26.electrontomographyinmaterialsscience electrontom · 2019-11-02 · electrontom 1279 partd|26...

Electron Tom1279

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26. Electron Tomography in Materials Science

Rowan K. Leary, Paul A. Midgley

This chapter illustrates how electron tomographyhas become a technique of primary importance inthe three-dimensional (3-D) microscopic analysisof materials. The foundations of tomography areset out with descriptions of the Radon transformand its inverse and its relationship to the Fouriertransform and the Fourier slice theorem. The ac-quisition of a tilt series of images is described andhow the angular sampling in the series affects theoverall 3-D resolution in the tomogram. The imag-ing modes available in the (scanning) transmissionelectron microscope are explored with reference totheir application in electron tomography and howeach mode can provide complementary informa-tion on the structural, chemical, electronic, andmagnetic properties of the material studied. Thechapter also sets out in detail methods for tomo-graphic reconstruction from backprojection anditerative methods, such as simultaneous itera-tive reconstruction technique (SIRT) and algebraicreconstruction technique (ART), through to morerecent compressed sensing approaches that aim tobuild in prior knowledge about the specimen intothe reconstruction process. The chapter concludeswith a look to the future.

26.1 Foundations of Tomography .............. 128026.1.1 The Radon Transform ......................... 128126.1.2 The Fourier Transform

and Fourier Slice Theorem .................. 1283

26.2 ET Acquisition ................................... 128426.2.1 Finite and Limited Angular Sampling .. 128426.2.2 Limited Sampling: Artefacts and

Reconstruction Resolution .................. 1286

26.3 ET Imaging Modes ............................. 128726.3.1 TEM or STEM for Tomography

in Materials Science ........................... 128826.3.2 BF-TEM Tomography .......................... 128826.3.3 STEM Tomography .............................. 128926.3.4 Aberration-Corrected

and Atomic-Scale TEMand STEM Tomography........................ 1292

26.3.5 Analytical Electron Tomography .......... 129226.3.6 Holographic ET .................................. 129826.3.7 Time-Resolved ET .............................. 1299

26.4 Tilt Series Alignment ......................... 1300

26.5 ET Reconstruction ............................. 130326.5.1 Backprojection .................................. 130326.5.2 Direct Fourier Inversion ...................... 130326.5.3 Algebraic Iterative Reconstruction ....... 130326.5.4 Algebraic Reconstruction Technique

(ART). ................................................ 130426.5.5 Simultaneous Iterative Reconstruction

Technique (SIRT) ................................ 130726.5.6 Advanced Reconstruction in ET ........... 1308

26.6 Segmentation, Visualization,and Quantitative Analysis .................. 1317

26.7 Conclusions ...................................... 1321

References ................................................... 1321

By recording images of an object at different orienta-tions, electron tomography (ET) provides a means toreconstruct that object in three dimensions, leading toa greater understanding of its internal structure, com-position, and physico-chemical properties. Whilst ETwas developed initially for the life sciences [26.1] it has,in the past 20 years or so, become an invaluable tech-

nique for the study of a vast range of materials acrossthe physical sciences [26.2].

Important length scales in materials systems extendover several orders of magnitude (Fig. 26.1) and com-plementary tomographic techniques may be used for3-D visualization. In this chapter, we focus on ET in the(scanning) transmission electron microscope, (S)TEM,

© Springer Nature Switzerland AG 2019P.W. Hawkes, J.C.H. Spence (Eds.), Springer Handbook of Microscopy, Springer Handbooks,https://doi.org/10.1007/978-3-030-00069-1_26

https://orcid.org/0000-0002-6817-458X

https://doi.org/10.1007/978-3-030-00069-1_26

PartD|26.1

1280 Part D Applied Microscopy

0.1 nm 1 nm 10 nm 100 nm 1 μm 10 μm 100 μm 1 mm(1 nm)³

(10 nm)³

(100 nm)³

(1 μm)³

(10 μm)³

(1 mm)³

(10 mm)³

(100 μm)³

Volume of material analyzed

Voxel dimensions/resolution

X-ray CT

Serial sectionmicrotome-SEM/(S)TEM

FIB-SEM

(S)TEM tomography

Single-particle analysis/atomic(S)TEM tomography

Atom probe tomography

Fig. 26.1 3-D imaging methodsshown as a function of the resolutionpossible and the volume of materialthat can be analyzed

which provides insight into the 3-D structure of ma-terials with sub-nm resolution across sub- m lengthscales.

There has been a rapid expansion in the numberof imaging modes that may be used for tomographicacquisition. High-resolution image series can now re-veal atomic structure in 3-D and a combination ofspectroscopic methods (e. g., energy-dispersive x-ray(EDX) and electron energy-loss spectroscopy (EELS))

and tomography leads to 3-D compositional, chemical,and electronic information. Moreover, a combinationof diffraction and tomography provides 3-D crystallo-graphic information and holotomography reconstruc-tions furnish a 3-D visualization of electro-magneticpotentials. In all cases, the data acquisition is limited,and much of the recent progress in ET has been facil-itated by improvements in the reconstruction routines,and these are explained below.

26.1 Foundations of Tomography

Although the literal meaning of the term tomographyrefers to the visualization of slices, transmission to-mography, such as (S)TEM tomography (referred tohere as ET) or x-ray tomography, can be considered asa method of reconstructing the interior of an object froma set of projections through its structure. In essence,this form of tomography is conventionally achieved byrecording a tilt series of projections, often about a singletilt axis. The ensemble of images is then used to forma reconstruction, or tomogram, via some operation thatcan essentially be seen as an inversion of the originalprojection process Fig. 26.2.

The mathematical framework for tomography orig-inates from a seminal paper by Radon in 1917 [26.3],in which the projection of an N-dimensional object intoa space of dimensions N � 1 was considered. That pro-jection, or transform, known now as the Radon trans-form and its inverse will be discussed in more detailbelow. In 1956, Bracewell [26.4] showed how tomog-raphy can be considered in terms of the more widelyknown Fourier transform, and the relationship betweenthe Fourier and Radon transforms was determined. Thefoundational principles of tomography stemming fromthese transforms are now well established, and further

Electron Tomography in Materials Science 26.1 Foundations of Tomography 1281Part

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a) b)Fig. 26.2 (a) Acquisitionof an angular series oftwo-dimensional (2-D)projections of an object,and (b) backprojectionof these images intoa 3-D space to obtaina reconstruction of theobject

coverage beyond that given here can be found in [26.5–10].

Although tomography is typically referred to asa 3-D reconstruction method, the most common sin-gle axis geometry permits the reconstruction process tobe addressed in terms of a series of, in principle, in-dependent 2-D reconstructions. In general, this is bothalgorithmically and computationally easier, and is usedfor the descriptions provided here. A possible disad-vantage of slice-by-slice 2-D reconstruction of a 3-Dvolume is that it may be difficult to fully exploit 3-Dprior knowledge during the reconstruction process, inwhich case fully 3-D reconstructions may be desirable.

26.1.1 The Radon Transform

The Radon transform R describes directly the projec-tion process, mapping a function f by line integralsalong all possible lines L. With increment ds along L,

Spatial domain

Projection(Radon domain)

f(I,θ)

θx

z

Is

L

f(x,z)

I = x cos θ + y sin θs = x sin θ + y sin θ

fFig. 26.3 The geometry of the Radontransform

the transformation may be defined as

Mf .l; �/ D Rf DZ

L

f .x; z/ ds (26.1)

the geometry of which is illustrated in Fig. 26.3 [26.6].The function f is defined here on the 2-D real spacecoordinates (x; z) and the Radon transform converts thedata into Radon space (l; � ), frequently referred to asa sinogram, where l is the line perpendicular to theprojection direction and � is the angle of projection(Fig. 26.4). Using polar coordinates (r; �), related toCartesian coordinates by

r Dpx2 C z2 and � D tan�1

� zx

�;

a point object in real space (x D r cos�, z D r sin�) isa line in Radon space (l; � ) linked by l D r cos.� ��/.

PartD|26.1


–r +r

θ = θ'(I', θ')

I = r cos (θ – Φ)

L

rI'

Φθ'

0

–90

900

θ (°)a) b)

c) d)

090

0

–90θ (°)

I

z

z

x

Ix

Fig. 26.4a–d The rela-tionship between (a,c) realspace and (b,d) Radonspace. (a) Point object,(b) sinogram of pointobject, (c) phantom,(d) sinogram of phantom

θ θ

kx

kz

f(I,θ) Projection(Radon domain)

f

Fourier transform

Fouriertransform

Fourier domainSpatial domain

x

zf(x,z) f(kx,kz)˜

Fig. 26.5 The Fourier slice theorem

Electron Tomography in Materials Science 26.1 Foundations of Tomography 1283Part

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z

x

kx

kz

I

θ

θ

k

f(x,y)

Radon transform,

f(I,θ)

Spatial domain

Fouriertransform

2

Interpolation

Radon domain (sinogram)

Fourier domain (Cartesian)Fourier domain (polar)

f(k,θ)

kx

kzf(kx,kz)

Fouriertransform

1

Fig. 26.6 Relationship between Fourier space and Radon space

In principle, the real space structure of the ob-ject f .x; z/, can be recovered from the Radon domainMf .l; �/ by an inversion of the Radon transform. Sincethe empirical sampling of an object by a projection isequivalent to a discrete sampling of the Radon integral,the goal in tomography is, then, to acquire a sufficientnumber of projections such that an inverse Radon trans-form, or some other means of reconstruction, can yieldan adequate approximation of the object.

26.1.2 The Fourier Transformand Fourier Slice Theorem

Closely related to the Radon transform is the Fouriertransform, which for the general case of an N-dimensional function f , may be written as

Qf .k/ D FNf D1Z

�1f .x/e�2 ikxdx ; (26.2)

where x D x1, x2, . . . , xN are the coordinates in realspace and k D k1, k2, . . . , kN the corresponding coordi-

nates in Fourier space. In words, the Fourier transformprovides an alternative means of representing a functionby decomposing it into (spatial) frequency components.Although perhaps most well known in the context ofanalyzing the characteristic periodicity in images ofcrystalline lattices, there is significance and practicalapplication in the tomographic imaging context too,which is embodied in the Fourier slice theorem, as de-scribed below.

The work of Bracewell [26.4] highlighted the im-portant relationship between projections formed via theRadon transform and the Fourier representation of anobject, which has become known as the Fourier slicetheorem [26.7]. The theorem states that

the Fourier transform of the projection of a func-tion, at a given angle, is equivalent to a centralsection, at that angle, through the Fourier trans-form of the function.

Figure 26.5 illustrates the theorem for 1-D projectionsof a 2-D object; an analogous relationship applies to

PartD|26.2


2-D projections of an object and central sections in its3-D Fourier space.

The Fourier slice theorem further aids the under-standing of tomographic reconstruction and the con-sequences of finite sampling: acquiring projections atdifferent angles is equivalent to sampling sections ofthe object’s Fourier space over the range of frequen-cies in each central section. However, most objects willnot be fully described by the frequencies in one section,or even a few sections, meaning that many projectionsat different angles are required to sample Fourier space

sufficiently such that, in principle, it would be possibleto obtain a satisfactory description of the object in realspace by direct inversion of the sampled Fourier space.

The relationships between real, Radon, and Fourierspace are shown explicitly in Fig. 26.6. From Fig. 26.6and the foregoing discussion it is apparent that the to-mographic reconstruction process can be approachedvia either a real space-based backprojection route, ora Fourier inversion route. The practicalities of ET how-ever, reviewed hereinafter, mean that either approach israrely straightforward to achieve.

26.2 ET Acquisition

The first example of 3-D reconstruction using TEMcame from DeRosier and Klug [26.11] who, har-nessing prior knowledge of the helical symmetry ofthe object, obtained a 3-D reconstruction of the tailof the T4 bacteriophage from a single 2-D image.They also outlined the principles of reconstructingarbitrarily-shaped 3-D objects from a series of 2-DTEM images; or more specifically, TEM projections.Along with two other seminal papers from the sameyear [26.12, 13], this is often seen as the starting pointof ET.

An ET investigation consists of a number of distinctbut inter-related stages, summarized in Fig. 26.7. Withthe exception of sample preparation, which is referredto inter alia, these are reviewed sequentially in the fol-lowing sections.

Sample preparation

Tilt-series aquisition

Pre-/postprocessing

Pre-/postprocessing

Pre-/postprocessing

Alignment

Reconstruction

Segmentation

Quantification

Visualization/interpretation

Fig. 26.7 Principal stages in an ETinvestigation

26.2.1 Finite and Limited Angular Sampling

The sampling theory outlined above suggests that thebest tomographic reconstructions will be achieved byacquiring as many projections over as large an angu-lar range as possible. For conventional reconstructions,without the application of prior knowledge (see later),regular tilt increments of 1�2ı are used. Other tiltingschemes are detailed in [26.5, 14]. However, severalfactors in ET always restrict the actual number of pro-jections obtained to being fewer than the ideal. Firstly,in contrast to many other tomographic techniques, ET isperformed in a highly restricted working space. In prac-tice, the sample is located between the polepieces of animmersion objective lens, as illustrated in Fig. 26.8a. Inorder to achieve high spatial resolution in each image,

Electron Tomography in Materials Science 26.2 ET Acquisition 1285Part

D|26.2

the polepiece gap must be kept as small as possible (ca.2�5mm) minimizing the effects of spherical and chro-matic aberration ([26.15] and references therein). Thisgap may limit the angular range over which the speci-men can be tilted.

ET is thus a limited-angle tomographic problem,with a large unsampled region in Fourier space, knownas the missing wedge (Fig. 26.9). The design of spe-cialized narrow profile tomography holders has enabledthe maximum tilt angle �max to approach 80ı. However,additional limitationsmay come into play at high tilt an-gles; this may be due to shadowing from the specimensupport grid or holder (in Fig. 26.8b,c), because the in-crease in projected specimen thickness (Fig. 26.8d)mayrender the projection unusable through blurring due tochromatic aberration in TEM or beam broadening, andthus loss of resolution, in STEM (Fig. 26.8e).

Where needed, any sample support grid used in anET experiment should be judiciously chosen, strikinga balance between sturdy support and occlusion of thesample during tilting. Selecting a wide grid bar spacing,to minimize shadowing, must be considered in conjunc-tion with the rigidity of the support film, determined byits type and thickness. Typically, for many nanoparticu-late specimens, a 200 mesh grid spacing (where smallermesh numbers correspond to wider bar spacing) pro-vides a reasonable compromise. Many manufacturersalso offer slot grids comprising elongated sample view-ing windows. Placing the long axis of these windowsperpendicular to the tilt axis then provides an extendedfield of view without grid bar occlusion.

To further minimize the missing wedge problem,dual-axis tomography may be used [26.17–20] which,by combining two mutually perpendicular tilt series,reduces the missing wedge to a missing pyramid. Insingle-axis ET, the effects of information loss due tothe missing wedge can be minimized by ensuring thatthe sample, or important features, are oriented alongthe tilt axis. Dual-axis tomography may be crucial incases where multiple feature orientations would meanthat information loss due to the missing wedge wouldbe intolerable [26.20, 21]. Disadvantages of dual-axisET include additional electron beam exposure, and op-erator time, required to record two tilt series, as wellas possible challenges in combining the two data setsaccurately.

In the physical sciences, dual-axis ET is used farless than the single-axis geometry. As an alternative,there has been a growing trend in the preparation ofneedle-shaped samples, which, using specialist holders,can be rotated through the full ˙90ı range, and avoidthe aforementioned problems of shadowing or thicknessincreases [26.22–25]. A prerequisite for this approachis that the sample must be amenable to fabrication into

Upper objectivelens polepiece

Lower objectivelens polepiece

e-beam

θmax

Tilt rangeca. ±70°

θmax

θmax

Specimensupport grid Specimen

holder

a)

b) c)

Thickspecimen

Extendedspecimen

θmax θmax

tmintmax

Out offocus

In focus

Out offocus

d) e)

Fig. 26.8a–e Reasons for limited angular sampling in ET:(a) restricted polepiece gap; (b) support grid or (c) holdershadowing; (d) specimen thickness; (e) limited depth-of-field

a needle shape using a focused ion beam or be attach-able by some means to a needle-shaped support. Thisis ideal for analysis of targeted features extracted frombulk specimens.

Even for samples that can be tilted over the fullangular range, undersampling still occurs because ofthe finite angular increment �� over which the sample

PartD|26.2


Fourier transform

Missingwedge

Missingwedge

1/D

∆θ

θmax

θmin

Missingwedge

Missingwedge

Slice through tomogramkz

kx

z

x

kz

kx

a) b)

c)

Fig. 26.9 (a) Illustration of finite and limited angular sampling of Fourier space in ET; (b,c) its manifestation in practice.In (a), each projection, of an object of diameter D, is a central section in Fourier space of thickness 1=D. The dashedline indicates the limit at which the information from adjacent projections just overlaps, corresponding to the Crowthercriterion [26.16] defined in (26.4)

is tilted between each projection. The chosen angularsampling is ultimately set by the electron dose thatthe specimen can withstand, as faithful tomographicreconstruction is reliant on the premise that the spec-imen does not change (significantly) during collectionof the image series. (An exception to this rule is dy-namic tomography, which seeks to track continuouslyoccurring changes, but is reliant on prior knowledge tobe able to predict the nature of changes; such methodshave not yet been developed in ET.) While pertinentin all (S)TEM studies, the repeated imaging of a re-gion in ET means that consideration of electron beaminduced changes and carbonaceous contamination areparamount. Every effort should be made to assess thepotential for and measures taken to mitigate beam dam-age and contamination, including sample preparationand strategy for the tilt series acquisition.

Automated or semi-automated tilt series acquisitionand low-dose procedures have facilitated the applica-tion of ET to beam-sensitive specimens that damageby inelastic processes (viz. heating and radiolysis), andhave been critical in the biological sciences [26.1].Likewise, operation below the threshold for knock-ondamage can open the door to analysis of specimens thatdamage predominantly by elastic scattering [26.26]. Al-though less critical for most specimens in the physicalsciences, the idea of dose fractionation [26.5] is one thatshould always be borne in mind when recording a tiltseries, optimizing the information acquired about thewhole specimen across the tilt series.

26.2.2 Limited Sampling: Artefactsand Reconstruction Resolution

The finite and limited angular sampling in ET can leadto serious artefacts in the tomographic reconstructions,which are well documented in the literature [26.2, 27–29]. In fact, structures may be partially or entirelyabsent from the tomogram if it is affected stronglyby the missing wedge [26.30]. In general, the missingwedge causes an apparent elongation e in the beam di-rection, determined by �max [26.31]

e Ds

�max C sin �max cos �max

�max � sin �max cos �max(26.3)

as exemplified in Fig. 26.10. Experimentally, it hasbeen shown that deviations away from the ideal tomo-graphic reconstruction fall to acceptably small valueswhen �max D ˙80ı [26.21, 28, 32]. Qualitatively, the ef-fect of finite angular sampling is to cause streakingartefacts, the severity of which increases with tilt in-crement (Fig. 26.10).

A consequence of the limited angular sampling isthat the reconstruction resolution will vary with direc-tion [26.33], the determining variables for which inthe context of ET were first considered by Crowther[26.16]. The resolution parallel to the tilt axis dy should,in principle, be equal to that of the input projections.Perpendicular to the tilt axis, the Crowther criterion

Electron Tomography in Materials Science 26.3 ET Imaging Modes 1287Part

D|26.3

+/– 90° tilt range +/– 70° tilt range +/– 50° tilt range +/– 30° tilt range

2° ti

lt in

crem

ent

5° ti

lt in

crem

ent

10°

tilt

incr

emen

t20

° ti

lt in

crem

ent

Fig. 26.10 The effect of finite tiltincrement and limited angular tiltrange on tomographic reconstructionof a spherical phantom (weightedbackprojection reconstructions). Themaximum tilt angle and tilt incrementused for each reconstruction isdenoted by the column and rowheadings, respectively

(which assumes �max D ˙90ı) gives the resolution dxas being determined by the number of projections Np

and the diameter of the region to be reconstructed D

dx D D

Np; (26.4)

the geometry of which is illustrated in Fig. 26.9a. Dueto the missing wedge, the reconstruction in the direc-tion of the electron beam dz is further degraded by theelongation factor exz

dz D dx exz : (26.5)

Although these expressions are often cited as usefulguides, the reconstruction resolution is often seen to beconsiderably better than predicted by this criterion whenconstrained reconstruction techniques are used ([26.5,Chap. 10] or [26.32, 34] for methods). As summarizedby Midgley and Weyland [26.2], the actual reconstruc-tion resolution is likely to depend on a combination ofthe effects of the sampling regime, noise characteristics,shape of the object to be reconstructed, and the nature ofthe reconstruction routine. A rough rule of thumb is thatthe achievable resolution in an electron tomographicreconstruction using conventional techniques will typi-cally be ca. 1=100 of the object diameter.

26.3 ET Imaging Modes

In the most basic sense, the transmission of the elec-tron beam through the specimen allows the (S)TEM tobe described as a structure projector. However, as ex-emplified by Hawkes [26.5, Chap. 3], and Midgley andWeyland [26.2], it is critically important to consider theextent to which the signals obtained in (S)TEM con-stitute a projection that is valid, or at least useful, fortomographic reconstruction.

While the ideal projection involves a sum integral ofsome physical property, as defined by the Radon trans-form, this is rarely achieved in (S)TEM. Instead, it isgenerally regarded as sufficient that a (S)TEM signal isa monotonic function of a projected physical quantity. Itis this more relaxed stipulation that is generally referredto as the projection requirement in ET. It is clear fromET studies to date that directly interpretable signals

PartD|26.3


satisfying the projection requirement are significantlyeasier to process, and the ET reconstructions are morereadily interpreted than those that do not. Nonetheless,ET is often performed using signals that do not fullysatisfy the projection requirement. In those cases espe-cially, careful interpretation of the resulting tomogramsmay be required to differentiate real structure from arte-facts, or special reconstruction approaches may need tobe adopted to achieve 3-D reconstruction.

26.3.1 TEM or STEM for Tomographyin Materials Science

Here it is worth contrasting specimen characteristicsand consequent ET practices in the physical and bio-logical sciences. Although the two fields share manyprinciples and practices, the specimen demands and thedominant imaging modes are, in the majority, quite dis-tinct. ET in the biological sciences has been practicedfor many years using bright-field (BF) TEM, and this isstill dominantly the case. BF-TEM is highly suitable forbiological specimens because they are often noncrys-talline and/or thin, weakly scattering objects. For these,mass-thickness or phase contrast are the main deter-minants in image formation, and the images obtained,perhaps after correcting for lens aberrations, may beconsidered true projections of the underlying structure.

For strongly scattering crystalline specimens on theother hand, as are common in the physical sciences,several factors (detailed below) can readily lead to vi-olation of the projection requirement in BF-TEM. Onthe other hand, specimens in the physical sciences areoften much more electron beam tolerant, permitting useof a range of imaging modes that would, in general, befar too damaging for biological structures. Alternativeimaging modes can both satisfy better the projectionrequirement for specimens in the physical sciences andenable measurement of not just morphological charac-teristics in 3-D, but also chemical, magnetic, electronic,and crystallographic properties.

While many insights have been made, and continueto be made, using TEM for ET in the physical sci-ences, and certain niche techniques inherently requireTEM-based techniques, STEM has become increas-ingly popular. Overwhelmingly, it is clear that majorreasons for the popularity of STEM are (i) the annulardark-field (ADF) imaging mode, which can often pro-vide both intuitive and high-fidelity analysis due to itsdirect interpretability; and (ii) the ability to acquire si-multaneously a range of signals, opening the door toextended multidimensional and multimodal analyses.

Table 26.1 summarizes the main signal modes thatare used in ET. The most important of these are re-viewed in the following sections.

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, 56]. In BF-TEM, the imageintensity, generally, does not show a monotonic depen-dence on the specimen thickness, depending stronglyand in an involvedmanner on defocus. As such, contrastreversals in the image can occur through the speci-men thickness or as a result of small changes in theelectron optical conditions. By their nature, BF-TEMimages typically yield only weak chemical sensitivity—a considerable drawback when seeking to investigatecomplex multi-element samples and/or when seekingto resolve fine-scale features against contrast gener-ated from any specimen support. For strongly scatteringcrystalline specimens, further complications in BF-TEM can be introduced due to strong Fresnel contrastand domination of the image by diffraction contrast(Bragg scattering). These signals carry a wealth of in-formation that is of interest in certain contexts, suchas diffraction contrast imaging of planar defects andstrain fields [26.55, Chaps. 25 and 26], but in other con-texts they can preclude a general facile interpretationthat a monotonically varying signal endows and causemarked problems for ET. Significant image complica-tions, so-called delocalization, may also arise at highresolution due to lens aberrations. Aberration-corrected(AC) optics and the use of “negative spherical aber-ration imaging” (Urban et al. in [26.57]) can permitcompensation of aberrations and allow more readilyinterpretable TEM images to be formed. However,direct information on chemical composition is still lack-ing.

Considering the effects outlined above, the suit-ability with regards to the tomographic projection re-quirement has been extensively discussed in founda-tional [26.35, 58], [26.10, Chap. 3] and review litera-ture [26.2, 20, 27, 28, 59–66]. There seems to be generalconsensus in the literature that BF-TEM is capableof approximately reconstructing the exterior shape ofconvex homogeneous crystalline objects, while the in-tensity of the interior may be subject to erroneousnonlinearity due to diffraction effects [26.65, 67, 68].

Another imaging mode, ADF-TEM, can be imple-mented using an annular aperture in the back focalplane. This can yield chemically sensitive tomogramsin a manner similar to ADF-STEM, and may haveparticular merits for fast acquisition and low-contrastsoft matter [26.42, 69]. However, for most samplesdark-field imaging in STEM can often provide a morepowerful approach. Interestingly, there are significantnew opportunities for TEM ET studies exploiting recentadvances with direct electron detectors.


D|26.3

Table 26.1 Principal (S)TEM imaging modes for ET of solid catalysts

Signal mode Contrastmechanism

Early/selected studies Statusa Suitable studies

Morphological imaging modesBF-CTEM Phase,

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

E WPOs, biological specimens,amorphous materials

ADF-STEM Atomic number(Z)

Midgley and Weyland (2001) [26.35] E Crystalline specimens, Z-contrast

Cs-CTEM Bar Sadan et al. (2008) [26.37] A Atomic-scale, WPOsCs-STEM Atomic number

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

A Atomic-scale; heavy metal nano-particles

Cc-TEM Baudoin et al. (2013) [26.40] A Thick biological specimensDF-TEM Angularly

selectivescattering

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

A Lattice defects,low-contrast soft matter

PrecessionBF-CTEM

Rebled et al. (2011) [26.43] A Crystalline specimens

BF-STEM Sousa et al. (2011) [26.44] A Thick specimens, polymers,biological sections

IBF-STEM Ercius et al. (2006) [26.45] A Thick specimensMAADF-STEM Sharp et al. (2008) [26.46] A DislocationsSTEM in ESEM Jornsanoh et al. (2011) [26.47] A Nonconductive

or hydrated specimens

Multidimensional imaging modesEFTEM Weyland and Midgley (2001) [26.48];

Möbus and Inkson (2001) [26.49]E Chemical segregation, optical

properties, bonding variationsEELS Inelastic scat-

teringJarausch et al. (2009) [26.22] A Chemical environment

Elemental distribution (core-loss)Optical properties (low-loss)

EDXS Secondaryx-ray emission

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

A Elemental distribution

Diffraction Kolb et al. (2007) [26.52] A Crystalline materialsHolography Reconstructed

phase andamplitude

Twitchett-Harrison et al. (2007) [26.53] A Mean inner potential,electrostatic and magnetic fields

Time resolved Kwon and Zewail (2010) [26.54] A New commercial TEMs

Signal mode Contrastmechanism

Early/selected studies Statusa Suitable studies

Morphological imaging modesBF-CTEM Phase,

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

E WPOs, biological specimens,amorphous materials

ADF-STEM Atomic number(Z)

Midgley and Weyland (2001) [26.35] E Crystalline specimens, Z-contrast

Cs-CTEM Bar Sadan et al. (2008) [26.37] A Atomic-scale, WPOsCs-STEM Atomic number

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

A Atomic-scale; heavy metal nano-particles

Cc-TEM Baudoin et al. (2013) [26.40] A Thick biological specimensDF-TEM Angularly

selectivescattering

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

A Lattice defects,low-contrast soft matter

PrecessionBF-CTEM

Rebled et al. (2011) [26.43] A Crystalline specimens

BF-STEM Sousa et al. (2011) [26.44] A Thick specimens, polymers,biological sections

IBF-STEM Ercius et al. (2006) [26.45] A Thick specimensMAADF-STEM Sharp et al. (2008) [26.46] A DislocationsSTEM in ESEM Jornsanoh et al. (2011) [26.47] A Nonconductive

or hydrated specimens

Multidimensional imaging modesEFTEM Weyland and Midgley (2001) [26.48];

Möbus and Inkson (2001) [26.49]E Chemical segregation, optical

properties, bonding variationsEELS Inelastic scat-

teringJarausch et al. (2009) [26.22] A Chemical environment

Elemental distribution (core-loss)Optical properties (low-loss)

EDXS Secondaryx-ray emission

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

A Elemental distribution

Diffraction Kolb et al. (2007) [26.52] A Crystalline materialsHolography Reconstructed

phase andamplitude

Twitchett-Harrison et al. (2007) [26.53] A Mean inner potential,electrostatic and magnetic fields

Time resolved Kwon and Zewail (2010) [26.54] A New commercial TEMs

aE D established, A D advanced

Despite the technical challenges, there have beencases where important information has been revealedusing BF-TEM tomography, e. g., [26.70] where dueconsideration is given to possible violation of the pro-jection requirement, or the effects are insignificant atthe level of interest in the tomogram. The morphol-ogy of polymer systems has been studied with BFtomography [26.71], though STEM is increasingly be-ing adopted for these too [26.72]. Thin carbonaceousor similar materials, i. e., those that are weakly scat-tering, have also been profitably studied with (AC)BF-TEM [26.26, 37]. Figure 26.11 illustrates the lo-cal nanoporosity obtained using BF-TEM ET [26.73]of SBA-15 mesoporous silica. Whilst Fig. 26.11ashows a single BF image, Fig. 26.11b–e show slicesfrom the tomographic reconstruction revealing, in fargreater detail than is possible with a single BF im-

age, the presence of locally disordered and mergedpores (Fig. 26.11b–e). The reconstructions were usedto prove that, in samples subjected to higher temper-ature hydrothermal treatments, pore fraction increasesdetected by nitrogen sorption were because of the in-crease in the number and volume fraction of disorderedmerged pores. A similar BF-TEM ET study [26.74]of hydrothermally treated zeolitic catalyst showed howimage analysis techniques may be used to reveal a hier-archical porosity [26.75].

26.3.3 STEM Tomography

STEM, using the ADF imaging mode, has becomethe most widely utilized technique for ET in thephysical sciences [26.2, 10, 27, 59–61, 75–77], [26.10,Chap. 12], [26.78, Chap. 8]. The motivation for collect-

PartD|26.3


100 nm

50 nm 50 nm 50 nm

100 nm100 nm

50 nm 50 nm 50 nm

100 nm

a) b)

c) d) e)

a) b)

c) d) e)

Fig. 26.11a–e BF-TEM imageof an ordered mesoporous silica(SBA-15). (b) Slice through anET reconstruction, showing localdisorder caused by merged pores,examples of which are highlightedin color in (c–e), which are slicestaken at different heights from theboxed region in (a). Reproducedfrom [26.73] with permission of TheRoyal Society of Chemistry (RSC)on behalf of the European Societyfor Photobiology, the EuropeanPhotochemistry Association, and theRSC

ing an ADF signal is that at high detection angles, andwith a large angular integration range, coherent contri-butions to the image from Bragg-scattered beams areminimized. With the detected signal then dominatedby Rutherford-like and thermal diffuse scattering, thescattering detected from each atom can be consideredas transversely incoherent. The signal intensity shouldthen vary monotonically with the thickness of the speci-men and the atomic number Z of the constituent atoms,approaching a Z2 relationship. The actual Z exponentlies somewhere in the region of Z1:3�2, depending (pri-marily) on the inner detection angle [26.79, 80]. Unlikethe phase contrast transfer function of BF-TEM, the op-tical transfer function of ADF-STEM does not oscillaterapidly with changing spatial frequency or defocus; it isthese characteristics that endow direct interpretabilityand high contrast.

It was realized in the early development of STEMin the 1970s (Pennycook’s historical review in Chap. 1of [26.78]) that high-contrast chemically sensitiveatomic resolution images can be obtained of heavymetal nanoparticles, clusters, or even single atoms onlow-Z support materials. These are characteristics ful-filled by many supported nanoparticulate catalyst sys-tems and also, in general, by heavy metal nanoparticlesdeposited on low-Z TEM sample support grids, andADF-STEM tomography has been applied to many cat-alyst and nanoparticle systems over the past decade orso ([26.2, 27, 59, 77], [26.78, Chap. 8], [26.81, 82]). Il-lustrative examples are shown in Fig. 26.12.

In 2001, it was shown [26.69, 70] that the char-acteristics of ADF-STEM also make it a particu-

larly successful imaging mode for 3-D imaging ofstrongly scattering crystalline specimens via ET. Itis widely agreed that for many specimens in thephysical sciences, ADF-STEM can satisfy the projec-tion requirement to a sufficient approximation. Thus,it is has often been concluded that ADF-STEMis the most suitable technique for ET nanometrol-ogy [26.65, 85, 86]. A clear example can be found ina study by Lu et al. [26.86], who found that BF-TEM substantially overestimated the constituent vol-ume fraction of carbon-black in polymer compositescompared to ADF-STEM, which provided acceptableaccuracy.

However, ADF-STEM is not immune to potentialdifficulties that can lead to complications in interpre-tation. It is often pointed out that for crystalline speci-mens, the intensity of the image may be modified due tostrong Bloch wave channeling when the crystal is nearzone-axis orientation, which tends to concentrate thebeam intensity onto atomic columns [26.87]. Primarily,this can increase the high angle scattering, and therebythe intensity in the image. In general though, this effecttends to occur only at a small number of crystal orien-tations across an ET tilt series, is more uniform acrossa crystal, and is less pronounced relative to diffractioncontrast in BF-TEM. Nevertheless, a decision may needto be made as to whether to discard images stronglyaffected by channeling, or to proceed with using themin the reconstruction. For relatively minor occurrences,the effects may be sufficiently negated by virtue of thecombination of many tilt series images during the re-construction process.


D|26.3

Concave

Convex

Saddle

(Pt,Ru)

20 nm

Au

TiO2

3-D reconstruction

(Pt,Ru)-SiO220 nm

a)

b) c)

Anatase(200)

Anatase(004)

Anatase(004)

Boundary

d = 0.24 nm

Au(111)

Fig. 26.12a–c ET reconstructions of supported heavy metal nanoparticles (a) (Pt,Ru) nanocatalysts supported on a dis-ordered mesoporous silica. The surface of the silica has been color-coded according to the local Gaussian curvature. Thenanocatalysts (red) appear to prefer to anchor themselves at the (blue) saddle-points (for details, [26.83]). (b) Surface-rendered visualization of Au nanocatalysts (red) supported on titania (blue). The nanocatalysts are located in the crevicesbetween titania crystallites; confirmed by the aberration-corrected STEM image in panel (c). (b,c) reprinted from [26.84],with permission from Elsevier

Caution should also be noted in that very large dif-ferences in atomic number may lead to signals thatcould readily exceed the dynamic range of the ADFdetector, and consequently impose restriction to lowcontrast of the low-Z component(s), or lead to con-trast saturation of those of high Z that would violate the

projection requirement. Detector saturation or contrastreversal can also result from very thick samples, the lat-ter due to scattering to high angles beyond the outerradius of the detector. These can lead to artefacts suchas voids or erroneous core-shell structures in ET recon-structions [26.45, 82], the nature of which are examined

PartD|26.3


in detail in [26.67]. Good practice is to assess the poten-tial for contrast saturation at different tilt angles beforeembarking on acquisition of the tilt series. Detector gainand offset (contrast and brightness settings) should notbe altered during the tilt series, as doing so would vio-late the projection requirement.

While ADF-STEM using high angles is currentlythe imaging mode that, arguably, is likely to best sat-isfy the projection requirement for the widest rangeof specimens, there may be certain scenarios whenother variants of STEM become more suitable, for ex-ample, the use of a type of BF-STEM imaging forparticularly thick specimens that would produce con-trast reversals in ADF-STEM. Ercius et al. [26.45]have shown that coherence artefacts can be mini-mized by using a large bright field detector, whosebroad integration area effectively suppresses diffrac-tion contrast, providing an incoherent bright-field (IBF)image that satisfies the projection requirement. In par-ticular, while ADF-STEM provides some composi-tional contrast through the Z dependence of the signal,where a more direct measurement of composition isneeded, the use of analytical signals capable of mea-suring composition (and other properties) directly isrequired.

26.3.4 Aberration-Correctedand Atomic-Scale TEMand STEM Tomography

AC TEM and STEM, in tandem with new advancedreconstruction schemes, have opened up opportunitiesfor atomic-scale ET, which have seen considerable de-velopment over the last 5 years or so. ET has pushedbeyond the long standing 1 nm3 gold standard [26.69,88], well into the atomic regime.

Although early pioneering studies showed the pos-sibility to achieve 3-D atomic-level detail [26.37, 89],the seminal study of Van Aert et al. using discrete con-straints and a regular atomic lattice was the first toachieve 3-D atomic-scale reconstruction of an Ag na-nocrystal [26.38]. More recently, the development ofnew reconstruction schemes (see later) has enabled 3-Dstudy of atomic-scale defects and subtle changes inatomic-scale morphology. These include crystal domain(grain) structure and atomic packing [26.90–95], crys-tallographic defects including dislocations, stackingfaults and vacancies [26.91, 92, 95, 96], atom-by-atomchemical distributions [26.93, 96–98], and atomic-scalestrain fields [26.92], offering exciting insights and op-portunities for materials science (Fig. 26.13).

Similarly, the crystallographic or noncrystallo-graphic structure of decahedral and icosahedral nano-particles has been a perplexing issue for decades, with

competing theories as to how strain is accommodated infivefold twinned geometry e. g., [26.99] and structuralcomplexities hidden in projection images. 3-D stud-ies by atomic-scale ET provide new means to addresslong-standing issues, opening the door, for example, tomapping the 3-D strain state as shown in Fig. 26.14.Here, a systematic lattice expansion is measured alongboth the x and z directions, but the expansion along z islimited to only the outer few atomic layers and showsasymmetry likely due to the decahedron resting on anamorphous carbon support [26.92].

26.3.5 Analytical Electron Tomography

Nano-analytical techniques undoubtedly play a signif-icant role in many (S)TEM investigations, enablingmapping of physical properties, such as local chem-istry. These may be extended to 3-D by utilizingthe signal in ET, providing a tomogram with one ormore additional signal dimensions beyond the spatialdomain (Fig. 26.15). Multidimensional or analyticalelectron tomography (AET) has grown significantly inrecent years, with EELS and energy-dispersive x-rayspectroscopy (EDXS) being the mostly widely im-plemented forms of AET to date. They can provideelement selective imaging for 3-D mapping of compo-sition [26.100–102] and, under suitable circumstances,chemistry (e. g., local valency [26.22]), electronic prop-erties [26.103], and optical properties [26.104]. Underfavorable circumstances, analytical tomograms can beinterrogated quantitatively, for example, to determinelocal elemental concentration [26.100]. Although moreelectron dose intensive than conventional structuralimaging techniques, recent developments in hardwarecoupled with data handling capabilities and advancedreconstruction algorithms have bought these signalmodes into feasibility for ET and generated signifi-cant activity to further develop rich new opportunities.While most AET studies to date have been on beam-resistant specimens, optimized and novel methodolo-gies can and should increasingly enable application ina wider range of contexts. Further techniques includingelectron holographic and crystallographic tomographyin (S)TEM add to a growing scope for multidimensionalAET investigations.

EFTEM and STEM-EELS Tomography(S)TEMs equipped with a post-column (or occasion-ally in-column) electron energy spectrometer offer theopportunity to pursue EELS tomography, recordingcharacteristic losses of the electron beam on interactionwith the specimen in the form of energy-loss images.This may be performed using energy-filtered TEM(EFTEM), where an energy selecting window is placed


D|26.3

2 nmfcc

L12

L1"[100]

[001]

[010]

[100] [010] [001]

a) b)

c)

L12

L10 A1

L12

L10 L12 L12 L12 L10

Fig. 26.13a–c 3-D determination of atomic coordinates, chemical species and grain structure of an FePt nanoparticle.(a) Overview of the 3-D positions of individual atomic species with Fe atoms in red and Pt atoms in blue. (b)Multisliceimages obtained from the experimental 3-D atomic model along the [100], [010], and [001] directions. Scale bar is 2 nm.(c) The nanoparticle consists of two large L12 grains, three small L12 grains, three small L10 grains, and a Pt-rich A1grain. From [26.98]

c)b)a) °0

yz x

yz x

zy x

–5% +5%

Fig. 26.14 (a) 3-D visualization of the tomographic reconstruction of a gold nanodecahedron; the arrows indicate planardefects inside the decahedron. (b,c) 3-D strain analysis. Slices through (b) the ©xx volume and (c) the ©zz volume. Reprintedwith permission from [26.92], published under ACS AuthorChoice License, permissions requests should be directed tothe ACS

PartD|26.3


Voxel

z

yxMultidimensional

analytical electron tomogram

Analytical dimension(s)

Nano-object

3-D space

Fig. 26.15 Principle of AET, in which each voxel in 3-Dspace contains additional signal dimensionality, in theform of, for example, an energy spectrum, or diffractionpattern

over a characteristic energy-loss feature, and an im-age formed using only electrons inside that window; orby STEM-EELS spectrum-imaging, where a spectrumcovering a chosen energy range is recorded pixel-by-pixel.

Depending on the particular energy-loss range, theimages may characterize different properties, requir-ing specific consideration of the projection requirement.The signal from core-loss ionization edges (whose on-set is determined by the characteristic energy requiredto promote an inner-shell electron of a particular type ofatom) can be obtained in EFTEM by using additional(usually two pre-edge) EFTEM images to enable sub-traction of the background under the edge. This thenyields an elemental map, and by acquiring a tilt series ofsuch maps for tomographic reconstruction, an element-sensitive tomogram can be obtained.

EFTEM ET has been a recognized method fora number of years (it was first demonstrated in2001 [26.48, 49]) and has proven to be of value in manycontexts [26.105, 106]. Figure 26.16a, for example,shows a 3-D iron elemental map revealing the morphol-ogy of an iron-based catalyst nanoparticle at the top ofa multiwall carbon nanotube (CNT). Determining the3-D form of each chemical constituent in the iron-filledCNTs is one of the key factors for understanding thegrowth mechanism and potential applications [26.105].

Si (O)

SiSi (Ti,N)

100 nm50 nm

Fe

C

a) b)

y

xz

Fig. 26.16 (a) EFTEM ET elemental mapping of aniron-filled multiwalled carbon nanotube. Reprintedfrom [26.105]. (b) STEM-EELS ET chemical state map-ping of silicon in a W-to-Si contact from a semiconductordevice. Reprinted from [26.22], with permission fromElsevier

However, the size of energy windows required formapping (typically 10�20 eV) limits analysis of finespectral information. With the development of fast andefficient spectrometers enabling acquisition in accept-able times and electron doses, there has been risingdevelopment of ET based on STEM-EELS. Advancesin spectrometers and in the design and wide availabilityof monochromators for the incident electron beam havealso brought improved energy resolution.

The spectrum at each pixel in a STEM-EELS spec-trum image can be analyzed post-facto, enabling mapsto be obtained from any energy-loss channel in a ver-satile manner. Similar to EFTEM, 3-D elemental mapscan be obtained using core-loss ionization edges acrossa tilt series of STEM-EELS spectrum images, and itis also possible to utilize the fine structure at theseedges to map specific phases, and electronic proper-ties and bonding across 3-D space [26.22, 107, 108].Figure 26.16b shows the bonding states of silicon ina semiconductor device in 3-D; the silicon is identi-fied either in its elemental form or as an oxide, or aspart of a metal silicide or silicon nitride [26.22]. In fa-vorable cases, it may be possible to directly map thevalency of certain elements. Materials with empty 3dand 4f shells are amenable to this because of a pro-nounced and intense EELS fine structure that can beused as a fingerprint to determine the valency. Theexample in Fig. 26.17a shows a ceria nanoparticle inwhich the particle surface is shown to be predominantly


D|26.3

Ce3+

Ce4+FeOFe3O43 nm 30 nm

a) b)

Fig. 26.17a,b STEM-EELS ET valence state mapping re-vealing different surface states of (a) a ceria nanoparti-cle. Reprinted with permission from [26.103]. Copyright2014 The American Chemical Society. (b) Changes in Fevalency in a FeO/Fe3O4 nanocube. Reprinted with per-mission from [26.107]. Copyright 2016 The AmericanChemical Society

Ce3C in character and the core Ce4C [26.103], andFig. 26.17b shows the iron distribution in an iron oxidecore-shell particle, distinguishing between the Fe(II)and mixed Fe(II) and Fe(III) contributions of the twooxides [26.107].

STEM-EELS and EFTEM in the low-loss regionmay also encode valuable electronic information suchas plasmonic behavior. The bulk, or volume, plasmonenergy depends on the local electron density, and plas-mon EFTEM images, utilizing narrow energy windowsca. 1�2 eV in width, may be used as input for tomo-graphic reconstructions. Figure 26.18a shows an earlyexample in which islands of silicon can be distinguishedfrom the silicon oxide matrix by the shift in the plasmonenergy [26.109]. More recently, the surface plasmonmodes of a Ag nanoparticle were reconstructed in3-D (Fig. 26.18b) [26.104] and a similar reconstruc-

α β γ δ ε10 nm

λ = 850 nm

λ = 650 nm

100 nm

a) b)

100 nm

c)

Fig. 26.18 (a) EFTEM tomography plasmon mapping, revealing the morphology of silicon nanoparticles embedded insilicon oxide. Reprinted from [26.109], with the permission of AIP Publishing. (b) 3-D rendering of surface plasmonmodes of a silver nanocube, reconstructed using STEM-EELS tomography [26.104]. (c) Cathodoluminescence tomogra-phy maps of a gold-polystyrene nanocrescent [26.110]

tion undertaken using cathodoluminescence to study themodes of a gold nanosphere [26.110].

Mapping elemental concentration directly is ideallysuited to meeting the projection requirement. How-ever, several factors can complicate STEM-EELS orEFTEM, restricting the range of amenable specimensor requiring careful protocols to mitigate their potentialimpact. At specimen thicknesses above a characteris-tic inelastic mean free path for the material (typically� 100 nm [26.111]), the influence of multiple inelas-tic and plural scattering becomes significant, and theapparent elemental signal may actually begin to fall.Thus the core-loss signal from thick specimens willno longer satisfy the projection requirement. Methodsto remove the effects of plural scattering, while avail-able, are not straightforward and are complicated bychanges in thickness with tilt, and have not been signif-icantly developed for ET to date. Diffraction contrast incrystalline specimens can similarly complicate STEM-EELS, and further complications may arise in aniso-tropic materials whose response will change markedlywith tilt.

The introduction of new spectrometer technologyenabling near simultaneous acquisition of both low-loss and core-loss spectra (dual EELS) [26.112] isa significant development, enabling the development ofSTEM-EELS ET in which many of the traditional chal-lenges can be overcome [26.108, 113]. It also opens thedoor to enhanced analyses and new possibilities, includ-ing absolute (as opposed to relative) quantification ofelemental concentration [26.108].

EDXS TomographyET utilizing energy dispersive x-ray spectroscopy(EDXS) has recently advanced significantly with the

PartD|26.3


availability of x-ray detectors for (S)TEM with signifi-cantly higher solid angles, detection efficiency, and pro-cessing capabilities. Early attempts at EDXS ET [26.50,114] were limited by the poor efficiency of conven-tional detectors. These subtend solid angles of only0:1�0:3 sr and are positioned on one side of the speci-men, which creates considerable problems from holdershadowing when tilting in ET. Modern silicon drift de-tectors with much higher solid angles (� 1 sr), fasterprocessing capabilities, and multiple detector chips po-sitioned symmetrically around the optic axis (such thatthere are always chips in the sight of the specimen) havestimulated new scope, and EDXS ET studies are rapidlygrowing in number, fidelity, and value (Fig. 26.19).

In essence, EDXS provides a signal well suited totomographic reconstruction, providing a direct measureof elemental concentration. With characteristic peaksthat lie on a relatively low background and few of theplural scattering problems that can complicate EELSof thick specimens, EDXS can in many regards pro-vide a much simpler means of compositional mappingin both 2-D and 3-D. Nevertheless, 3-D mapping withEDXS ET is not without its challenges, which maycompromise the projection requirement and require cor-

Au 7%Ag 93%

Au 40%Ag 60%

Zn KαC KαAu LαPt LαAg Lα

10 nm 10 nm

y

xz

a)

b)

300 nm

Fig. 26.19 (a) STEM-EDXS elemen-tal tomograms of Au-Ag bimetallicnanorings, showing (left) irregular Ausurface segregation and (right) moreuniform Ag segregation. Reprintedwith permission from [26.100] pub-lished under ACS AuthorChoiceLicense, permissions requests shouldbe directed to the ACS. (b) STEM-EDXS tomograms revealing 3-Delemental distributions in an organ-ic/inorganic core/multishell nanowire.Reproduced from [26.101], publishedunder CC-BY 4.0 license

rection for high-fidelity ET and especially for quantita-tive analysis in 3-D. Even with new detectors providingvastly improved collection efficiency and positionedsymmetrically around the specimen, detector shadow-ing can be a major consideration. Recent efforts havesought to characterize experimental configurations andparameterize the most important determining factors,including holder and detector geometry as a functionof tilt angle, to enable corrections to be applied fordetector shadowing [26.115, 116]. A second potentialchallenge that may require correction is x-ray absorp-tion. Absorption correction procedures for ET havebeen proposed using the Cliff–Lorimer [26.115] and�-factor [26.117] methods, which can also incorporateshadowing. Another approach has been to progressivelyrefine absorption correction in an iterative reconstruc-tion process [26.101]. In thick specimens, additionaleffects such as x-ray fluorescence, may become sig-nificant. Samples fabricated into needles by focusedion beam milling overcome holder shadowing [26.102,118]. By reducing the volume of material on the x-raypath to detectors relative to a slab geometry they alsoreduce, though do not eliminate, absorption or fluores-cence effects.


D|26.3

In many modern instruments, it is possible to col-lect the EDXS and EELS signals simultaneously. Thegreat advantage of simultaneous acquisition is that theEELS signal is especially sensitive to light elements,and the EDXS to heavier elements, so the combinationcan provide a more complete 3-D chemical picture. Fig-ure 26.20 shows 3-D elemental maps of an Yb-dopedAl-5wt% Si alloy using simultaneous (right) STEM-EELS and (left) STEM-EDXS tomography [26.102].Though such studies have been limited to date, there isconsiderable scope for advancedmultidimensionalmul-timodal analysis.

In both STEM-EELS and EDXS ET, signal acqui-sition within a reasonable electron dose and/or timeis still not easy. Methods that enable robust recon-struction from fewer images may be the only way toopen up these imaging modes to less beam resistantspecimens.

As well as the spectroscopic signals, the past 5years or so have seen the inception, or progression, ofa number of other advanced signal modes for ET. Whileremaining primarily the practice of a select numberof groups, holographic and diffraction techniques havenow been confirmed as valuable ET signal modes (theyare reviewed in [26.59, 119] and [26.120], respectively).

0

0.4 0.6 0.8 1.0 1.20

50

100

150

Counts

X-ray energy (keV)

2 4 6 80

200

400

600

800

1000

1200

1400Counts

X-ray energy (keV)

1400 1600 1800 20000

200

400

600

800

Counts

Energy loss (eV)

EDXS EELS

20 nm 20 nm

Yb M5

Al K

Yb M4Yb M3

Si K

Yb Mα Al K α

Si KαYb Lα

(Co Kα)(Fe Kα)

O Kα

(Cu Lα)Ga Lα

Yb Mζ

Fig. 26.20 3-D elemental maps of Yb (green), Si (red), and Al (blue) from an Al-5wt% Si alloy (6100 ppm Yb), obtainedfrom simultaneous (right) STEM-EELS and (left) STEM-EDXS tomography. Local spectra are shown from the Yb-richprecipitates. Reproduced from [26.102] with permission of the Royal Society of Chemistry

Indeed, automated acquisition has been developed forboth techniques.

Crystallographic ETDiffraction contrast that arises in TEM can, under favor-able circumstances, be profitably utilized in dark-fieldTEM ET, using the objective aperture in the backfocal plane of the objective lens to select a partic-ular Bragg reflection to contribute to the image. Byensuring that the diffraction conditions remain approx-imately constant at each tilt angle, it has been shownthat the projection requirement can be satisfied suffi-ciently well. Although the acquisition is challenging,this technique can be highly sensitive to small changesin crystalline orientation and has been used for imagingof defects such as precipitates [26.121] and dislocations(using the weak-beam dark-field technique) [26.41] orburied structures such as quantum dots [26.122].

For polycrystalline samples, the 3-D distributionof grains within the specimen volume may be recon-structed by acquiring a large number of dark field (DF)images varying sample tilt and scattering angle. By ac-quiring an ensemble of ca. 100 k DF images the 3-Dgrain distribution in a polycrystalline Al specimen wasreconstructed [26.123].

PartD|26.3


An alternative approach involves raster-scanninga near-parallel beam and recording a full 2-D diffrac-tion pattern at each point in the raster—a technique thatis sometimes known as scanning electron diffraction(SED). By repeating such a scan at a series of specimentilt angles crystallographic information from a volumeof material may be recovered (Eggeman et al. [26.124]and more recently Meng and Zuo [26.125]). Such tiltseries data can be extremely information rich, allow-ing for versatile 3-D crystallographic analysis. Thediffraction patterns can be analyzed computationallypost facto, and virtual dark-field or component imagesmay be formed (the latter using multivariate statisti-cal analysis (MSA) methods), which can then be usedto reconstruct, in 3-D, both real and reciprocal spaces,as illustrated in Fig. 26.21b. By interrogating subvol-umes to retrieve local 3-D crystallography, it is possibleto determine, for example, the orientation relationshipsbetween grains or phases and across interfaces. Thescanned diffraction ET data sets also provide a promis-ing means for 3-D mapping of crystallographic strain atthe nanoscale [26.126].

26.3.6 Holographic ET

As discussed elsewhere in this book, electron hologra-phy (both in-line and off-axis holography) is exquisitelysensitive to changes in the phase of the wave broughtabout by variations in the sample’s electrostatic or mag-netic potential. Combining holography and tomographyshould, then, offer a route to exploring that electro-magnetic potential in 3-D. In general, we can describe

100 nm

b*

a*

Fig. 26.21 3-D crystallographic reconstruction of an Ni-based superalloy from scanning precession electrondiffraction tomography showing a faceted metal carbide(blue), ˜-phase (green), and surrounding matrix (orange).At each point in real space (left), reciprocal-space informa-tion is available, as shown (right) for the lath-like ˜-phase.The colored overlay of spots is the auto-correlation of thezero-order Laue zone reflections with reciprocal lattice ba-sis vectors marked [26.124]

the reconstructed phase image as a projection of a 3-Dpotential through the equation

' .x; y/ D CE

ZV .x; y; z/ dz� e

¯“

B .x; y/ dS ;

(26.6)

where the first integral is made parallel to the beam di-rection, z, V is the crystal potential, B is the magneticinduction and related to the magnetic potential A byB D rotA, and S is normal to the area mapped out bythe trajectories of the electrons going from source todetector; CE is a wavelength-dependent constant.

In the absence of magnetic fields and diffractioncontrast, and in the absence of stray fields outside thespecimen, the phase change can be related directly tothe mean inner potential V0 of the specimen

'.x; y/ D CEV0 .x; y/ t.x; y/ ; (26.7)

where t is the specimen thickness.This has been used to investigate changes in the

electrostatic potential in 3-D in semiconductor devicescontaining a p-n junction where variations in the deple-tion region near the junction were revealed [26.53].

Mapping magnetic fields (or the magnetic induc-tion) in 3-D is possible in principle, but the vectorialnature of the magnetic induction B makes this chal-lenging, as three components of B must be found ateach reconstructed voxel. The theoretical basis for un-dertaking such vector tomography has been extensivelyset out [26.127–130]. The reconstructed phase fromthe hologram will be sensitive to both electrostatic andmagnetic components and, in order to separate the elec-trostatic and magnetic phase shifts, a tilt series overa full 360ı tilt range is needed. (Alternatively, twotilt series may be acquired, one before and one afterreversing the direction of magnetization in the speci-men, e. g., using the TEM objective lens or flipping thesample up-side down.) If two tilt series over 360ı areacquired with mutually perpendicular tilt axes, two in-dependent components of the magnetic induction maybe reconstructed. Application of the no monopole con-dition rB D 0 enables the third component of B to befound from the other two.

The first successful reconstruction of the full 3-Dmagnetic induction and vector potential was achievedby Phatak et al. [26.131], using the transport of in-tensity (TIE) approach to study a magnetic permal-loy plate (Fig. 26.22). More recently, Wolf andcoworkers [26.132, 133] used a simplified experimentalscheme, with off-axis electron holography, to achievea quantitative reconstruction of one component of B forneedle samples, with the component parallel to the nee-dle (and tilt) axis, Fig. 26.23.


D|26.3

B = ∆ × A Incomingelectron wave

ψ = a(r)eik∙r

Exitingelectron waveψ = a(r)eik∙r+φ~

0.5 μm

zy

x

z,ω

xy

b)a)

g)f)

c) d) e)

Fig. 26.22a–g ET reconstruction ofthe magnetic vector potential outsidea multidomain permalloy island.(a) The magnetic vector potentialA (red arrows) and the magneticinduction (blue). (b) Acquisitionscheme to achieve data for thevector field ET. (c–e) Fresnelimages at a single tilt angle (under,in, and over focus, respectively).The location of a central vortex ismarked with an arrow. (f,g) 3-Dvisualization of the reconstructedmagnetic induction and vectorpotential, respectively. Reprinted withpermission from [26.131]. Copyright2010 by The American PhysicalSociety

26.3.7 Time-Resolved ET

Compared to x-ray tomography, there has been rela-tively little use made of time-resolved ET. One keyleap forward was made in the ultrafast electron com-munity in a paper by Kwon and Zewail [26.54], inwhich a stroboscopic pump-probe technique was ableto capture the 3-D dynamic changes in the vibration ofa nanowire. For nonstroboscopic measurements, therehas been some progress in acquiring tilt series overever smaller time periods, allowing the possibility ofdynamic tomography, especially in the BF-TEM mode,where use can be made of the remarkable sensitivity and

efficiency of new direct electron detectors. One demon-stration of this was done by Migunov et al. [26.134],who, via continuous tilting, recorded a rapid low-dosetilt series in just a few seconds.

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

PartD|26.4


a) b)

c) d)

e)

f)

Co2FeGa nanowireCo nanowire

100 nm 5 23V

11 22V

0 1.5T

+1.0 –1.0TClosuredomain

Magnetic dead layer

22

21

20

23A (V)

19

1.41.21.0

1.6B (T)

nm300250200150100500 350

0.8

60 nm

60 nm

Fig. 26.23a–f 3-D electron holographic reconstructions of cobalt and Co2FeGa magnetic nanowires (NWs). 3-D volumerendering of electric potential (in volt) (a,b) and axial (predominant) B-field component (in tesla) (c,d) inside the NWs.(e) Axial B-field component inside the Co NW obtained from micromagnetic simulation. The arrow plots visualize theout-of-plane components showing the twist of magnetic induction. (f) Line scans in the axial direction through the centreof the NW from the tip to the back. (a,c,e) reprinted with permission from [26.132] published under ACS AuthorChoiceLicense, permissions requests should be directed to the ACS. (b,d,f) reprinted with permission from [26.133]. Copyright2016 The American Chemical Society

26.4 Tilt Series Alignment

Automated feature tracking during acquisition, com-puter control of goniometers, and improvements instage design have greatly facilitated the successful ac-quisition of ET tilt series (ensuring that the specimenremains close to the centre of the field of view in eachimage). Nonetheless, post-acquisition alignment of theprojections to a common tilt axis is almost always re-quired and should, ideally, be with sub-pixel accuracy.

Where specimens have distinctive features, com-mon in the physical sciences, the alignment is usuallycarried out using cross-correlation [26.2, 62] and [26.5,Chap. 6]. An alternative is to place high-contrast mark-ers on the specimen or support film, usually goldnanoparticles, and to track these in each image to deter-mine the required shifts [26.5, Chap. 5]. This approachis more common in biological applications, where thespecimens often show lower contrast and, where fea-tures are more sporadically distributed throughout thereconstruction volume, may suffer less from obstructionby the markers. A number of software packages alsoprovide facilities for manual adjustments to be made.

Alignment by cross-correlation is illustrated in Fig.26.24. The cross-correlation determines the match be-tween two images across all lateral and vertical dis-

placements and provides an output image (Fig. 26.24c)whose intensity peak indicates the shift required to bringthe features from the two images into coincidence. Of-ten the sharpness of the cross-correlation peak, and,therefore, the accuracy of the determined shifts, can beimproved by applying one or more filtering processesto emphasize or reduce the influence of certain featuresin the images. In Fig. 26.24d–f, for example, a muchsharper cross-correlation peak has been obtained by useof a Sobel filter to highlight edges. Since the projectedview of the specimen is similar but not identical at suc-cessive tilts, the cross-correlation match will never beexact, and this may be particularly so for slab-like speci-mens and/or where additional objects enter into the fieldof view. Foreshortening of features in projection at suc-cessively higher tilts can be significant in extended slab-like specimens (Fig. 26.25). This can be alleviated byapplying a linear stretch of 1= cos � to the projections,perpendicular to the tilt axis [26.135], restoring the spa-tial correspondence between successive projections.

An advantage of marker-based alignment is that, aswell as determining the required shifts, it also enablesdetermination of the position and angle of the tilt axis,whereas with standard cross-correlation approaches this

Electron Tomography in Materials Science 26.4 Tilt Series Alignment 1301Part

D|26.4

a) c)b)

d) f)e)

100 nm

Fig. 26.24a–f Determining the relative shift between two tilt series images by cross-correlation. (a,b) Successive tiltseries images and (c) the corresponding cross-correlation indicating their relative shift. (d,e) Sobel filtering of the imagesto yield a sharper cross-correlation peak (f)

Field of view

Projection plane

θ

l

l cos θ

Fig. 26.25 The spatial relationshipbetween sample features at successivetilt angles, leading to foreshorteningin projection. The correspondencebetween these projections can berestored by a stretch of 1= cos � , asindicated by the red arrows

has to be performed by other means. Often, projectinga tilt series along the z-direction by summation, or othermeans such as maximum intensity projection, can pro-vide an initial coarse estimate of the angle and lateralposition of the tilt axis. Features of an object locatedat some distance from a fixed tilt axis of rotation ap-pear, 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 exempli-fied in Fig. 26.26. Features lying directly on the tilt axisappear stationary in location through a tilt series, andwhere such trackless features are seen, and from wheretracks appear to emanate, reveals the lateral position ofthe tilt axis.

Alignment of the tilt axis is also critical for high-fidelity reconstructions. Figure 26.27 illustrates howthis can be achieved through minimizing the tarcing

PartD|26.4


15°

a) b)Fig. 26.26a,b Tilt axis identificationfrom z-direction projection. (a) Asingle HAADF-STEM image, at 0ıtilt, of gold nanoparticles on a carbonsupport. (b) Maximum intensityz-projection of the full tilt series andits Fourier transform (inset), wherethe particle tracks reveal the tilt axisangle, 15ı from the vertical, andlocation

Central slice

Bottom slice

Top slice

b)b) Tilt axis

c) Axis angle misalignment

d) Axis shiftmisalignment

d)c)

(((

( ( ( ( ( (

( ( (

( ( (

a)

Fig. 26.27a–d Tilt axis alignment by minimization of arcing. (a)HAADF-STEM image of gold dog-bone nanoparticles,showing the location of a central and top/bottom slices chosen for preliminary reconstruction and tilt axis adjustment.(b) WBP reconstructions of each slice when the tilt axis is correctly positioned. (c) Incorrect tilt axis angle manifestsas arcing artefacts in opposite directions in each of the top/bottom slices (as indicated schematically in the lower right-hand corner). (d) Incorrect lateral position of the tilt axis results in arcing artefacts in a common direction in all threereconstructed slices

of features if the axis is misaligned. It is important toemphasize that accurate alignment is fundamental forhigh-fidelity and high-resolution ET reconstructions tobe obtained. Indeed, critical to extending the achievable

resolution and fidelity in recent years has been the de-velopment of specialized alignment procedures such ascentre-of-mass-based approaches and refinement dur-ing iterative reconstruction [26.136–138]. These have

Electron Tomography in Materials Science 26.5 ET Reconstruction 1303Part

D|26.5

been especially pertinent in atomic-scale ET and arealso of importance to enable a growing trend for ETreconstruction from very few tilt series images. Thereis also ready opportunity for alignment procedures and

software implementations developed in the biologicalsciences, such as feature or local patch tracking [26.5,Chap. 6], [26.139] to be applied in physical science con-texts [26.140, 141].

26.5 ET Reconstruction

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

1. Direct transform methods, including backprojectionand Fourier techniques

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

Comprehensive mathematical description of themethods discussed can be found in [26.5–9], [26.10,Chap. 2]. A notable concise summary of the manydifferent forms of ART and SIRT algorithms is givenin [26.142].

26.5.1 Backprojection

The reconstruction method favored by the ET commu-nity has for many years been the weighted backprojec-tion (WBP) algorithm, owing in large part to its speedof execution and because the algorithm is well under-stood. In the most basic description [26.62], [26.78,Chap. 8], backprojection consists of smearing each pro-jection from a tilt series back into space at the angleat which it was originally formed. By backprojectinga sufficient number of projections, the summation of thebackprojected rays in the space will generate the orig-inal object; such direct backprojection was illustratedschematically in Fig. 26.2.

However, ET reconstructions from simple back-projection appear blurred because the radial samplingregime of ET (Fig. 26.9) leads to relative undersam-pling of higher spatial frequencies (Fig. 26.28b). Thiscan be corrected using a ramp-like weighting filter, usu-ally applied to the projections in Fourier space. Theresult is a WBP [26.143], [26.5, Chap. 8], as shown inFig. 26.28c. While this filtering process has the benefitof enhancing edges, it can complicate any quantitativeanalysis of the voxel intensities in the tomogram.

26.5.2 Direct Fourier Inversion

Fourier-based reconstruction methods exploit theFourier slice theorem outlined in Sect. 26.1.2, Fourier-

Based Methods. Essentially, for an N-dimensional re-construction, Fourier reconstruction entails applicationof an (N � 1)-dimensional discrete Fourier transformacross the spatial dimension of the projections to obtainradial Fourier data. An N-dimensional inverse Fouriertransform is then applied to this data set to recover thefunction in real space. However, as was indicated inFig. 26.6 and is shown explicitly in Fig. 26.29, the datain the Fourier domain of the function lies on Cartesiancoordinates, whereas the radial Fourier data is on a po-lar grid. To convert the data between the two coordinatesystems requires some form of interpolation or griddingprocess. This step is challenging and can result in poorquality reconstructions if simple interpolation (such asbilinear) is used. As such, direct Fourier inversionmeth-ods have generally been disregarded in ET.

Nonetheless, several Fourier-based reconstructionmethods using sophisticated nonuniform Fourier trans-form, or gridding, procedures have been proposedrecently in the biological ET and single particle mi-croscopy context, with potential performance enhance-ments; reviewed by, for example, Penczek [26.9].A more recent approach, in both biological [26.144,145] and physical [26.91, 94, 104, 146–149] sciencesET has been to combine sophisticated Fourier-based op-erators with iterative reconstruction.

26.5.3 Algebraic Iterative Reconstruction

In a qualitative description, algebraic iterative recon-struction (AIR) techniques in ET operate by constrain-ing the reconstruction to match the original projections,with the match being improved at successive iterations(Fig. 26.30). A difference reconstruction is obtained viaa comparison of projections of the reconstruction withthe original projections, either by division in multiplica-tive techniques or subtraction in additive techniques.The current reconstruction is then updated via multipli-cation or addition of the difference, respectively. Thiskind of iterative refinement by projection and reprojec-tion can also be described mathematically in terms ofprojection onto convex sets [26.150].

Formally, AIR is based on the discretization ofthe projection process into a finite number of basisfunctions (n in total), as illustrated in Fig. 26.31. The

PartD|26.5


a) Phantom b) Backprojection c) Weighted backprojection

Fourier transform

Uneven recovery

Oversampled

Undersampled

Pseudo-even recovery

Fig. 26.28a–c The application of a weighting filter in backprojection reduces the blurring effects brought about throughrelative under-sampling of high spatial frequencies

projection system is represented by the matrix ˆ, andthe vector x contains the function in n discrete points inspace. The vector b contains the ray sums, correspond-ing to a discretized sinogram, with m entries in total.The tomography reconstruction process can then be for-mulated as a system of linear equations

bi DnX

jD1

�i;jxj fori D 1; : : : ;m ; (26.8)

where each �i;j is often calculated as the fraction of thej-th basis function intersected by the i-th projection ray,implying 0 � �i;j � 1. Equation (26.8) can be abbrevi-ated as

b D ˆx : (26.9)

These equations represent an inverse problem, wherethe task is to estimate x given the data b and the projec-tion matrix ˆ. In ET, the limited number of tilt seriesprojections means that there are far fewer equations

than unknowns (i. e., m n), and the system of equa-tions is underdetermined, implying there is an infinitenumber of solutions consistent with the projection data.This is compounded by the ill-posedness arising fromdata imperfections, such as noise, projection misalign-ment, or diffraction contrast. AIR techniques such asART [26.151] and SIRT [26.152] were proposed in thecontext of ET in the 1970s. With advances in computa-tional power and efficient algorithmic implementationse. g., [26.153], they have been the mostly widely uti-lized reconstruction methods in materials science. Dueprimarily to greater stability when the projections arenoisy, the SIRT algorithm has usually been preferred toART in ET and, in the physical sciences, is generallyseen as the established standard.

26.5.4 Algebraic Reconstruction Technique(ART)

The classic AIR technique to solve (26.9) is that ofKaczmarz [26.154]. In the literature, this is often re-ferred to simply as ART, but it is important to realize


D|26.5

a) b)

c)

Convolution kernel

kz kz

kx kx

kz

kx

Fig. 26.29(a) Radial and(b) Cartesian datapoints; and (c) aninterpolation(gridding)process toconvert betweenthe two

that ART also refers to a class of AIR techniques.Kaczmarz’s method can be expressed with the additiveupdate scheme

OxkC1 D Oxk C �k bi �

D�i; Oxk

E

k�ik2`2�i ; (26.10)

where each �i represents a row of ˆ in (26.9), andhence

D�i; Oxk

E

denotes the standard inner product of the vectors®i andOxk. The index i addressed at the k-th iteration is given byi D .kmodm/C 1; � is the relaxation parameter, whichinfluences the sensitivity of the update to noise and maybe fixed (�) or vary at each iteration (�k).

ART was introduced to the ET community by Gor-don et al. [26.151], who also presented a multiplicativeform (sometimes referred to as MART). An update

scheme can be written as

xkC1j D

0@ biD

�i; OxkE1A

�k 'i;j

xkj ; (26.11)

where 0��i;j�1 and again i D .kmodm/C 1.Another form of ART that has been extensively

discussed in the ET literature is block-ART, first in-troduced by Eggermont et al. [26.155], which for Oblocks of P equations (where O P D m) may be writtenas [26.10, Chap. 2], [26.5, Chap. 7]

OxkC1 D Oxk C �k .okC1/PXiDok PC1

bi �D�i; Oxk

E

k�ik2`2�i ; (26.12)

where ok (0�ok�O) is the index of the block to be usedin the k-th iterative step.

As outlined by Kuba and Herman [26.10, Chap. 2],the essential difference between block-ART and the

PartD|26.5


Project(electron microscope)

Iterative loop

Differencecalculation

Reprojections Original projections

Reprojections

Reproject

Difference projectionsOriginal projections

Use asreference

Backproject Backproject

Initial reconstruction New currentreconstruction

Differencereconstruction

Currentreconstruction

Differencereconstruction

Final reconstruction

Updatereconstruction

k = kmax

k = k +1

Fig. 26.30 Principle of iterative tomographic reconstruction. (Shown specifically for a SIRT-type algorithm; based onthe diagram of Weyland and coworkers [26.62])

more conventional ART methods is that in the for-mer the update proceeds by taking into account groups(blocks) of measurements that come from a particu-lar projection, compared to the latter dealing with onlyone measurement at a time (i. e., one ray integral). If

P D m (and, hence, O D 1), then the method is said tobe fully simultaneous [26.156, p. 100] and is closelyrelated to SIRT-type methods. An intermediate case iswhen the blocks are formed by all the equations asso-ciated with a single projection, an example of which


D|26.5

Grid of basisfuncions

x1 x4x3x2 x5

xj

bi +1

bi

Fig. 26.31 Discrete representationof a tomographic experiment. Thefunction f is represented by n discretebasis functions, which can be writtenas a vector x. Each ray sum is denotedby bi, with the set of ray sums formingthe vector b. Linking b and x is theprojection matrix ˆ, where eachelement �i;j describes the contributionof the j-th basis function xj to the i-thprojection ray bi

is the simultaneous algebraic reconstruction technique(SART) [26.157].

26.5.5 Simultaneous IterativeReconstruction Technique (SIRT)

The other major class of AIR algorithms in ET areSIRT-type methods. As the name suggests, informationfrom all the equations (projections) is used at the sametime for the update process. This accounts for SIRT of-ten being less sensitive to noise than ART.

SIRT methods can be written in the general additiveform

OxkC1 D Oxk C �k ‡ˆ��.b�ˆ Oxk/ ; (26.13)

where ˆ� is the (conjugate) transpose of ˆ. The ma-trices � and ‡ are symmetric and positive definite,and for most implementations ‡ is the identity trans-form ([26.142] for examples of functional roles playedby these matrices). The most common variant is theLandweber method [26.158]

OxkC1 D Oxk C �k ˆ�.b�ˆ Oxk/ (26.14)

which corresponds to setting� D ‡ , the identity trans-form in (26.13). It is well known that in overdeterminedcases (i. e., m> n) for which there is a unique solution,SIRT effectively solves a (weighted) least-squares prob-lem of the form [26.159]

Ox D argminOx

kˆ Ox� bk2`2 : (26.15)

However, in underdetermined problems (typical of ET)an infinite number of solutions Ox may exist that yield

minimal discrepancy in (26.15) (or other AIR algo-rithms). As will be discussed later, in such cases, itcan be distinctly advantageous to apply additional con-straints during the iterative reconstruction that help toselect from the possible solutions.

Further, in highly ill-posed scenarios, the standardAIR algorithms can exhibit marked semi-convergence,whereby initial iterations tend towards better ap-proximations of the solution, but at some pointmay start to deteriorate to a poorer approximation(Fig. 26.32) [26.142, 160] and references therein]. Thiscan be particularly problematic in ET, where semi-convergent type behaviors may occur when there isa high noise level in the projections or other significantinconsistencies such as projection misalignment, whichbecome exacerbated at large iteration numbers. Impor-tant aspects that remain to be adequately addressed inthis regard are optimal choice of the variables in the ba-sic AIR algorithms, namely the relaxation parameter �and the total number of iterations kmax, both of whichare important parameters influencing the outcome ofthe algorithm. Determination of the optimal number ofiterations in ET often requires reconstruction for differ-ent iteration numbers and some form of qualitative orquantitative (e. g., [26.34]) assessment. Some ET soft-ware packages do not even allow � to be altered. Theseparameters have been discussed in the past, but robustautomated (i. e., nonempirical) methods for choosing orintelligently varying them (e. g., [26.142, 159, 160]) areyet to find marked endorsement in ET.

Slight variants of the conventional ET reconstruc-tion algorithms include dual-axis SIRT [26.102, 161],WSIRT [26.162], and DIRECTT [26.163, 164]. Dualaxis SIRT [26.161] has been less popular, possibly due

PartD|26.5


x2ˆ

x1ˆ

Reconstruction error

Number of iterations

x5ˆ

x10ˆ

x40ˆx300ˆ

xkoptˆ

x

Fig. 26.32 Principle of semi-convergence in tomographicreconstruction: the initial iteratestend to better approximation of theexact solution, but above a certainnumber of iterations they begin todeteriorate

to the inherent difficulties in aligning dual axis tiltseries and the added computational demands but hasmore recently been advocated as yielding potentiallyvaluable resolution enhancement compared to single-axis ET, even for structures not affected by missingwedge artefacts [26.102]. WSIRT, proposed by Wolfet al. [26.162], combines WBP and SIRT, showing im-proved convergence, resolution, and reconstruction er-ror compared to SIRT alone, including a reduced pointspread in the missing wedge direction. The DIRECTTalgorithm of Lange et al. [26.163, 164] resembles SIRT,but at each iteration only a selected portion of vox-els in the reconstruction is updated, based on eithertheir gray level error or local contrast. This favors high-density/contrast features, and the gradual introductionof voxel updates acts as a regularizing mechanism.

26.5.6 Advanced Reconstruction in ET

Compressed Sensing Electron TomographyThe relative paucity of data in ET experiments meansthat to achieve higher-fidelity reconstructions requiresadvanced methods that make best use of that data dur-ing tomographic reconstruction. The highly underde-termined and ill-posed nature of the ET reconstructionprocess implies that seeking data fidelity alone will beinsufficient. In this case, it is well-known, from the field

of inverse problems, that to improve the fidelity or qual-ity of a tomographic reconstruction, some form of priorknowledge constraints (often called regularization) canbe introduced during the reconstruction process. Theregularization selects out of the possible solutions tothe underdetermined system of equations those whichadditionally satisfy the prior knowledge characteristics,and therefore, in principle, should reduce the number ofprojections required for reconstruction.

In general, as the level of undersampling increases,so must the strength or efficacy of the prior knowledgeconstraints, if reconstruction fidelity is to be main-tained. Caution should be noted in this regard though,as the fidelity of the outcome depends on validity ofany prior knowledge constraints imposed. Stronger con-straints can be introduced to bias the results towardsa particular outcome, but this outcome will only be ofhigh fidelity if the constraints are valid. The ideal sce-nario is one in which the prior knowledge constraintsare relatively liberal but effective during the optimiza-tion process and accurately describe the object. In somecases though, it may be necessary to sacrifice some de-gree of reconstruction fidelity to obtain a reconstructionthat possesses other desirable characteristics. For ex-ample, a reconstruction that has been biased so thateach of its constituent objects possess homogeneousdensity and sharp boundaries—whether this is true or


D|26.5

an approximation of the object—may make it easier toidentify and analyze those objects.

With advances in computational power and math-ematical methods there have been considerable recentdevelopments in bringing reconstruction methods in-corporating prior-knowledge constraints to ET. Onemethod of signal recovery from undersampled data thathas seen huge growth in interest and application re-cently is compressed sensing (CS, also referred to ascompressive sensing or compressive sampling) [26.165,166]. By exploiting the sparsity implicit in many sig-nals, CS methods are able to recover signals withremarkably high fidelity from far fewer measurementsthan traditionally would have been necessary. CS ormore generally sparse regularization and related ap-proaches have now gained significant attention in thecontext of ET and have provided high-fidelity tomo-graphic reconstructions even from very few projec-tions [26.146, 147].

The application of recovery methods exploitingsparsity is growing rapidly, including in areas such asx-ray computed tomography [26.167], magnetic reso-nance imaging (MRI) [26.168, 169] and single particlemicroscopy [26.170, 171]. CS harnesses principles oftransform coding and sparse approximation that arewell established from their use in image compressionalgorithms. For example, for the ubiquitous JPEG andJPEG-2000 image compression standards, sparse rep-resentation is provided by the discrete cosine transform(DCT) and the discrete wavelet transform (DWT), re-spectively [26.172].

Formally, the representation of a signal x (such asan ET reconstruction) in a basis ‰ is said to be sparseif there are few (s in total) nonzero coefficients in thatrepresentation, i. e., s n, where n is the full dimen-sion of the signal in its native domain. In this case,only s coefficients in the basis contain all the informa-tion about x. If x can be well approximated by s nnonzero coefficients, x is said to be compressible in ‰;here, there may be many small negligible coefficients,which can be set to zero, and only s significant coeffi-cients. A compressible representation of x in the basiscaptures only the most important information about x ins coefficients. A wide variety of transforms are availablefor this task, offering scope for sparse representationapproaches to be wide reaching. A simple, illustrative,example of sparse representation is shown in Fig. 26.33.

Consider the approach first for image compression.An image is first fully sampled and then transformedinto a chosen domain (e. g., a wavelet domain). If thetransform has been chosen correctly, the number of sig-nificant transform coefficients will be relatively smallwith many less important ones being discarded. Thus,the amount of stored information representing the im-

age is reduced or compressed. However, such datareduction, if carried out correctly, should not lead to anysignificant loss of fidelity in the recovered image.

Within the CS framework, however, we keep inmind the possibility of using transform sparsity andcompressibility during the initial acquisition, with theaim to record a relatively small number of samples butthat are sufficient to capture the important informationin the signal. In other words, we aim to record the signaldirectly in compressed form.

Unlike image compression methods, where thesparse coefficients are known, with a CS approach theseneed to be recovered by searching for the sparsest sig-nal in the transform domain that is consistent with themeasured data. This can be performed very effectivelyusing the `1-norm, defined as the sum of the absolutevalues. A popular formulation is

Ox� D argminOx

�kˆOx� bk2`2 C� k‰ Oxk`1

�(26.16)

where � is a parameter that weights the relative impor-tance of the transform domain sparsity versus the datafidelity in the reconstruction. As further illustrated inFig. 26.33, CS recovery seeks the sparsest signal in thetransform domain that is also consistent with the mea-sured data.

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

A strong focus of ET is in the 3-D reconstruc-tion of nanoscale objects, which are often restrictedin one, two, or even three dimensions. As such, many(S)TEM images and ET reconstructions may be con-sidered sparse in the image domain itself, and thus thesparsifying transform ‰ is simply the identity trans-form. The finite and limited angular sampling in ETcan lead to prominent streaking artefacts in the recon-struction, especially in the missing wedge direction. Byimposing sparsity in the image domain such artefactsmay be reduced [26.146] (Fig. 26.34). This has provento be effective even in atomic resolution STEM to-mography of gold nanorods [26.39], where each atomicpotential may be considered as sufficiently localizedin space such that an atomic scale ET reconstructionshould be inherently sparse. Sparsity constraints in theimage domain work well only if the background is zero,and so any background intensity should be excludedfrom reconstructions. If that is not possible, or if theobject of interest occupies a large portion of the field ofview, then an image domain sparsity constraint is less

PartD|26.5


a) b)

Signal, x

Ψ Ψ* Ψ Ψ

Prior knowledgeof sparsity

1-D FTs

Projection data(noisy, imperfectly

aligned)

Radial Fourier data Image estimate at iteration k

(Sparse) transform domaina)–e) Nonlinear optimization Sparse solution

Electronmicroscope

Missing

wedge

θmax

θmin

NUFFT*

NUFFT

Missingwedge

kmax

x, reconstructed imageˆ

c) d) f)

ai) e) fi)

Fig. 26.33a–f The key operators, domains, and prerequisites in CS-ET. The image in (a) can be represented sparsely inthe gradient domain (ai), via a spatial finite differences operator. In ET a limited number of images (b) are acquired inthe transmission electron microscope over a finite tilt range (leaving a missing wedge of unsampled information (c)). CSrecovery proceeds (c–e) by minimizing the number of nonzero coefficients in the (sparse) transform domain (e), whilstensuring consistency with the measured data (c) yielding a reconstruction (f) that is sparse in the transform domain (fi)

50 nm

y y

x

CS-ET SIRT

z

x

x

z

x

a) b)

Fig. 26.34a,b Comparison between the reconstructions of an ensemble of nanoparticles using (a) CS and (b) SIRTmethods. The reduction in fan artefacts in particular is clearly seen in both orthoslice orientations


D|26.5

applicable and other sparsifying transforms should beconsidered.

An alternative, and increasingly popular, sparsi-fying transform is spatial finite differences. In thistransform, a constraint is imposed on the number ofdiscontinuities in the image and the homogeneity ofobjects. The `1-norm of the spatial gradients of theimage, often referred to as the total variation (TV)-norm [26.174], penalizes many small variations in theimage intensity, but allows a limited (i. e., sparsely dis-tributed) number of large gradients. A TV constraint isespecially suitable for images that consist of homoge-neous regions with sharp boundaries, often referred toas piecewise constant, and in the physical sciences isideal for reconstructing small numbers of homogeneousphases, such as nanoparticle systems [26.39]. In anearly application of CS-ET [26.146], TV-minimizationwas applied simultaneously with image domain spar-sity to reconstruct with high-fidelity concave iron oxidenanoparticles using just nine projections.

Another useful sparsifying transform is that ofDWT [26.175], in which wavelet coefficients captureboth spatial position and spatial frequency information.Wavelets are, therefore, able to represent smooth, andpiecewise smooth, signal content, including nonperi-odic features such as jumps and spikes. There are nowa number of studies using DWTs, including denoisingof biological ET reconstructions [26.176] and singleparticle images [26.177], orientation determination insingle particle microscopy [26.178], and single particle3-D reconstruction [26.171].

Finally, a discrete Fourier transform may be usedto provide a sparse representation of an image contain-ing periodic features (such as a crystal lattice). In realsystems, however, defect structures or finite periodicitywill decrease the sparsity of any Fourier representa-tion [26.179]. DCT, a variant of the discrete Fouriertransform, may be applied locally and provide sparserepresentation of locally oscillating textures in naturalimages [26.180]. However, although providing sparserepresentations, the Fourier and DCT domains are notincoherent with the signal domain used for ET (theRadon/sinogram or Fourier domain). As such, thesetransforms are generally not suited for ET reconstruc-tions, but they may be of use for pre/postprocessing inET, for example, CS-based in-painting of fiducial mark-ers [26.179].

Discrete TomographyAnother way of incorporating prior knowledge duringtomographic reconstruction that has been quite exten-sively developed in the context of ET is the method ofdiscrete tomography [26.8, 181], which can be used toprovide high-quality and high-fidelity reconstructions if

features of the specimen can be considered in discreteterms. A specimen could, for instance, be consideredto consist of a discrete number of constituents of uni-form density [26.182] or to consist of discrete elementsthat lie on a regular grid, such as atomic positionsin a (perfect) nanocrystal [26.38, 183]. The applica-tion of such techniques in ET have been advanced byBatenburg and coworkers in particular, using a class ofalgorithm known as the discrete algebraic reconstruc-tion technique (DART [26.182]; a more mathematicaldescription is given in [26.184]). These have shownprofitable results, including reduction of missing wedgeartefacts [26.25, 182] and reconstruction from few pro-jections [26.146] in nanoscale ET, as well as enablingatomistic ET studies [26.38, 185].

DART has received quite wide recognition in thephysical sciences, and a number of variants have beenproposed by the Batenburg group and others [26.186–188]. At their core, these algorithms harness SIRT,but additionally introduce thresholding and gray levelassignment during the iterative refinement. A consider-able advantage of discrete approaches is that objects aresegmented during the reconstruction process, as theyare assigned to a particular discrete group. In many re-gards, this partitioning into homogeneous regions withsharp boundaries is very similar to a total variation con-straint in CS-ET, but is stricter in forcing regions toa specific gray level, rather than still permitting smallvariations. Figure 26.35 shows conventional SIRT andDART reconstructions of a bamboo-like carbon nano-tube containing an iron catalyst.

However, the ET practitioner must consider thatmany real samples may not fully satisfy discrete con-straints. Even if a high-quality discrete reconstructionof such samples can be obtained, it may not be of highfidelity. Moreover, often such strong prior knowledgeis not available, although methods for automatic graylevel selection may help in this regard [26.187]. Torelax the strict constraints and increase the level of au-tomation, further modifications of DART that have beendeveloped have included partial discreteness [26.189],adaptivity [26.186], and combination with TV regular-ization [26.190]. Other variants of discrete tomographyadvocated for ET include the binary algebraic recon-struction technique (BART, [26.191]) and the Bayesianapproach of Wollgarten and Habeck [26.192]. Fig-ure 26.36 shows a comparison of SIRT, total variation,and discrete tomography reconstructions of facetted na-noparticles, where it can be seen that elongation inthe missing wedge direction present in the SIRT re-construction is largely negated with total variation ordiscrete tomography. The total variation regularizationpromotes broadly homogeneous intensity in the nano-particles and sharp boundaries, compared to the binary

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Fig. 26.35a–fComparison of (a,b,c)SIRT reconstructionand (d,e,f) DART re-construction of part ofa bamboo-like carbonnanotube. Reprintedfrom [26.182], withpermission from Elsevier

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Fig. 26.36a–iComparison of(a,d,g) SIRT reconstruc-tion, (b,e,h) TV-basedreconstruction and(c,f,i) a DART reconstruc-tion of a series of facettednanoparticles. The double-headed arrow in (g) showsthe elongating effectof the missing wedge,and (i) the arrow showsa possible artefact at thesurface in the DART-based reconstruction.Reprinted from [26.97],with permission fromElsevier


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discrete tomography reconstruction. One slight prob-lem noted in this study is that diffraction contrast waspresent in some of the tilt series images. As noted previ-ously this, in general, is bad for the tomographic recon-struction, as it breaks the purely thickness dependenceof the signal. The discrete tomography reconstructionstruggled to deal with this, hindering correct estimationof the particle boundaries and leading to the small arte-facts indicated by the arrows [26.97].

Geometric TomographyAnother class of advanced algorithms comes from thefield of geometric tomography [26.193]. These arechiefly concerned with recovering the shape of ob-jects and mainly incorporate prior knowledge regard-ing convexity and homogeneity. As shown by Saghiet al. [26.67], these approaches can be valuable whennonlinearities, such as diffraction contrast or detectorsaturation, are prevalent in the tilt series projections. Inaddition to a geometric surface tangent algorithm pro-posed by Petersen and Ringer [26.194], a selection ofgeometric algorithms from the mathematical literaturehas been explored by Alpers et al. [26.191], includingreconstruction from very few projections when stronggeometric prior knowledge is available. Limitations ofgeometric algorithms are that the mass-density distribu-tion is neglected, i. e., they assume homogeneity and/orrecover only external or internal shape or edges, and ina number of cases, the object to be reconstructed mustbe convex.

Fourier-Based MethodsThe past few years have seen a resurgence of Fourier-based ET reconstruction, with major advances be-ing made through combination with iterative refine-ment and implementation of constraints. This in-cludes Fourier-based implementations of CS-ET (whichcan also be performed using a real space projec-tion operator) and the development and applicationof an approach known as equally sloped tomography(EST) [26.94, 148]. A distinct feature of EST is pref-erence for acquisition of projections at equal slopeincrements, as opposed to conventional equal angu-lar increments (or other schemes such as the Saxtonscheme [26.195]). Equally sloped sampling, in princi-ple, enables high-accuracy implementation of a pseu-dopolar fast Fourier transform (PPFFT) to convert be-tween the pseudopolar coordinates of the projectionsand Cartesian coordinates. However, significant effi-cacy of the algorithm arises because of the combinationof a PPFFT with oversampling and iterative refinement,during which constraints such as positivity and finitespatial support can be imposed. As outlined by Miaoet al. [26.148], the iterative process in EST can result in

filling in some information in the missing wedge due tocorrelation among Fourier components.

EST has been used in a series of studies at theatomic scale, combining the reconstruction with meth-ods for identification of atom positions to show defectstructures in small nanoparticles [26.91, 94, 149]. Morerecently, a generalized Fourier iterative reconstruction(GENFIRE) algorithm has been developed to incorpo-rate various physical constraints and refinements, suchas tilt angles, and was used in the aforementioned re-construction of a bimetallic FePt nanoparticle to revealchemical order/disorder (Fig. 26.13) [26.98].

Single-Particle ReconstructionsAnother notable development has been the applica-tion of single-particle microscopy techniques in thephysical sciences. Well established in the biologicalsciences e. g., [26.5, 196, 197], single-particle methodsinvolve recording many thousands of images of iden-tical particles (e. g., viruses) at different orientationsand using the ensemble of images to reconstruct theparticle in 3-D. Single-particle methods in materialsscience have been more limited by virtue of the factthat significant populations of identical or near identi-cal specimens (usually nanoparticles) are less common.However, there do exist certain magic number atomicclusters and nanoparticles whose configurations can beparticularly stable, and for which single-particle ap-proaches can be used. While the identification of magicnanoparticle morphologies from comparison of 2-D(S)TEM images to model structures has long been per-formed, Azubel et al. [26.90] recently adapted low-doseaberration-corrected TEM and SPM approaches to de-termine the 3-D atomic structure of Au68 nanoparticles.

Park et al. [26.95] determined the structure of few-nm Pt nanoparticles at near atomic resolution usingsingle particle methods by exploiting free rotation of thenanoparticles in a graphene liquid cell to obtain multi-ple viewing angles. This approach not only enabled thestudy of unique nanoparticles and their defects, but isalso significant in extending 3-D electron microscopyto address in-situ contexts. The use of direct electrondetection to obtain high-quality, low-dose, and rapidlyacquired images of particles (that are undergoing mo-tion) mirrors the successful exploitation of this newtechnology in biological contexts [26.198, 199].

Machine LearningAn increasingly pertinent need is to extract informationcontent efficiently from potentially large multidimen-sional ET data sets. Often, this involves reducing thedata down to a more manageable size. Here multivari-ate statistical analysis or machine learning methodscan be of particular value. These can be used for im-

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Fig. 26.37 (a) A HAADF STEM image of a region close to the tip of the needle and 2-D element maps of the majoralloying elements extracted from EDX spectra. (b) A scree plot of the 50 principal components. (c) Independent com-ponent weightings (IC#0–IC#4) at �30ı tilt and (d) their corresponding component spectra. From [26.200], publishedunder CC-BY license J

e) IC#0 IC#1γ'γ

z

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f)Fig. 26.37 (e) Orthoslices of thereconstructions of two independentcomponents. (f) Segmentation andvisualization of the ”0 strength-ening phase after ellipsoid fitting.From [26.200], published underCC-BY license

provement of signal to noise, but also provide powerfulseparation of significant components in data in a moreobjective manner. Examples of use in ET include prin-cipal component analysis and independent componentanalysis [26.107, 200], and nonnegative matrix factor-ization [26.104, 124, 201]. As an example, Fig. 26.37shows how component analysis can be carried out suc-cessfully on a tilt series of EDX spectrum images of anNi base superalloy to discover that there are six com-ponents of interest in the sample with correspondingloading maps at each tilt. Those loading maps may beused as input to reconstructions so as to form a 3-Dloading map indicating where that EDX componentdominates in the specimen volume.

Model-Based ReconstructionsAnother way in which electron tomographic recon-struction may be improved is through better modelingof the electron–specimen interactions. Typically, as al-luded to already, the signal used for conventional tomo-graphic reconstruction should be a monotonic functionof a projected physical quantity, such as composi-

tion or thickness—the projection requirement, How-ever, as ET expands into increasing use of analyticalsignals, probing perhaps more complex properties orwith more complex electron–specimen interactions, de-viations from the projection requirement will need tobe accounted for. Thus, the recent expansion of ETreconstruction methods incorporating constraints is be-ing accompanied by more comprehensive modelingof signal formation processes as part of the recon-struction scheme [26.202]. Model-based reconstructionapproaches offer the ability to account for and uti-lize signals that are not simple projections. As well asimproving the fidelity of established imaging modes,where violation of the projection requirement has beentolerated, they offer scope for significantly broaden-ing the range of properties and phenomena that canbe studied by ET. Figure 26.38, for example, summa-rizes a model-fitting approach developed by Collinset al. [26.201], matching simulated and experimentalEEL spectra, for refinement of the underlying chargedensity of the individual localized surface plasmoneigenmodes.

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Electron Tomography in Materials Science 26.6 Segmentation, Visualization, and Quantitative Analysis 1317Part

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Fig. 26.38 (a) Summary of key processing steps in eigenmode tomography. (b) Electron tomography and tilt seriesEELS of the surface plasmon modes of a silver right bipyramid on a MoO3 substrate. Surface representation of a CS-ET reconstruction with the zero tilt HAADF micrograph and a plan-view perspective of the right bipyramid. The insetscale bar is 25 nm. Surface mesh used for simulations. Spectral NMF decomposition factors corresponding to surfaceplasmon mode excitations. HAADF images, experimental NMF component maps, and simulated loss probability maps atthe corresponding tilt angles. (c) Calculated corner (dipole) eigenmode and corresponding surface charge reconstructionsfor a silver right bipyramid on a MoO3 dielectric substrate. Reprinted with permission from [26.201] published underACS AuthorChoice License, permissions requests should be directed to the ACS J

26.6 Segmentation, Visualization, and Quantitative Analysis

In order to analyze a tomogram quantitatively, forexample to determine surface area, volume fraction,crystallography, or porosity, it must first be segmented.This involves assigning each voxel in the tomogram toa feature of interest, for instance a nanoparticle, thevacuum, or the substrate. Segmentation is based uponclassifying the voxels based on some similarity metric,such as their intensity, spatial location, or local charac-teristics such as the image gradient or textural patterns.The difficulty in achieving segmentation of ET recon-structions has meant that, often, they have been treatedonly in a qualitative manner. Alternatively, in a numberof cases where segmentation has been achieved, it hasonly been through labor intensive manual procedures,in which the identification of features and delineationof their boundaries is open to individual interpretation.Indeed, segmentation can be the most time-consumingpart of the ET work flow. Image processing techniquesmay facilitate automated or semi-automated segmenta-tion, and recognition of their important role in ET hasgradually grown over recent years in both the biologicaland the physical sciences.

Segmentation is one of the most difficult aspects ofET to summarize because of the wide variety of differ-ent methods employed. The segmentation requirementswill depend on both the nature and the quality of theET reconstruction, and therefore often need to be devel-oped on a case-by-case basis. However, the key aspectsof image processing-based segmentation can be gen-erally applicable to many similar data sets with smalladjustments.

Several advanced segmentation methods have re-ceived attention [26.203–205], [26.10, Chap. 11],[26.78, Chap. 8], [26.5, Chaps. 11–15], mainly originat-ing from the biological ET community. These includedenoising by anisotropic nonlinear diffusion [26.206],watershed transformation [26.198], and gradient vec-tor flows [26.207]. Examples of more recently proposedadvanced segmentation techniques are noise reductionutilizing Beltrami flow [26.208], application of fuzzy

set theory [26.209], and segmentation of thin structuresusing orientation fields [26.210].

Segmentation procedures used in physical scienceET investigations are sometimes well described in par-ticular studies [26.211–214]. Fernandez [26.203] re-cently reviewed computational methods for ET, includ-ing segmentation techniques in the physical sciences,and a number of the aspects covered are recountedhere. Many of the image processing operations thatare readily applicable to materials science ET recon-structions can also be found in other tomographiccontexts or general reviews of 3-D tomographic dataanalysis [26.10], as well as in standard image pro-cessing texts [26.215]. Often, effective segmentationschemes consist of a number of standard image pro-cessing operations strung together, as exemplified inFig. 26.39 [26.211].

Typically, for most ET reconstructions to date,segmentation begins with a procedure for denois-ing and/or enhancing the features of interest. Thiscould involve basic regional averaging such as low-pass [26.212] or median filtering [26.70], histogramequalization [26.213], edge enhancement such as So-bel filtering [26.216], difference of Gaussians [26.211],or unsharp masking [26.146, 147], or more sophisti-cated processes such as anisotropic nonlinear diffu-sion [26.206]. Denoising or feature enhancement istypically followed by feature extraction based on a sim-ilarity metric. One of the most widely applicable simi-larity metrics in materials science, where many samplesconsist of regions of homogeneous density (e. g., na-noparticles), is simply the voxel intensity [26.25, 146,214]. In this case, features can be differentiated byglobal thresholding on the image gray level histogramor by local spatially aware thresholding, for which a va-riety of threshold selection methods exist [26.217].

Optimal threshold selection in tomography is an ac-tive area of research [26.218], and the success will stillbe dependent on the structural complexity of the systemunder consideration and the quality of the reconstruc-

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Fig. 26.39a–m Semi-automated segmentation routine applied to an ET reconstruction of unsupported nanoparticulateGa-Pd catalysts, to enable identification of individual and agglomerated nanoparticles constituting the densely populatedcluster. (a–l) The principal stages of the routine on a 2-D slice from the x–y plane of the tomogram. (m) A 3-D voxelprojection visualization. (l,m) The final segmented tomogram, where individual voxels have been given a color accord-ing the nanoparticle or group of strongly agglomerated nanoparticles to which they belong. Reprinted with permissionfrom [26.211]. Copyright 2012 The American Chemical Society

tion. Applicable to sufficiently high-quality data wherethere is a clear intensity difference between features andbackground, Otsu’s method [26.219] is one of the mostwell-known automated threshold selection techniquesand seeks the optimal separation based on minimizingthe intraclass variance in the image histogram. Ac-cordingly, a number of ET studies have used the Otsuor multilevel Otsu method [26.146, 214, 220]. Moresophisticated threshold selection procedures proposedspecifically for tomography have involved analysis ofedge profiles [26.221] or projection data error mini-mization [26.222].

Alternative procedures for feature extraction mightinclude detection of specific shapes. Such methods haveprimarily arisen in the biological field (e. g., for ex-tracting membranes and filaments [26.223]), but can beequally powerful in materials contexts too, for examplesphere extraction [26.212]. Discrete or partially discretereconstruction algorithms [26.182, 184, 186–188] thatincorporate gray level assignment a priori as part of the

reconstruction process could be classed as a distinct ap-proach to segmentation.

Subsequent to initial feature identification, addi-tional procedures may be used to better delineate ordifferentiate identified features. These might include,for example, morphological operations to denoise orregularize the boundaries of objects [26.146, 147] or theWatershed transform [26.224] to separate mildly touch-ing objects [26.211, 212, 214] and/or to locate theircentroids [26.83, 212].

With segmented components of a reconstructionthere are a variety of quantitative measures that canbe obtained, which can be of high value in the cat-alytic context. Thus, in spite of the challenges, exam-ples of quantitative catalytically relevant data obtainedfrom ET are growing in number. Examples includesize, shape, and local distribution of nanoparticulatecatalysts [26.70, 211, 214], determination of porosity,surface area, local curvature, and fractal dimension ofcatalyst supports [26.83]. Crystallographic analysis can

Electron Tomography in Materials Science 26.6 Segmentation, Visualization, and Quantitative Analysis 1319Part

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also be undertaken on segmented or unsegmented tomo-grams, and there are numerous examples revealing thecrystallography of catalytically relevant nanoparticlesthat was unobtainable from 2-D projections [26.141,225].

It is important to emphasize that success of segmen-tation routines and the accuracy of quantitative analysisis often dominated by the quality of the input data.In this regard, the appropriate choice of signal mode,acquisition geometry and reconstruction technique cangreatly facilitate the segmentation/quantification pro-cess. A high-quality reconstruction will facilitate sim-pler segmentation procedures. Indeed, in a follow-upstudy to the one shown in Fig. 26.39 [26.76], whichobtained a higher fidelity tomogram using CS-ET(Fig. 26.34) [26.147], the segmentation procedure couldbegin by thresholding directly on the raw tomogramwithout need for postreconstruction edge enhancement.Despite indications to the contrary in some particularcases [26.21, 28, 32], the effects of the missing wedgeand finite sampling often cannot be ignored in quanti-tative analysis. Methods that overcome or negate theseproblems, such as the use of needle samples [26.22–25]or advanced reconstruction algorithms, may provide theonly routes to truly reliable quantitative ET. The devel-opment of more widely applicable methods to tacklethese issues is one of the most worthy areas of devel-opment in ET.

Moving beyond 2-D images to a data volume in-herently adds additional complexity to visualization,requiring an extended set of methods. Different visu-alization techniques will convey the data in the volumein different ways and must be selected carefully to showthe desired information. Often, more than one techniqueis needed to enable a complete interpretation to be es-tablished. Visualization can be performed before and/orafter segmentation.

Volume rending, also known as voxel projection,provides a versatile means of visualizing the intensitydistribution through a reconstruction volume. As thename suggests, the rendering is in essence achieved bycomputing a projection through the volume, analogousto the original projection process on the microscope.Forming a 2-D projection once again may at first seemcounterintuitive, but with the reconstructed volume onthe computer, it is possible to control various proper-ties to enhance information of interest. These includemanipulating the view to give a sense of depth orperspective, changing the type of projection (such asmaximum intensity projection), using color, or alter-ing the level of transparency. It is also possible to viewthe reconstruction from any chosen direction and torestrict the visualization to just a subvolume. Tech-niques for volume rendering can reach a high level of

sophistication, enabled by modern computer graphicscapabilities. While offering significant versatility, careshould also be taken that significant information is notlost or overlooked when generating a volume rendering.For reconstructions suffering from tomographic arte-facts, some compromise may need to be made betweenartefact and object visibility.

Surface rendering can often provide a more dis-tinctive visualization of the 3-D form of objects. Here,a surface is rendered around features in a reconstructionproviding an intuitive visualization of their morphol-ogy. A common approach is to generate a surfacearound voxels at a given intensity, forming an isosur-face. For a high-fidelity reconstruction where the objectof interest has a well-defined intensity distribution, suchan isosurface should reveal its structure in a meaning-ful way. In practice, as with volume rendering, somecareful choices may be needed to visualize genuinestructure and not artefacts. Color, transparency, orienta-tion, and lighting effects can all be used to give a senseof 3-D structure and to display multiple components ofthe reconstruction simultaneously.

In many regards, the most objective and definitivemanner in which to interrogate a tomographic recon-struction is to display 2-D slices from the volume.These show a plane cut at a chosen position in the3-D volume and should not be confused with projec-tions through it. Figure 26.40 illustrates this distinction.Often, the slices are taken orthogonally to the primaryreconstruction axes and are known as orthoslices, buta slice can be computed (using some form of 3-D inter-polation) at any arbitrary cutting plane.

To illustrate further, Fig. 26.41 shows different as-pects of an ET reconstruction of a hierarchical macroand mesoporous SBA-15 silica selectively loaded withsmall (� 2 nm) Pt nanoparticles in the mesoporesand larger (� 6 nm) Pd nanoparticles in the macrop-ores [26.226]. The volume rendering (Fig. 26.41a) givesan overall feel of the structure, showing simultaneouslyall of the Pd and Pt nanoparticles and the general formof the macropore and mesopore structure. However,similar to the original projections from the microscope,the overlap of the complex pore structures in projectionmakes them difficult to discern. Here, surface render-ings provide a more suitable approach. Figure 26.41breveals the macropore structure with a surface renderingof the outer morphology of the SBA-15. Figure 26.41balso displays the volume rendering at the same time,showing that the larger Pd nanoparticles are locatedin the macropores. Figure 26.41c reveals the internalmesopore structure by combining an opaque surfacerendering of the mesopores along with the surface fromFig. 26.41b shown with semitransparency. Still, to de-termine the location of the smaller Pt nanoparticles

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Fig. 26.41a–c Different visualizations of an ET reconstruction of a hierarchical macro and mesoporous SBA-15 silicaselectively loaded with small (� 2 nm) Pt nanoparticles in the mesopores and larger (� 6 nm) Pd nanoparticles in themacropores. (a) Volume rendering, (b) volume rendering combined with surface rendering of the outer morphology ofthe SBA-15, (c) semitransparent rendering of the surface in (b) along with surface rendering of the internal mesoporestructure

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definitively requires (ortho)slices. Figure 26.42a usesa surface rendering to show the position of an or-thoslice in the ET reconstruction of another part ofthis hierarchical catalyst. As shown in Fig. 26.42b, theorthoslice clearly reveals the hexagonal arrangement

of internal mesopores, as well as the Pt nanoparticleswithin those mesopores. The multidirectional form ofthe mesopores, however, requires slices to be computedin various directions in order to understand fully the 3-Dstructure.

26.7 ConclusionsOver the past 20 years or so, ET has changed from be-ing a rather niche technique practised by very few inthe physical sciences to one that is now a routine and awidely-used method to determine the nanoscale struc-ture of materials in three dimensions. In parallel, therehas been a remarkable growth in the number of imag-ing modes that have been used in a tomographic wayto reveal not only the morphology of the region of in-terest but also composition, chemistry, electro-magneticproperties, optical properties, and local crystallography.

The growth of tomography in materials science, us-ing not only electrons but also x-rays and to a lesserextent other forms of radiation, has been aided also bythe extraordinary rise in computational power and theadvent of new reconstruction algorithms with which totake advantage of it. In particular, the use of CS andrelated methods holds great promise to improve notonly the fidelity of the tomogram but also to enablea model-based approach to reconstruction, which al-lows materials properties to be recovered that are notimmediately accessible through conventional backpro-jection routes.

The introduction of more efficient and faster spec-trometers and cameras enables analytical tomographyto be performed more quickly and with greater use ofwhat signal is provided by the electron–specimen inter-action. Even with materials science specimens, whichare generally more robust than those in the life sciences,care needs to be taken to ensure beam damage does notaccumulate to unacceptable levels over the time to ac-quire a tilt series.

There is now the technical capability to automatemuch of the image acquisition process and with ma-chine learning and AI, and the possibility of analyzing

the data as it is acquired. Modern cameras and spec-trometers enable vast quantities of data to be acquiredin short timescales and on-the-fly processing will un-doubtedly become a common feature in years to come.Such processing also allows judgement as to whetherjust enough data has been acquired to move to a newregion or to tackle a new problem and thus enablemore efficient use of the microscope, a feature thatis perhaps more critical with tomography than otherTEM techniques, given the necessarily long acquisitiontimes/large doses required.

The complexity of modern materials and deviceswill only increase in the future and much of thatcomplexity will be three-dimensional and chemicallyheterogeneous. ET is a technique that will continue toprogress rapidly, with improved spatial resolution andreconstruction fidelity and with the application of newanalytical methods.

Acknowledgments. The research leading to these re-sults was possible through funding from the Euro-pean Union Seventh Framework Program under GrantAgreement 312483-ESTEEM2 (Integrated Infrastruc-ture Initiative–I3), from the European Research Councilunder the European Union’s Seventh Framework Pro-gram (FP/2007–2013)/ERC Grant Agreement 291522–3-DIMAGE, and funding from the EPSRC, grantnumber EP/R008779/1. R.K.L. acknowledges a Ju-nior Research Fellowship at Clare College. The au-thors acknowledge the many people with whom theyhave worked, including most recently Sir John MeurigThomas, Francisco de la Pena, Sean Collins, Adam Lee,Emilie Ringe, Alex Eggeman, Jon Barnard, DuncanJohnstone, and David Rossouw.

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Rowan K. Leary

Dept. of Materials Science & MetallurgyUniversity of CambridgeCambridge, [email protected]

Rowan Leary obtained his degrees in materials science and engineering at theUniversity of Leeds, and his PhD from the University of Cambridge. He joined ClareCollege as a Junior Research Fellow and is now a Royal Society Tata UniversityResearch Fellow at the Department of Materials Science and Metallurgy, Universityof Cambridge. His research involves the development and application of atomic-scaleelectron microscopy and ET.

Paul A. Midgley

Dept. of Materials Science & MetallurgyUniversity of CambridgeCambridge, [email protected]

Paul Midgley is a Professor of Materials Science and the Director of the ElectronMicroscopy Facility at the Department of Materials Science andMetallurgy, Universityof Cambridge. He has studied a wide variety of materials by electron microscopy andhelped develop a number of novel electron microscopy techniques, including ET andprecession electron diffraction.

26.electrontomographyinmaterialsscience electrontom · 2019-11-02 · electrontom 1279 partd|26...

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