Scholar - K. Batenburg, J. Sijbers: DART: A practical reconstruction algorithm for discrete tomography, IEEE Trans. Image Process.
**20**(9), 2542–2553 (2011)CrossRefGoogle Scholar - S. Bals, M. Casavola, M.A. van Huis, S. Van Aert, K.J. Batenburg, G. Van Tendeloo, D.L. Vanmaekelbergh: Three-dimensional atomic imaging of colloidal core-shell nanocrystals, Nano Lett.
**11**(8), 3420–3424 (2011)CrossRefGoogle Scholar - F.J. Maestre-Deusto, G. Scavello, J. Pizarro, P.L. Galindo: ADART: an adaptive algebraic reconstruction algorithm for discrete tomography, IEEE Trans. Image Process.
**20**(8), 2146–2152 (2011)CrossRefGoogle Scholar - K.J. Batenburg, W. van Aarle, J. Sijbers: A semi-automatic algorithm for grey level estimation in tomography, Pattern Recognit. Lett.
**32**(9), 1395–1405 (2011)CrossRefGoogle Scholar - A. Zürner, M. Döblinger, V. Cauda, R. Wei, T. Bein: Discrete tomography of demanding samples based on a modified SIRT algorithm, Ultramicroscopy
**115**, 41–49 (2012)CrossRefGoogle Scholar - T. Roelandts, K.J. Batenburg, E. Biermans, C. Kübel, S. Bals, J. Sijbers: Accurate segmentation of dense nanoparticles by partially discrete electron tomography, Ultramicroscopy
**114**, 96–105 (2012)CrossRefGoogle Scholar - X. Zhuge, W.J. Palenstijn, K.J. Batenburg: TVR-DART: A more robust algorithm for discrete tomography from limited projection data with automated gray value estimation, IEEE Trans. Image Process.
**25**(1), 455–468 (2016)CrossRefGoogle Scholar - A. Alpers, R.J. Gardner, S. König, R.S. Pennington, C.B. Boothroyd, L. Houben, R.E. Dunin-Borkowski, K.J. Batenburg: Geometric reconstruction methods for electron tomography, Ultramicroscopy
**128**, 42–54 (2013)CrossRefGoogle Scholar - M. Wollgarten, M. Habeck: Autonomous reconstruction and segmentation of tomographic data, Micron
**63**, 20–27 (2014)CrossRefGoogle Scholar - R.J. Gardner:
*Geometric Tomography*(Cambridge Univ. Press, Cambridge 2006)CrossRefGoogle Scholar - T.C. Petersen, S.P. Ringer: Electron tomography using a geometric surface-tangent algorithm: Application to atom probe specimen morphology, J. Appl. Phys.
**105**(10), 103518 (2009)CrossRefGoogle Scholar - W.O. Saxton, W. Baumeister, M. Hahn: Three-dimensional reconstruction of imperfect two-dimensional crystals, Ultramicroscopy
**13**(1/2), 57–70 (1984)CrossRefGoogle Scholar - Y. Cheng: Single-particle cryo-EM at crystallographic resolution, Cell
**161**(3), 450–457 (2015)CrossRefGoogle Scholar - M. Shalev-Benami, Y. Zhang, D. Matzov, Y. Halfon, A. Zackay, H. Rozenberg, E. Zimmerman, A. Bashan, C.L. Jaffe, A. Yonath, G. Skiniotis: 2.8-Å cryo-EM structure of the large ribosomal subunit from the eukaryotic parasite Leishmania, Cell Rep.
**16**(2), 288–294 (2016)CrossRefGoogle Scholar - E. Callaway: The revolution will not be crystallized: A new method sweeps through structural biology, Nature
**525**, 172–174 (2015)CrossRefGoogle Scholar - N. Grigorieff: Direct detection pays off for electron cryo-microscopy, eLife
**2**, e00573 (2013)CrossRefGoogle Scholar - D. Rossouw, R. Krakow, Z. Saghi, C.S.M. Yeoh, P. Burdet, R.K. Leary, F. de la Peña, C. Ducati, C.M.F. Rae, P.A. Midgley: Blind source separation aided characterization of the γ' strengthening phase in an advanced nickel-based superalloy by spectroscopic 4D electron microscopy, Acta Mater.
**107**, 229–238 (2016)CrossRefGoogle Scholar - S.M. Collins, E. Ringe, M. Duchamp, Z. Saghi, R.E. Dunin-Borkowski, P.A. Midgley: Eigenmode tomography of surface charge oscillations of plasmonic nanoparticles by electron energy loss spectroscopy, ACS Photonics
**2**(11), 1628–1635 (2015)CrossRefGoogle Scholar - S.M. Collins, S. Fernandez-Garcia, J.J. Calvino, P.A. Midgley: Sub-nanometer surface chemistry and orbital hybridization in lanthanum-doped ceria nano-catalysts revealed by 3D electron microscopy, Sci. Rep.
**7**, 5406 (2017)CrossRefGoogle Scholar - J.-J. Fernandez: Computational methods for materials characterization by electron tomography, Curr. Opin. Solid State Mater. Sci.
**17**(3), 93–106 (2013)CrossRefGoogle Scholar - N. Volkmann, J.J. Grant: Methods for segmentation and interpretation of electron tomographic reconstructions, Methods Enzymol.
**483**, 31–46 (2010)CrossRefGoogle Scholar - R. Narasimha, I. Aganj, A.E. Bennett, M.J. Borgnia, D. Zabransky, G. Sapiro, S.W. McLaughlin, J.L.S. Milne, S. Subramaniam: Evaluation of denoising algorithms for biological electron tomography, J. Struct. Biol.
**164**(1), 7–17 (2008)CrossRefGoogle Scholar - J.-J. Fernández, S. Li: An improved algorithm for anisotropic nonlinear diffusion for denoising cryo-tomograms, J. Struct. Biol.
**144**(1/2), 152–161 (2003)CrossRefGoogle Scholar - C. Bajaj, Z. Yu, M. Auer: Volumetric feature extraction and visualization of tomographic molecular imaging, J. Struct. Biol.
**144**(1/2), 132–143 (2003)CrossRefGoogle Scholar - J.-J. Fernandez: TOMOBFLOW: feature-preserving noise filtering for electron tomography, BMC Bioinformatics
**10**, 178 (2009)CrossRefGoogle Scholar - E. Garduño, M. Wong-Barnum, N. Volkmann, M.H. Ellisman: Segmentation of electron tomographic data sets using fuzzy set theory principles, J. Struct. Biol.
**162**(3), 368–379 (2008)CrossRefGoogle Scholar - K. Sandberg, M. Brega: Segmentation of thin structures in electron micrographs using orientation fields, J. Struct. Biol.
**157**(2), 403–415 (2007)CrossRefGoogle Scholar - R. Leary, Z. Saghi, M. Armbrüster, G. Wowsnick, R. Schlögl, J.M. Thomas, P.A. Midgley: Quantitative high-angle annular dark-field scanning transmission electron microscope (HAADF-STEM) tomography and high-resolution electron microscopy of unsupported intermetallic GaPd
_{2}catalysts, J. Phys. Chem. C**116**(24), 13343–13352 (2012)CrossRefGoogle Scholar - R. Thiedmann, A. Spettl, O. Stenzel, T. Zeibig, J.C. Hindson, Z. Saghi, N.C. Greenham, P.A. Midgley, V. Schmidt: Networks of nanoparticles in organic-inorganic composites: Algorithmic extraction and statistical analysis, Image Anal. Stereol.
**31**(1), 27–42 (2011)CrossRefGoogle Scholar - C.J. Gommes, K. de Jong, J.-P. Pirard, S. Blacher: Assessment of the 3D localization of metallic nanoparticles in Pd/SiO
_{2}cogelled catalysts by electron tomography, Langmuir**21**(26), 12378–12385 (2005)CrossRefGoogle Scholar - R. Grothausmann, G. Zehl, I. Manke, S. Fiechter, P. Bogdanoff, I. Dorbandt, A. Kupsch, A. Lange, M.P. Hentschel, G. Schumacher, J. Banhart: Quantitative structural assessment of heterogeneous catalysts by electron tomography, J. Am. Chem. Soc.
**133**(45), 18161–18171 (2011)CrossRefGoogle Scholar - J.C. Russ, F.B. Neal:
*The Image Processing Handbook*, 7th edn. (CRC, Boca Raton 2015)Google Scholar - H. Li, H.L. Xin, D.A. Muller, L.A. Estroff: Visualizing the 3D internal structure of calcite single crystals grown in agarose hydrogels, Science
**326**(5957), 1244–1247 (2009)CrossRefGoogle Scholar - M. Sezgin, B. Sankur: Survey over image thresholding techniques and quantitative performance evaluation, J. Electron. Imaging
**13**(1), 146–168 (2004)CrossRefGoogle Scholar - W. van Aarle, K.J. Batenburg, J. Sijbers: Optimal threshold selection for segmentation of dense homogeneous objects in tomographic reconstructions, IEEE Trans. Med. Imaging
**30**(4), 980–989 (2011)CrossRefGoogle Scholar - N. Otsu: A threshold selection method from gray-level histograms, IEEE Trans. Syst. Man Cybern.
**9**(1), 62–66 (1979)CrossRefGoogle Scholar - J.C. Hindson, Z. Saghi, J.-C. Hernandez-Garrido, P.A. Midgley, N.C. Greenham: Morphological study of nanoparticle-polymer solar cells using high-angle annular dark-field electron tomography, Nano Lett.
**11**(2), 904–909 (2011)CrossRefGoogle Scholar - H. Friedrich, S. Guo, P.E. de Jongh, X. Pan, X. Bao, K.P. de Jong: A quantitative electron tomography study of ruthenium particles on the interior and exterior surfaces of carbon nanotubes, ChemSusChem
**4**(7), 957–963 (2011)CrossRefGoogle Scholar - K.J. Batenburg, J. Sijbers: Optimal threshold selection for tomogram segmentation by projection distance minimization, IEEE Trans. Med. Imaging
**28**(5), 676–686 (2009)CrossRefGoogle Scholar - M.N. Lebbink, W.J.C. Geerts, T.P. van der Krift, M. Bouwhuis, L.O. Hertzberger, A.J. Verkleij, A.J. Koster: Template matching as a tool for annotation of tomograms of stained biological structures, J. Struct. Biol.
**158**(3), 327–335 (2007)CrossRefGoogle Scholar - N. Volkmann: A novel three-dimensional variant of the watershed transform for segmentation of electron density maps, J. Struct. Biol.
**138**(1/2), 123–129 (2002)CrossRefGoogle Scholar - H. Katz-Boon, C.J. Rossouw, M. Weyland, A.M. Funston, P. Mulvaney, J. Etheridge: Three-dimensional morphology and crystallography of gold nanorods, Nano Lett.
**11**(1), 273–278 (2011)CrossRefGoogle Scholar - C.M.A. Parlett, M.A. Isaacs, S.K. Beaumont, L.M. Bingham, N.S. Hondow, K. Wilson, A.F. Lee: Spatially orthogonal chemical functionalization of a hierarchical pore network for catalytic cascade reactions, Nat. Mater.
**15**, 178–182 (2016)CrossRefGoogle Scholar