Microsc. Microanal. 20, 548–560, 2014doi:10.1017/S1431927614000075

© MICROSCOPY SOCIETYOF AMERICA 2014

Combined Scanning Transmission ElectronMicroscopy Tilt- and Focal SeriesTim Dahmen,1 Jean-Pierre Baudoin,2,† Andrew R. Lupini,3 Christian Kübel,4 Philipp Slusallek,1 andNiels de Jonge2,5,*

1German Research Center for Artificial Intelligence GmbH (DFKI), 66123 Saarbrücken, Germany2Department of Molecular Physiology and Biophysics, Vanderbilt University School of Medicine, Nashville, TN 37232-0615,USA3Oak Ridge National Laboratory, Materials Science and Technology Division, Oak Ridge, TN 37831-6071, USA4Karlsruhe Institute for Technology (KIT), 76344 Eggenstein-Leopoldshafen, Germany5Leibniz Institute for New Materials (INM), 66123 Saarbrücken, Germany

Abstract: In this study, a combined tilt- and focal series is proposed as a new recording scheme for high-angleannular dark-field scanning transmission electron microscopy (STEM) tomography. Three-dimensional (3D)data were acquired by mechanically tilting the specimen, and recording a through-focal series at each tiltdirection. The sample was a whole-mount macrophage cell with embedded gold nanoparticles. The tilt–focalalgebraic reconstruction technique (TF-ART) is introduced as a new algorithm to reconstruct tomograms fromsuch combined tilt- and focal series. The feasibility of TF-ART was demonstrated by 3D reconstruction of theexperimental 3D data. The results were compared with a conventional STEM tilt series of a similar sample. Thecombined tilt- and focal series led to smaller “missing wedge” artifacts, and a higher axial resolution than obtainedfor the STEM tilt series, thus improving on one of the main issues of tilt series-based electron tomography.

Key words: STEM, tomography, 3D, focal series, whole cell, nanoparticle, SART, 3D reconstruction, back projection

INTRODUCTION

The three-dimensional (3D) organization of the cellularultrastructure is primarily studied using tilt series transmis-sion electron microscopy (TEM) (Hoenger & McIntosh,2009; Kourkoutis et al., 2012). A 3D volume is reconstructedfrom images recorded at several projections obtained bymechanically tilting the sample stage. The resolution is in therange of 2–20 nm, thereby filling a critical length scalebetween the atomic resolution of X-ray crystallography andsingle particle electron tomography (Frank, 2006) on the onehand, and high-resolution confocal light microscopy (Hell,2007) and X-ray microscopy (Meyer-Ilse et al., 2001) on theother. The tilt range strongly influences the resolution of the3D reconstructions (Koster et al., 1997; Fernandez, 2012).Therefore, one would ideally acquire tilted images coveringthe entire angular range of ±90°. However, in practice, themaximum tilt range is usually only about ±60–78° due tomechanical limitations of specimen holders and because theeffective thickness of the specimen as seen by the electronbeam increases as the section is tilted. The tomographicreconstruction then suffers from missing information andlimited resolution on account of this so-called “missingwedge.” Different solutions have been explored. The missingwedge can be reduced to a missing pyramid using double-tilttomography (Penczek et al., 1995). The limitation of missingvertical information is more severe for imaging samples

thicker than several mean free path lengths for electronscattering, typically a few hundreds of nanometers for bio-logical samples. The resolution of TEM tomography is thenreduced by electron–matter interactions. First, (multiple)elastic scattering events lead to an angular broadening of theelectron beam, especially in areas behind high-densityobjects. Second, inelastic scattering spread the energy spec-trum of the electron beam leading to chromatic blurring ofthe TEM objective lens (Reimer, 1998). Chromatic blurringin thick biological samples can be reduced by introducingenergy filtering (Koster et al., 1997) or chromatic aberrationcorrection (Baudoin et al., 2013).

Another approach to image thick samples is to usescanning transmission electron microscopy (STEM) tomo-graphy (Engel & Colliex, 1993; Yakushevska et al., 2007;Aoyama et al., 2008). Particularly the high-angle annulardark-field (HAADF) imaging mode has been shown to besuitable for tilt series reconstructions (Weyland & Midgley,2003; Kübel et al., 2005). Using the dynamic focus functionof STEM microscopes, the focal plane can be tilted togetherwith the specimen (Zemlin, 1989). This helps to avoid blur-ring as parts of the specimen are rotated out of the depth offield. The atomic number contrast of HAADF STEM makesit possible to detect the 3D locations of gold nanoparticlesattached to proteins within whole cells (Sousa et al., 2007;Baudoin et al., 2013). Bright-field STEM can be used toimage thick sections because its signal suffers less from beambroadening by elastic scattering (Hohmann-Marriott et al.,2009). Avoiding tilting altogether is also possible. Serialsectioning with a focused ion beam and subsequent 3D

*Corresponding author. niels.dejonge@inm-gmbh.de†Current address: La Timone Hospital and Medicine School, Marseille, France

Received September 19, 2013; accepted January 2, 2014

reconstruction (Heymann et al., 2006) or serial block facescanning electron microscopy (Denk & Horstmann, 2004)present routes for the intermediate resolution range. Asecond approach is the recording of focal series to obtain 3DSTEM information (Frigo et al., 2002; Borisevich et al., 2006;Behan et al., 2009; de Jonge et al., 2010; Dukes et al., 2011).Hereby, axial information is retrieved from a stack of imageswith different focal values and the 3D data set is pre-ferentially deconvolved (Ramachandra & de Jonge, 2012).But the focal series data suffers from a vertically elongatedpoint spread function (PSF), and the vertical resolution iseven further reduced by shadowing effects below stronglyscattering objects (Behan et al., 2009; de Jonge et al., 2010).

In this work, we present a combined tilt- and focal seriesas a new recording scheme for 3D STEM imaging. In thisscheme, the specimen is rotated in relatively large tilt incre-ments over the possible tilt range, and for every tilt direction,a focal series is recorded. The feasibility of the technique isdemonstrated by imaging and generating a 3D reconstruc-tion of a thick biological sample consisting of a whole-mountcell with gold nanoparticles. Our main contributions are(1) the introduction of the combined tilt- and focal series as anew recoding scheme for the image acquisition, (2) theintroduction of a newmethod for axial alignment of confocalSTEM images, and (3) the introduction of the tilt–focalalgebraic reconstruction technique (TF-ART), which con-sists of customized forward- and back projection operatorsand allows the application of the Kaczmarcz algorithm asa method for tomographic reconstruction of a combinedtilt- and focal series. By comparing the TF-ART recon-struction to a reconstruction with the sequential algebraicreconstruction technique (SART) of a standard tilt series, wedemonstrate that a combined tilt- and focal series leads to amore uniform resolution of the tomogram and especially to areduction of the axial elongation artifact.

MATERIALS AND METHODS

Sample PreparationMacrophages derived from monocytes (THP-1 cells, Amer-ican Type Culture Collection) were grown directly on elec-tron transparent silicon nitride TEM windows supported bysilicon microchips (Ring et al., 2011). The growth occurredin phorbol-12-myristate-13-acetate supplemented medium(Jerome et al., 2008). Native low-density lipoprotein (LDL)was conjugated to 16± 3 nm (n = 20) or 7± 1 nm (n = 20)gold nanoparticles. Cell samples (Baudoin et al., 2013b) wereincubated with 16 nm LDL gold for the first day and with7 nm LDL gold for the second day. The incubation took placeat 37°C in 1% fetal bovine serum medium with an equivalentconcentration of 8 µg/mL LDL. To prepare the samples forelectron microscopy, the cells were rinsed with phosphatebuffer saline, fixed with 2.5% glutaraldehyde in 0.1 Msodium cacodylate buffer/0.05% CaCl2, postfixed with anultra-low concentration (0.001%) of osmium tetroxide,gradually dehydrated with ethanol, and finally critical point

dried with liquid carbon dioxide. To increase resistance toelectron beam damage, a layer of about 20 nm of carbon wasevaporated on the samples (Dukes et al., 2011). The carbonwas applied using an electron beam evaporator with a basepressure of 5 × 10− 7 torr for 45 min. Additional stainingwith, e.g., lead was avoided to be able to image through theentire cell.

Data AcquisitionSTEM images were recorded at 300 kV with a transmissionelectron microscope equipped with a probe corrector forspherical aberrations (Titan 80-300, FEI, Hillsboro, OR,USA). Images were acquired at 160,000 × magnification andusing a pixel size of 2.3 nm. The objective aperture semi-angle α was 49.1 mrad. The focal series consisted of 41images separated by 50 nm in axial direction, of which the20 images in the middle vertical range were selected forfurther processing. Images of 512 × 512 pixels were recordedwith an acquisition time of 12 µs/pixel. The tilt series wererecorded with specimen tilts ranging from −40 to +40° at 5°increments (higher tilt angles were not possible because theedges of the viewing window in the microchip masked thespecimen at higher tilt angles). The data set thus contained atotal of 697 images. A script (written in Java) controlled theFEI microscope for an automated acquisition of the focalseries. After changing the tilt angle, the region of interest wasrealigned, and the probe was refocused.

SoftwareAll tomographic reconstructions were performed using asoftware package of local design, named Ettention. Thesoftware was implemented as a library of compute kernelsthat were programmed in C++ and OpenCL. The OpenCLkernels were used to leverage the massively parallel nature ofgraphic processing units (GPUs) for maximal computationperformance. The individual kernels were combined intoreconstruction algorithms for execution as command linetools or integrated in the IMOD software (Kremer et al.,1996). The algorithms SART (Andersen & Kak, 1984) andsimultaneous iterative reconstruction technique (SIRT)(Gilbert, 1972) were implemented in the Ettention softwarein previous work (unpublished).

IMOD 4.3.7 was used to perform the lateral alignmentof combined tilt- and focal series. ImageJ (version 1.46r,64 bit, NIH) was used for various image processing andfiltering tasks. TomoJ 2.21 (Messaoudii et al., 2007) was usedto automatically detect nanoparticle chains for axial align-ment. All statistical analysis was performed using IBM SPSSStatistics 21. One sample t-tests and independent two samplet-tests were used for statistical testing, p-values <0.05 wereconsidered statistically significant.

Coordinate SystemIn the following, we introduce the coordinate setup that willbe used throughout this text (see Fig. 1). Positions inside the

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volume are identified by the coordinates x, y, and z, mea-sured in nanometers. During a tilt series, the volumewas rotated by the tilt angle β around an axis, which is byconvention the y-axis of the volume. The image was recordedby a convergent electron beam with an opening semi-angle α.In the combined tilt- and focal series, for each tilt direction athough-focus series was recorded. Each pixel in the data set istherefore identified by a 4D coordinate x′, y′, i, and β.Hereby, x′ and y′ refer to the lateral (horizontal) pixel posi-tion in the image stack, i is the vertical index of the image inthe stack. z′ annotates the axial (vertical) position measuredin nanometers, which was computed from i during the axialalignment described below.

Each image in a focal series was recorded with a differ-ent vertical focus position f, measured in nanometers, whichwe express by convention relative to the tilt axis. The verticalfocus position of the first image in each stack measured fromthe tilt axis is annotated as f0 and the distance betweenconsecutive focal positions isΔf. The focus position fi of eachimage can then be expressed as: fi = f0 + i Δf.

AlignmentA (S)TEM tilt series needs to be aligned to correct for stageshifts occurring during tilting. One method is implementedin IMOD (Kremer et al., 1996), consisting of a coarse auto-matic alignment and an iterative refinement procedure ofsemi-automatic fiducial marker alignment. This methodworks well for a tilt series with one image per tilt angle.However, the method could not be directly applied to ourcase with a focal series for each tilt angle. It was observed thatthe alignment was sufficient within each focal stack but theimage positions shifted as the tilt angle was changed. Theproblem was that these shifts did not only consist of a lateral(x′y′) shift but also of an axial shift. It was thus unknownhow the indices of different frames of the focal series corre-sponded to the focus positions f, and it was not possible tofind this relation with the existing algorithm. Therefore, two

alignments were performed. First, we determined the affinetransforms for all images in the series to bring the differentprojections into a common coordinate system, i.e. lateralalignment. Second, we needed to find the parameters f0 andΔf of the spatial positions of the focus planes, i.e. axialalignment.

Lateral AlignmentFor lateral alignment we used the following procedure.A vertical projection was computed for the image stack of eachfocal series, averaging the intensities for each pixel x′y′ of thefocal series. The vertical projections were then combined intoan image stack representing a conventional tilt series. Next, wecomputed the affine transformations for the alignment using astandard method (Kremer et al., 1996). The determinedtransformations per tilt angle were applied to each image inthe focal series for the corresponding tilt angle. This methodwas possible because the focal series at each tilt angle did notcontain noticeable lateral shifts in themselves. For cases with asignificant lateral shift within a series, an additional alignmentstep could be added before the projection.

Axial AlignmentIn the following, we describe the procedure for axial align-ment. The goal of the axial alignment was to find the relationbetween the index i of the image in the stack, the corre-sponding vertical focus positions of the first image f0, and thefocal distance between consecutive images Δf. We did so bysearching first for nanoparticles in images corresponding toadjacent tilt directions (nanoparticle chain detection). Oncethe position of a certain nanoparticle was known in morethan one image, we estimated its 3D position by means oftriangulation. Finally, the algorithm detected which imagein the stack was closest to focus for that nanoparticle andcorrelated the focus position to the 3D position of thenanoparticle.

a b

Figure 1. The coordinate system used during reconstruction and alignment. a: A tilted object to be reconstructedis imaged with a conical electron beam of scanning transmission electron microscopy. α denotes the beam openingsemi-angle and β the tilt angle. One image stack is acquired per tilt direction. b: The focal series recorded at a certaintilt angle is stored in an image stack with each image having a different focal position. Voxels in the reconstructedvolume refer to the original (nontilted) object, and are identified by the coordinates x, y, and z. Pixels in an image stackat a certain tilt angle are annotated with the lateral positions x′ and y′, and the vertical location in the stack z′ at focalplane index i.

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Particle Chain DetectionA procedure for axial alignment was developed based on theautomated identification of nanoparticles with high contrastin the images. First, the tilt–focal data were projected intoa tilt series as described above. A high-pass filter with a10 pixel cutoff was applied (ImageJ). After this, the stack wasopened in TomoJ, using the normalization setting “ElectronTomo.” Background removal was performed with a rollingball radius of 15 pixels and smoothing enabled. Next, chainsof objects were generated with a method described in detailelsewhere (Sorzano et al., 2009).

In short, nanoparticles were detected by searching forlocal maxima in an image. Those nanoparticles were thensearched for in adjacent images in the tilt series. The searchwas performed by predicting the nanoparticle position usingaffine transformations based on the tilt angle difference andconsecutive local optimization of the correlation index. If thecorrelation index was greater than a given threshold, theregions in the two adjacent images were accepted to repre-sent the same nanoparticle. A so-called particle chain wasthus generated from the local positions of the nanoparticle inconsecutive images. A set of chains was generated aiming totrack as many different nanoparticles as possible.

The particle chain generation was performed in thesoftware TomoJ using the following settings: algorithm =“critical points− local maxima,” number of seeds = 20,number of best points to keep in each image = 40, length oflandmark chain = 11, patch size in pixel = 14, minimaneighborhood radius = 8, fiducial markers = yes. Thefound nanoparticle positions in the images were exported forfurther processing.

Nanoparticle Position TriangulationNext, the algorithm estimated the 3D position of an indivi-dual nanoparticle by triangulation. It selected two images inwhich the 2D positions of the same nanoparticle wereknown. Based on both known lateral positions and the tiltangle, 3D lines were created by assigning an origin and adirection. The line origin was the nanoparticle position in theimage plane while the line direction was the tilt directionassociated with that image.

In the ideal case of perfect alignment, the two linescorresponding to the same nanoparticle would intersect in

the center of that nanoparticle. In practice, the lines did notintersect due to alignment errors. Therefore, the line segmentof closest distance between them was calculated. The mid-point of this segment provided an estimate of the 3D positionx,y,z of the nanoparticle, relative to the tilt axis. The length ofthe segment was considered as a measure of the alignmenterror. The above procedure was repeated for all possible linepairs of the same particle chain. The midpoints were thenaveraged to obtain the most precise position estimate. Theprocedure was applied to all particle chains, each giving the3D position of a different nanoparticle.

Estimation of Focal ValuesThe algorithm computed the parameters for the axial align-ment. For a given tilt direction, the software considered alldetermined nanoparticle center positions using the 2D x′,y′position of the nanoparticle from the particle chain. For eachposition, it searched all frames in the focal series, applying alow-pass filter with a radius of 2 pixels in the lateral directionto suppress noise. The image i with the highest intensity atthe pixel x′,y′ was defined as the best focus for this nano-particle. The corresponding index was called ifocus (seeFig. 2a). This procedure was repeated for all known nano-particles, and the algorithm created (in principle) a plotdisplaying the axial distance of the nanoparticle to the tiltaxis z′ against ifocus (see Fig. 2b). Hereby, z′ was obtained byrotating the 3D position x,y,z of the nanoparticle with respectto the tilt angle of the stack under consideration. A lineartrend was fitted through the plot using linear least squaresregression. Since the data were recorded with identical focussteps between the images in a focal series, the slope of thetrend gave the relative distance between consecutive focuspositions Δf, while the intersection with the f-axis gave thefocus position of the first image with respect to the tilt axis f0.The operation was repeated per tilt direction, so the softwarecomputed one value for f0 and Δf for every image stack.

Volume ReconstructionIn the following section, we present TF-ART, a new methodof volume reconstruction applicable to a combined tilt- andfocal series. For parallel beam tilt series, different recon-struction algorithms such as the ART (Gordon et al., 1970),

a b

Figure 2. Estimation of focal positions for axial alignment. a: Plot of the pixel intensity of an individual nanoparticleversus the focus index i as a measure of the focus index where the nanoparticle was best in focus ifocus. b: Plot of thevertical position z′ versus ifocus for all nanoparticles. A fitted linear trend is also shown. The slope of this trend equalsΔf. The intersection with the f-axis gave the location of the first image plane f0.

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SART (Andersen & Kak, 1984), and SIRT (Gilbert, 1972)have been well established. Common to all of these algo-rithms is the basic principle to express the reconstructionproblem as an ill-posed system of linear equations and solvethe system by application of instances of the Kaczmarczalgorithm (ART, SART) or the Landweber iteration (SIRT-type algorithms).

In either case, the method starts with an initial volumeand generates a virtual projection using a forward projectionstep. Residuals are computed by per-pixel subtraction of thevirtual projection and the measured projection. A backprojection is then used to correct the volume by distributingthe residuals back into the volume. Figure 3a depicts theprinciple. One application of this correction step to all pro-jections of an image set is referred to as “one iteration” andthe algorithm typically uses several iterations to allow thesolution to converge to an optimum.

Reconstruction LoopOur algorithm looped over all tilt directions in randomorder. For each direction, it stepped through all focus posi-tions sequentially. The residuals for all focus positions of onetilt direction were computed and stored on a stack. Then, thevolume was corrected for all residuals of this direction byiterative execution of the back projection operator beforemoving to the next direction. Figure 3b depicts the algorithmas a block diagram.

Forward Projection by Ray CastingAlgebraic reconstruction methods operate on a systemmatrix A that reflects the structure of the forward- and backprojection operator. While the formulation of the underlyingKaczmarcz algorithm guarantees convergence with relativelyfew assumptions on A (e.g., Gregor & Benson, 2008 or Zeng& Gullberg, 2000 for the case of unmatched projection/backprojection pairs), in the field of electron tomography onetypical assumption is that the intensity of a pixel in a verticalprojection through a 3D volume represents the integral alongthe line to that pixel (Radon transform). In this case, the line

integral of the volume can be efficiently computed using raycasting (Levoy, 1990). For each pixel in the forward projection,the ray through that pixel is generated and the volume issampled at uniform intervals along that ray. Trilinear inter-polation or Blobs (Marabini et al., 1998) are used between thevoxel centers. If the sampling interval chosen is small enough(≤ 1 =

2 voxel size), the average gray value at the sampling posi-tions gives a good estimate of the line integral through thevolume. This method is accurate for the approximately parallelillumination in TEM tomography, or for STEM tomographywith a small beam convergence angle leading to a large depthof field with respect to the sample thickness.

Simulation of Convergent Beams by Cone TracingHowever, in aberration-corrected STEM, the electron beamis convergent with a focal depth of typically a few nano-meters (Borisevich et al., 2006; Lupini & de Jonge, 2011), sothat the line model of the forward projection is no longer agood approximation. For this reason, the convergence of theelectron beam was taken into account in our choice of theprojection operators. The probe shape in the forward pro-jection used a model consisting of a double cone, as a simplemodel for the PSF. Any lateral cut through this double coneresults in a circular disc. In this model, the value of a pixel inthe projection equaled the volume integral of the gray values ofthe voxels inside the double cone, weighted by the local currentdensity. We assumed that the probe current was homogeneouswithin each disc and zero outside. As a consequence, the localcurrent density changed with the reciprocal of the disc area, i.e.was larger close to the focus plane.

In order to compute the volume integral of the doublecone, our algorithm used a Monte Carlo technique, approx-imating the integral with multiple individual line integrals.The lines were chosen so that they formed a specific samplingpattern as described below. Each line integral was computedusing the ray casting technique. The individual lines were thenaveraged to estimate the volume integral of the double cone.Figure 4a depicts the principle. The sampling method wasinspired by the way the focal depth of optical camera systemsis simulated using Whitted-style ray tracing (Whitted, 1980).

a b

Figure 3. The overall reconstruction algorithm. a: Visualization of the dataflow. Starting from an initial reconstructionvolume, residuals were generated by a forward projection and per-pixel subtraction from a real projection. The volumewas then corrected using a back projection operator. b: The nesting of the reconstruction loops. Operations imple-mented as kernels on the graphic processing units are marked as light gray. All virtual projections from one directionwere generated before a correction was applied to the volume.

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This technique is frequently used in the field of realistic imagesynthesis, for example, in movie production.

Stratified Rejection SamplingAs a method to randomly place the individual lines in thedouble cone while maintaining roughly uniform samplingover the domain, stratified sampling (Cook, 1986) was used.Each line was specified by two points. The first point was thefocus point of the double cone. In order to specify the secondpoint a horizontal cut through the cone was consideredspecifying a circular disc. A 2D grid was placed over this discand within each grid cell one point was placed at a pseudo-random location. If the point happened to be outside thedisc, the sample was rejected and the corresponding line wasnot considered during integration. The method is calledrejection sampling (Robert & Casella, 2005) and is frequentlyused together with ray casting. Figure 4 depicts the principle.

In summary, a forward-projected image was generatedby stepping through all pixels. For every pixel, the probe wasapproximated by a double cone. Inside this double cone, thevolume integral of the voxel gray values, weighted by thelocal current density of the beam (because rays converged),was computed. The computation was performed by a conetracing implementation based on ray casting and stratifiedrejection sampling. The resulting projection corresponded toa single slice of the focal series, assuming a simplified PSFand the absence of aberrations, alignment issues, noise, anddrift. Also, in this version, the forward projection did not yettake into account beam blurring occurring in thick samplesby the interaction of the electron beam with the specimenand reducing the axial resolution of focal series data(Ramachandra et al., 2013).

Back Projection by Regularization with Distancefrom Focal PlaneThe volume reconstruction also needed a back projectionalgorithm. It was not possible to use the same algorithm asused for the forward projection, because this would require

an inverted PSF that was not available. Instead, we correctedthe effect of the PSF with a new regularization method.The back projection operator corrected the volume for anindividual residual image, corresponding to one tilt directionand focal value. In order to correct the volume for one tiltdirection the operator was executed consecutively for everyfocal value. An individual application of the back projectionwas implemented by looping over all voxels in the relevantreconstruction volume. The center of each voxel was pro-jected to the image plane of the projection by multiplicationwith the 4 ×4 matrix representing the parallel projectioncorresponding to the tilt angle. Bilinear interpolation wasused to look up the residual value at the projected pixelposition. The voxel was then corrected with the residualvalue, modified by a regularization factor described below.An efficient implementation of the back projection operatorwould ignore voxels that have zero contribution, which caneasily be determined in advance.

This method is functionally equivalent to looping overthe residuals pixel-by-pixel and projecting the pixel value tothe reconstruction volume using ray casting. However, thevoxel-by-voxel approach maps better to current GPU hard-ware because it avoids scattered memory-write operationsand synchronization issues and thus results in higher per-formance (Xu et al., 2010).

The back projection was limited to that part of thevolume where the corresponding projection contained themost high frequency information, i.e. around the focus. Weachieved this by introducing a regularization factor Γ. Inalgebraic reconstruction methods, a regularization factor ismultiplied to the correction applied to a voxel in order toimprove the convergence characteristics of the method or asa means to incorporate prior knowledge. We introduce thefollowing regularization factor Γ(x,y,z,β).

Γðx; y; z; βÞ :¼ λ � 0 if z0 - fij j>Δf1 - z0 - fij j

Δf else

((1)

where Γ(x,y,z,β) is a function of the position x,y,z of thecenter of the voxel that is currently being corrected, and the

Figure 4. The forward projection operator. a: The electron beam was modeled as a double cone. The intensity of apixel was computed by integrating the gray values of the volume inside the double cone. The electron beam wasapproximated by several rays integrated using ray casting and averaged. b: In the stratified sampling scheme, a grid wasplaced over the circle representing a horizontal slice through the volume. In every cell of the grid, one sample (whitedot) was placed at pseudo-random position. Samples outside the disc were rejected (black dots). c: The per-pixel errorof a forward projection plotted over the number of rays per pixel that was used to approximate the double cone.

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tilt direction. λ is the relaxation parameter typically used inalgebraic reconstruction methods (Gordon et al., 1970). Weused λ = 0.3 for all experiments. The idea behind theformula for Γ was that information from a focal plane shouldonly influence the region of the volume close to the focalplane. Between the individual planes, linear interpolationwas used to achieve a smooth transition.

Figure 5 is a schematic representation of this regular-ization scheme. Note that z′− fi is the axial distance from avoxel center to the focal plane i. The regularization factor Γ iszero everywhere except in a slice of thickness Δf in bothdirections from the current focal plane, so an individualapplication of the back projection operator only corrects aslice of thickness 2Δf, the remainder of the volume remainsunchanged. In order to correct all voxels in the volume, theback projection was executed once for every residual imagefrom one tilt direction as determined by the reconstructionloop. After the execution of a tilt direction, every voxel in thevolume was changed twice, once for each of the two focalplanes closest to the voxel, resulting in a linear interpolationbetween the two relevant residual values for each voxel.

RESULTS AND DISCUSSION

Acquisition of Combined Tilt- and Focal SeriesWith STEMA combined tilt- and focal series data set was acquired asdescribed in the “Materials and Methods” section. Thesample was a whole-mount macrophage cell with goldnanoparticles coated with LDL taken up into vesicles.Figures 6a and 6b show two images acquired at -40° tilt angleand at two different focus positions (the entire combinedtilt- and focal series is shown in Supplementary Movie 1).Different clusters of nanoparticles are visible in the image,depending on the focus position. The white arrows mark therespective positions that are in focus. Note that the samplewas optimized for the imaging of nanoparticles throughout awhole cell, and the staining level of the cellular ultrastructurewas kept to a minimum. Furthermore, the acquisition of thecombined tilt- and focal series was not optimized for thelowest possible electron dose but served to test the 3Dreconstruction of this type of data.

Supplementary Movie 1

Supplementary Movie 1 can be found online. Please visitjournals.cambridge.org/jid_MAM.

AlignmentThe combined tilt- and focal series was aligned in lateraldirection using a standard method to a precision of ~1 pixel.Hereafter, the alignment in axial direction was performedusing the newly developed procedure involving particlechains (see “Materials and Methods” section). In total, 134particle chains were generated, containing a total of 1,516known 2D particle positions. For every tilt direction anaverage of 160 (standard deviation σ = 14) different 2D

Figure 5. The regularization factor used in the back projectionoperator. The regularization factor Γ(x,y,z,β) maps a voxel of thespace of the reconstruction x,y,z into the space of the original datax′,y′,z′ at a certain β. The function equals one exactly inside thefocal plane and drops linearly to the adjacent focal planes.

a b

Figure 6. Projections of the original combined tilt- and focal series high-angle annular dark-field scanning transmissionelectron microscopy (STEM) data (input). The sample was a whole-mount macrophage cell containing gold nanoparticlesof two different sizes distributed in clusters throughout its volume. The data were recorded for a tilt range of −40 to +40°in 5° increments, with a focal series at each tilt angle using focus steps of 50 nm, and for α = 49.1mrad. a: STEM imageof a combined tilt- and focal series at the tilt direction −40°. b: Image recorded at the same tilt angle but at a 780 nm(13 steps of 60 nm) different focus. The white arrows indicate sections of the images that are in focus.

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nanoparticle positions were known, which allowed us later toreach the required precision in axial alignment.

The 2D nanoparticle positions were used to determinethe focal parameters f0 and Δf (Fig. 1b) by fitting a lineartrend as explained in the “Materials and Methods” section.Hereby, Δf was considered constant across tilt directions.However, the fitting of the linear trend did not reach thesame measure of confidence for all tilt directions on accountof the difference in vertical separation of the nanoparticles,i.e. at higher tilt angles, the nanoparticles as seen from theelectron beam were spread out more than at lower tilt angles,leading to a more accurate estimate of Δf. For the purpose ofestimating Δf, we thus considered the 11 directions with thehighest measure of confidence only, which were the directionswith the highest tilt angles. By combining those values, wecalculated a mean value ofΔf = 60 nm (σ = 3 nm). This valuediffers significantly (p< 6.6 × 10− 6) from the nominal value ofΔf = 50 nm that was expected from the microscope control.We used the computed value for Δf in all reconstructions.

The value of f0 had to be calculated independently pertilt direction. The precision of the measurement of f0 wasquantified by calculating σ of the constant coefficient of thelinear fit using statistical standard methods. For f0, we cal-culated σ = 18 nm, which corresponds to a precision of±37 nm assuming a 95% confidence level. Because the elec-tron beam had an opening semi-angle of α = 41 mrad, thisaxial alignment error corresponded to a blurring with aradius of 1.5 nm (≈0.7 pixel) that was considered to have noobservable impact on the reconstruction quality. The align-ment procedure relies on the presence of gold nanoparticles,but fiducial gold markers commonly used for STEM or TEMtomography (Lawrence, 1992), would work as well.

3D RECONSTRUCTION

The aligned image stacks of the combined tilt- and focalseries were reconstructed using TF-ART. TF-ART is an

instance of the variable block ART algorithmic scheme(Censor, 1990), generalized to feature an unmatched pro-jection/back projection pair as discussed in Zeng & Gullberg(2000). It can be expressed in the form:

Xðk + 1Þ ¼ Xk + SXm

i¼1wi B

TðP -CXkÞ (2)

where X is the volume, Pi, i = 1… m are the pixels in all ofthe measured projections, C is the matrix expressingthe forward projector, B is the matrix expressing the backprojector, wi are weights that control at which granularityand in which order the measured projections are processed.

In algebraic reconstruction methods, forward projectionoperators are used to generate projections of the inter-mediate tomograms, called “virtual projections.” Ourforward projection operator used a double cone as a simplemodel for the PSF of the electron beam as explained in the“Materials and Methods” section (Fig. 4). The effect of thedouble cone model of the electron beam can be observed asvarying amounts of geometric blurring of the differentclusters of particles, depending on the focal plane. The modelaccommodates the blurring that occurs in the real imagesdue to the convergent shape of the probe.

In order to determine the required number of rays perpixel for the forward projection, we tested experimentallyhow quickly the computational error dropped as the numberof rays used to approximate the double cone was increased(convergence rate). We performed forward projections usingdifferent numbers of rays per pixel. Figure 4c shows a plot ofthe mean pixel error as percentage of the maximum intensityover the number of rays per pixel. In order to determine theper pixel error, a ground truth image was generated using avery high sample count (100,000 samples/pixel). It wasconfirmed that at this high sampling, adding further samplesdid lead to no difference in the result within computationprecision and the ground truth image was used as referenceto measure the sampling error of the forward projection. Inthe case that 25 rays/pixel were used to approximate the

a b

Figure 7. Intermediate results of the back projection operator. a: Intermediate tomogram of the same measured dataset as shown in Figure 6, after all residuals from one tilt direction were processed. The focal planes are displayed aswhite lines. b: Intermediate tomogram after the processing of a second direction. The white arrow marks the location ofa nanoparticle.

Combined STEM Tilt- and Focal Series 555

double cone, the remaining mean error was 0.28% of themaximum intensity. We found that this pixel error hadno measurable impact on the overall reconstruction quality,and 25 rays/pixel were thus used for all reconstructions. Thevirtual projection was subtracted pixel-by-pixel from themeasured projections to compute residual images. The inter-mediate volume was corrected with respect to those residualsusing our back projection operator and regularization factordescribed in the “Materials and Methods” section.

Figure 7a shows the intermediate volume after allimages of one tilt direction were processed. After only onedirection, nanoparticles were reconstructed as elongatedstreaks in the tilt direction. The focal planes were perpendi-cular to the tilt direction and are shown as white lines in theimage. Between the focal planes, linear interpolation wasused. After two iterations had been processed (Fig. 7b), thepositions of the nanoparticles started to show as intersectionsof the streaks.

The reconstruction was performed using 120 iterationsof TF-ART. The resulting tomogram is shown in Figure 8a,the white arrow marks the position of the same nanoparticlein all three directions. The gold nanoparticles are clearlyvisible against the background, and their spherical shape isreconstructed accurately considering the limited tilt range,i.e. as somewhat oval shapes as seen from yz or xz projection.Examination of the xz slice shows some additional artifacts,especially a star-like structure in the direction of themaximum tilt directions. This artifact is typical for STEMtomograms and might be reduced in future reconstructionalgorithms, for example by application of compressedsensing approaches (Goris et al., 2012). Figure 8c shows aperspective rendering of the tomogram, with a transferfunction applied for coloration. An animation showing a360° rotation of the tomogram is shown in SupplementaryMovie 2. The total thickness of the sample as measured fromthe locations of lowest and highest nanoparticles andamounted to 516± 24 nm.

Supplementary Movie 2

Supplementary Movie 2 can be found online. Please visitjournals.cambridge.org/jid_MAM.

The required number of 120 iterations to reach anoptimal solution for TF-ART was considered as a highnumber compared with standard tomographic reconstruc-tion methods. One possible cause is that our back projectionoperator does not use an inverted PSF, so that the decon-volution of the projection images takes place implicitlyduring the tomographic reconstruction. Consequently, afuture reconstruction algorithm that uses an explicitlyinverted PSF might be expected to exhibit faster convergencecharacteristics.

Future research could further involve optimization ofTF-ART, for example, by implementing a more realistic PSFmodel for the forward projection using a 3D probe calculated

a

b

c

Figure 8. Comparison of the results of the reconstructions of thecombined tilt–focal series scanning transmission electron micro-scopy (STEM) data with the reconstruction of an STEM tilt series.a: Tomogram of the tilt–focal algebraic reconstruction technique(TF-ART) reconstruction from the combined tilt–focus seriesshown in Figure 6. b: Tomogram of the sequential algebraicreconstruction technique reconstruction from an STEM tilt seriesof a similar sample. The data were acquired using a tilt range of−37 to +37° in 4° increments, and with α = 2 mrad. The whitearrows mark the same nanoparticle in each of the three views.c: Perspective rendering of the tomogram of the TF-ART recon-struction with transfer function for coloration.

556 Tim Dahmen et al.

from measured aberrations (Lupini & de Jonge, 2011). ThePSF could also be defined as a function of the vertical posi-tion to account for beam broadening by elastic scattering(Ramachandra et al., 2013). The data presented here servesas proof of principle but the recording scheme could beoptimized toward highest axial resolution or toward bestcompromise between electron dose and resolution. Theoptimization also depends on the desired resolution range,for example, if one aims for sub-nanometer resolutionin samples ranging in thickness up to a few hundreds ofnanometers, or if thicker samples are being studied, forwhich a resolution of several nanometers is sufficient.Samples of several hundreds of nanometers thickness couldalso be imaged at higher tilt angles than 40° as was used here.The reconstruction of a dual-axis tilt- and focal series wouldbe a straight-forward extension of the technique. Discretereconstruction techniques can possibly be employed tofurther reduce the electron dose while maintaining spatialresolution (Batenburg et al., 2009).

STEM TomographyIn order to evaluate the TF-ART reconstruction, we com-pared the results with a conventional tomographic SARTreconstruction of an STEM tilt series of a similar sample(Baudoin et al., 2013) containing nanoparticles of similar butslightly smaller sizes (5 and 14 nm) and recorded using asmaller pixel size of 0.67 nm than the 2.3 nm used here.When comparing a combined tilt- and focal series to aconventional tilt series, the choice of the beam semi-angle isa crucial issue. STEM tilt series are best recorded with a verysmall beam opening semi-angle such as 2 mrad as proposedby Hohmann-Marriott et al. (2009). This is the case becausethe limited depth of field resulting from a larger beam semi-angle cannot be compensated and would lead to geometricblurring. On the other hand, in a combined tilt- and focalseries, information from different focal planes is used toenhance the tomogram. Consequently, a large beam semi-angle is favorable. We thus compared tomograms that wererecorded under conditions chosen for the respective method,i.e. we used 2 mrad for the tilt series and compared it to acombined tilt- and focal series recorded with 41 mrad beamsemi-angle. For the comparison we used a tilt series record-ing of a data set of whole cells containing LDL-coated goldnanoparticles (Baudoin et al., 2013). The sample was imagedwith STEM with a tilt range of 76° (−38° to +38°) in 4° tiltincrements and reconstructed using SART. The resultingtomogram is shown in Figure 8b.

Axial ResolutionOne main issue with tilt series-based tomograms that wewanted to improve on was axial elongation, i.e. the effect thattomograms have a lower resolution in the axial directionthan in lateral direction. This effect is typically quantified bythe axial elongation factor (eyz), which is defined as the ratioof axial resolution and lateral resolution. We followed this

approach and selected small nanoparticles in the tomogram.For each nanoparticle, the full-width at half-maximum(FWHM) in axial and lateral direction was measuredautomatically. The axial elongation factor was then definedas the ratio of those values. For the tilt series, we measuredeyz = 2.8± 0.5. For the combined tilt- and focal series wemeasured eyz = 2.2± 0.5. The experiment shows that acombined tilt- and focal series gives significantly(p< 7.0 × 10− 4) better axial elongation than a tilt series. Ourmeasurement for a tilt series differs from the literature,which reports eyz = 2.5 (Baudoin et al., 2013). We attributethis difference to the fact that in the literature one of thesmallest particles was selected manually while here werandomly selected a total of 17 small nanoparticles andcomputed the mean. However, even when comparing to thevalue reported in the literature, this combined tilt- and focalseries still results in an improved axial elongation.

Directional Dependence of Elongation FactorWe generalized the concept of the axial elongation factorstarting from a measurement of FWHM of a nanoparticle inlateral direction to a measurement of the elongation factor asa function of the angle between the viewing direction and thevertical axis. A second axis in the xz slice was chosen, thatalso intersected the center of the nanoparticle and formed anangle γ with the lateral axis. FWHMwas measured along thisaxis and we defined the angle-dependent elongation factor(exγ) as the ratio of those values. For γ = 0°, exγ is 1 bydefinition. For γ = 90°, exγ becomes the axial elongationfactor. A plot of exγ over γ is shown in Figure 9. It can be seenthat the tilt series has a larger elongation factor than thecombined tilt- and focal series around the axial direction(90°), which is in line with our earlier measurements of axialelongation factor. The elongation factor was up to 1.27 timeslarger for a combined tilt- and focal series between 110 and135° direction. This effect was present only on one side (theopposite range of 45–80° did not suffer from this effect), so itis presumably the result of an alignment error.

Frequency DomainTo further evaluate the information obtained from the combinedtilt- and focal series, two xz slices of the tomograms of the

Figure 9. Analysis of the gain of information in vertical direction.Plot showing the direction-dependent elongation factor (exγ) overthe angle to the lateral plane, compared between the combinedtilt- and focal series, and the tilt series.

Combined STEM Tilt- and Focal Series 557

combined tilt- and focal series, and of the tilt series weretransferred into the frequency domain (Fourier transform).In the case of the tilt series (see Fig. 10a), sharp streaks arepresent corresponding to the tilt directions. The “missingwedge” effect is also clearly visible, and the vertical directioncontains hardly any signal components. The informationobtained in the vertical direction is thus very limited. In thecase of the combined tilt- and focal series (see Fig. 10b), thestreaks corresponding to the tilt directions are less pro-nounced and spread over an angular region equal to thebeam-opening angle, so additional information is presentbetween the tilt directions. Note that the streaks in the tiltseries would be less pronounced for a data set containingmore tilt planes with a smaller tilt angle spacing. The missingwedge is still visible for the combined tilt- and focal series butin the central vertical region low spatial frequency signalcomponents are now present (white arrow). Thus, thecombined tilt- and focal series results in additional infor-mation in the axial direction compared with a pure tilt series.

CONCLUSIONS

Combined tilt- and focal series 3D data were recorded usingHAADF STEM. The sample was a whole-mount macro-phage cell containing gold nanoparticles coated with LDL.The reconstruction of tomograms from a combined tilt- andfocal series is not possible using standard methods. There-fore, we introduced a new reconstruction method calledTF-ART. It considered the convergent shape of the electronprobe by using a model consisting of a double cone. For theback projection, a new regularization factor was introducedto limit the correction for every image to the part of thevolume, where this image was best in focus. The combinedtilt- and focal series is an alternative for tilt series STEMtomography with an improved axial resolution for a giventilt range. The improved directional distribution of

information implies a more truthful representation of 3Dshapes, simplifying the interpretation and analysis of STEMtomograms.

ACKNOWLEDGMENTS

We thank Lukas Marsallek and Steve Pennycook fordiscussions, and Eduard Arzt for support through theLeibniz Institute for New Materials. Electron microscopywas performed at the SHaRE user facility at Oak RidgeNational Laboratory, sponsored by the Office of BasicEnergy Sciences, US Department of Energy, and at theKarlsruhe Nano Micro Facility, a Helmholtz research infra-structure at the Karlsruhe Institute of Technology. Thisresearch was supported by the US Department of Energy,Basic Energy Sciences, Material Sciences and EngineeringDivision, and by NIH grant R01-GM081801.

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