Electron lambda-tomographyEric Todd Quintoa,1, Ulf Skoglundb, and Ozan Öktemc,2

aDepartment of Mathematics, Tufts University, Medford, MA 02155; bDepartment of Cell and Molecular Biology, Karolinska Institutet, SE-171 77 Stockholm,Sweden; and cSidec AB, Torshamnsgatan 28A, SE-16440 Kista, Sweden

Edited by Roger D. Kornberg, Stanford University School of Medicine, Stanford, CA, and approved October 14, 2009 (received for review June 12, 2009)

Filtered back-projection and weighted back-projection have longbeen the methods of choice within the electron microscopy com-munity for reconstructing the structure of macromolecular assem-blies from electron tomography data. Here, we describe electronlambda-tomography, a reconstruction method that enjoys the ben-efits of the above mentioned methods, namely speed and ease ofimplementation, but also addresses some of their shortcomings. Inparticular, compared to these standard methods, electron lambda-tomography is less sensitive to artifacts that come from structuresoutside the region that is being reconstructed, and it can sharpenboundaries.

local tomography | limited angle tomography | microlocal analysis

T his article describes a reconstruction method applicableto electron tomography (ET). The rigorous mathematical

description of the method and its application to ET is given inref. 1. Here we concentrate on the functionality of the methodin an experimental setting with tests on real ET data. Further-more, we derive a heuristic explanation for its advantages andguidelines for its usage. Finally, we compare it with the mostwidely used methods in the field, namely filtered back-projection(FBP) and weighted back-projection (WBP). In this context, itshould be mentioned that other reconstruction methods havealso been developed and applied to ET. Iterative methods, suchas algebraic reconstruction technique (ART) and simultaneousiterative reconstruction technique (SIRT) (2, 3), became prac-tically applicable to ET only after regularization through earlystopping. A clever discretization, based on Kaiser–Bessel windowfunctions (blobs), was combined with strongly over-relaxed ARTand then applied to ET data in refs. 4–6. Another approach isbased on variational regularization where in refs. 7 and 8 relativeentropy regularization is applied to ET. For more on these otherapproaches and their merits, we refer to refs. 9, (section 10.2), 10,and 11.

We begin with a very brief introduction to ET, including adiscussion of the various data collection geometries and a mathe-matical formulation of the structure determination problem inET. This is followed by a brief outline of the FBP and WBPmethods. We then move on to our algorithm, electron lambda-tomography (ELT), which is based on two-dimensional lambdatomography (12–14). However, ELT is also valid for a broad rangeof three-dimensional data acquisition geometries. It is a methodthat maintains the main benefits of the FBP and WBP methods,namely speed and ease of implementation, while addressing someof the shortcomings. In particular, ELT is generally less sensitiveto artifacts that come from structures outside the region of interest(ROI) than these other methods. We conclude by providing exam-ples of reconstructions obtained by ELT from real and simulatedET data.

Basic Notation. We now introduce notation used throughout thepaper. We let R denote the set of real numbers and R

+ the setof positive real numbers. The three-dimensional space is denotedby R

3 and the unit sphere in R3, i.e. the set of all orientations in

three-dimensional space, is denoted by S2. Furthermore, “:=” inequations will mean “defined as.”

Next, given a function f defined in three-dimensional space, theprojection P(f ) of f is defined as

P(f )(ω, x) :=∫ ∞

−∞f (x + tω)dt. [1]

In the mathematics literature, P(f ) is called the X-ray transformof f . Note that when f is represented by its voxel values in three-dimensional space, then P(f )(ω, x) is essentially the sum of thevalues of f in the voxels that lie on the line through the point xthat has direction given by ω.

Finally, in some cases we choose to express formulae explicitlyin a specific coordinate system (x, y, z) in R

3. In such case we willmake use of the following convention: the x axis is parallel the tiltaxis and the z axis is parallel to the optical axis of the microscopeat 0◦ tilt-angle.

Electron TomographyData Collection Geometry. Many tomographic experimentalsetups, including ET, yield data recorded on a detector that attainsdifferent orientations with respect to the specimen whose inter-nal structure we seek to recover. In the case of ET, each recordedtransmission electron microscope (TEM) image is associated witha tilt angle which in turn uniquely specifies an orientation of thespecimen with respect to the optical axis of the TEM. Hence,the tilt angle can equally well be re-interpreted as an orienta-tion of the TEM detector with respect to the specimen. The datain a tilt series constitutes a series of TEM images where the tilt-angle lies on a curve S of directions in three-dimensional space.Below, we explicitly describe this curve for each of the standarddata acquisition geometries in ET.

Single-axis tilting. Here the specimen is rotated around a singleaxis perpendicular to the optical axis of the TEM. Then, the curveS is part of a longitude circle on the sphere, i.e. expressed in the(x, y, z) coordinates

S := {(0, sin(θ), cos(θ)) : −θmax ≤ θ ≤ θmax}. [2]

In Eq. 2, θmax corresponds to the largest tilt angle, which is ≈60◦.

Multiaxis tilting. In this case more than one single-axis tilt dataseries are taken. The curve S is given as the union of a number ofsingle axis curves (see Eq. 2) rotated around the z axis. Dual axistilting corresponds to the case where two single-axis data sets aretaken and fused in the above manner.

Slant tilting. For fixed 0 < α < π/2, the curve S is the set ofangles α radians from the vertical z axis. Hence, S is a latitudecircle of the sphere. To get such data, one places the specimen ina plane of angle π/2 − α from the electron beam and rotates thespecimen in that plane around a fixed point.

Author contributions: E.T.Q., U.S., and O.O. designed research; E.T.Q., U.S., and O.O. per-formed research; E.T.Q. and U.S. contributed new reagents/analytic tools; E.T.Q., U.S., andO.O. analyzed data; and E.T.Q. and O.O. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence may be addressed. E-mail: todd.quinto@tufts.edu.2Present address: Center for Industrial and Applied Mathematics, Royal Institute ofTechnology, SE-100 44 Stockholm, Sweden.

21842–21847 PNAS December 22, 2009 vol. 106 no. 51 www.pnas.org / cgi / doi / 10.1073 / pnas.0906391106

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The Model for Image Formation and the Reconstruction Problem. Theidea that data from a TEM image can be interpreted as a “projec-tion of the specimen” (15) underlies all current models for imageformation used in ET. This is valid under certain approximationswhich are discussed in e.g., refs. 1, 9, and 16. Given these approx-imations, a processed tilt series can be regarded as a finite sampleof projections P(f )(ω, x) which are given by the tilt angles ω con-tained in the curve S and x in the detector plane. Here, the functionf : R

3 → R is the “density” to be recovered and it is proportionalto the electrostatic potential, which in turn describes the structureof the molecules in the specimen (9, 17). Furthermore, we canrepresent the detector plane for tilt angle ω by

ω⊥ := {x ∈ R3 : x · ω = 0}

which is the plane perpendicular to ω through the origin (the“physical” detector plane is a translate of this plane). If weintroduce

MS := {(ω, x) ∈ S2 × R3 : ω ∈ S and x ∈ ω⊥}, [3]

then MS represents the collection of all lines with directions par-allel to vectors in S (for each ω ∈ S and x ∈ ω⊥, (ω, x) representsthe line parallel ω through x). Therefore, a tilt series will be a finitediscrete sampling on MS and the reconstruction problem in ETcan be stated as follows.

The reconstruction problem in ET. After suitable processing of thetilt series, the reconstruction problem in ET can be reformulatedas the problem of determining the real-valued function f , or someproperty thereof (e.g., shape of the structures), from a finite sam-pling of P(f ) on MS in which the directions are contained in thecurve S ⊂ S2 are given by the tilt angles.

ET is often compared to medical tomography. However, theET reconstruction problem has two important difficulties that donot arise in standard reconstruction problems in medical X-raytomography. First, since traditionally only a small portion of thespecimen is exposed to the electron beam, we are dealing with alocal tomography problem, i.e. the tomographic data originatesfrom a small ROI rather than from the entire specimen. This inturn means that, unless prior knowledge is being used as in ref.18, structures in the specimen can not be exactly recovered evenif one where to have access to noise-free continuum data (19). Inref. 20 this issue is referred to as the unit cell being only partiallydefined. Second, due to the limited angular range of the tilt-angle,all of the standard data acquisition geometries described aboveyield incomplete data in sense that the curve S yields tilt anglesthat do not circumscribe the specimen. This is known to introducesevere instability into the reconstruction problem.

To summarize, mathematically exact reconstruction for limitedangle region of interest data is not possible even in cases when onehas noise-free continuum of data, and the reconstruction problemis severely unstable. However, as we shall see in Theorem 1, cer-tain important features of the specimen can be reconstructed fromsuch data.

Established Analytic Reconstruction SchemesBoth FBP and WBP are well-established analytic reconstructionschemes frequently used within the ET community for solving thereconstruction problem in ET.

For tomographic data that satisfies a completeness conditionOrlov’s condition (ref. 10, section 6.1.1); it can be shown that FBPyields a smooth approximate version of f (see e.g., ref. 21, section2.34) if the data is complete in a sense that can be made precise.Now, in ET all the data collection geometries mentioned beforegive rise to data that are not complete (10), hence for such data,FBP will not even recover a smooth approximate version of f .

The WBP approach is the Fourier space formulation of theFBP approach. For the same reasons, WBP will not recover an

approximation of f from such ET data; the details are given in ref.10, section 6.1.1 (see also ref. 21, section 6.2.4).

Practical Considerations. In the practical usage of FBP and WBPon real ET data, one also has to consider how the reconstructionoperator is discretized and how to deal with noisy data.

First, in real applications of the FBP method, it is not desirableto attempt to recover the function exactly. The reason is simple;recovering f from finitely many samples of P(f ) on MS is an inher-ently unstable problem even if the data were complete in the senseof Orlov (10). Hence, a useful reconstruction method must includesome kind of regularizing step even in the case of noise-free data,and in the FBP method this is achieved by choosing the filter suchthat the recovered function is a smooth approximation to f . Thusan accurate and reliable recovery of f using the FBP scheme ispossible only for complete data, which for the aforementioneddata collection schemes for ET is only possible in special cases inwhich there is some intrinsic symmetry of the object.

In the particular case of single-axis tilting, each plane orthogo-nal to the tilt axis can be dealt with separately. This allows one toreduce the three-dimensional reconstruction problem to a stack oftwo-dimensional reconstruction problems. Each of these is, how-ever, a limited angle region of interest problem in the plane due tothe limited range of the tilt angle. Still, in the current usages of theFBP method given single-axis tilting data, one simply chooses thefilter in each slice as if one had complete data following the guide-line in refs. 19 and 22. In the electron microscopy community, thisguideline is frequently referred to as Crowther’s criterion (ref. 17,page 316). The reconstruction is also often low-pass-filtered inorder to further regularize the solution.

Electron Lambda-TomographyELT is an analytic reconstruction scheme. Therefore, it is based onthe same assumptions as the FBP/WBP methods, namely that onecan rephrase the reconstruction problem in ET as the problem ofrecovering a function f from projections P(f ) sampled on MS.

In ELT we don’t attempt to reconstruct f itself. Instead, wereconstruct a three-dimensional structure containing the infor-mation about f that can be stably retrieved, which turns out to becertain boundaries of the molecules. Furthermore, the recoveredthree-dimensional structure also shows what is inside and whatis outside these boundaries, see Theorem 1 for a more precisestatement.

The Reconstruction Operator. The ELT reconstruction operatorthat we consider is defined in ref. 1 and is designed for projectionssampled on MS. It reads as

LS,μ(f ) := P∗S

((−D2S + μ

)P

)(f ) [4]

and we now explain the meaning of the above expression. P∗S

denotes backprojection operator which is formally defined as

P∗S (g)(x) :=

∫S

g(ω, x − (x · ω)ω

)dω [5]

where g is a function in data space representing the projected data.Expressed in plain words, P∗

S (g)(x) is the sum of the data takenover all lines passing through x (since

(ω, x − (x · ω)ω

)represents

the line parallel ω and passing through x). The backprojection P∗S

is a fundamental part of the FBP and WBP methods as well asELT.

The derivative D2S is a second order differentiation in the

detector plane along the tangential direction to the curve S, i.e.

D2Sg

(ω, y

):= d2

ds2 g(ω, y + sσ

)∣∣∣s=0

[6]

where σ is the unit tangent to S at ω ∈ S. D2S is chosen in this

specific direction for mathematical reasons (1, 23).

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In contrast to the FBP method, LS,μ(f ) is not an approximationof f ; LS,μ(f ) replaces the filters in FBP/WBP by the simpler filter(−D2

S + μ).The microlocal regularity principle stated now describes which

object boundaries are visible from ET data and what boundariesare well-recovered by LS,μ(f ). For the mathematically preciseformulation and proof, see ref. 1.

Theorem 1. One can reconstruct a (molecular) boundary at a point xwhenever there is a projection in MS (the continuum data set) along aline (electron path) through x that is tangent to this boundary. More-over, such “visible” boundaries of f will be boundaries of LS,μ(f ).Finally, the recovery of such visible boundaries is mildly ill-posed soit is possible to stably detect them in practice.

This principle is illustrated our reconstructions in the Examplessection and we will discuss this in the last section of the article.

Note that LS,μ is useful for ET because the algorithm recon-structs two important types of features of f . The pure Lambdareconstruction term (derivative term),

P∗S

(−D2SP

)(f ), [7]

emphasizes differences in data that occur at boundaries. In otherwords, the pure Lambda term picks up visible boundaries as givenby Theorem 1. However, it does not distinguish interiors from exte-riors since the derivative in these areas is typically small. The purebackprojection reconstruction term (μ term),

P∗S (μP) (f ), [8]

is an averaged version of f . To see this, assume the value of f is large(resp. small) near a point x. Then the data on lines through x will,in general, be large (resp. small) and the μ term P∗

S (μP)(f ) will belarge (resp. small). Thus theμ term adds contour to the reconstruc-tion and allows one to distinguish objects from their surrounding.One can see this mathematically by noting that μP∗

SP(f )(x) is aconvolution of f with c/|x| on a cone representing the lines in thelimited data set (see refs. 1 and 23 for specific calculations). Thesum in Eq. 4 defining LS,μ(f ) will therefore sharpen boundariesand highlight interiors of objects, as we will show using simulatedand real data.

Practical Considerations. So far, we have introduced two impor-tant parameters: μ and the width of the derivative kernel. In theactual implementation of LS,μ in the ET setting, the D2

S operatorin Eq. 4 is evaluated using a filter that is a smoothed version of thesecond derivative (a smoothed central second difference), and thehalf-width of the filter is determined by the noise characteristicsof the data, the sensitivity of the detectors and the contrast of thespecimen as given in the discussion on “Reconstruction Protocol”below.

The other important parameter is μ. In ref. 14, a paradigm tochoose μ is given where a feature is selected and μ is chosenso that the LS,μ reconstruction is closest to flat inside a specificfeature. This is possible, in general, because the pure Lambdareconstruction term, Eq. 7 curves down inside regions and thepure backprojection term Eq. 8, term curves up. The first author’sstudent T. Bakhos has tested this, and it works well with low noisedata and as long as there is only one main region of interest in thereconstruction region. However, in ET, noise and sparsity of datasuggest that more ad hoc methods are better.

Because the ET data are so noisy, we convolve on the detec-tor plane in the direction perpendicular to σ (see Eq. 6), and thisbrings up a third parameter, the kernel width of this convolution,which for single-axis tilting is just averaging over slices. As with thederivative kernel, this width is correlated with the noise character-istics of the data. This corresponds to the normal way to regularizeFBP or WBP reconstruction. A starting point for setting the width

Fig. 1. A three-dimensional surface plot of the PSF in FBP (Left) and in ELTwith μ = 0 (Right) for single-axis tilting. The beam direction is vertical, andthe tilt axis is perpendicular to the plane of the picture. The maximum tiltangle is 60◦ from the beam direction. We see that the ELT PSF in the rightimage is more localized than the standard FBP PSF in the left image.

of these kernels could be calculated according to Crowther’s Cri-terion on each slice. This step can be done either slice by sliceor after the fact, in 3D, for the whole reconstruction. The recon-struction method is fast enough to allow extensive experiments tocalculate optimal parameters.

Advantages of ELTThe FBP/WBP methods have been extensively used over the yearsand they are well known to deliver a decent result when reasonableparameters are given. The improvement of ELT over FBP/WBPis nevertheless clear and based on the intrinsic differences of themethods. Mathematically, FBP and WBP require complete data toaccurately reconstruct, but ET data are not complete. ELT beinglocal, does not require complete data (although it will be difficultfor any algorithm to image the invisible singularities described inTheorem 1).

One can see the advantages of locality in several ways. If onecompares the convolution kernels for ELT and FBP as done infigure 1 of ref. 1. The ELT kernel is local—it is zero away fromthe origin, but the FBP kernel is not. In fact, the FBP kernel inthat figure has oscillations on the interval where the ELT kernelis zero that are ≈7% of the maximum amplitude. This illustratesthe fact that ELT needs only data through the ROI to reconstructthe structures but FBP needs all data on the detector plane (whichis not given in ET). Comparing point spread function (PSFs) inFig. 1, one sees the ELT reconstruction operator has a much morelocalized PSF than FBP. The “X” or wings at the end of the angularrange on both the FBP PSF and the ELT PSF are expected in anylimited angle backprojection algorithm. In addition, because thePSF is less localized in the FBP case, the signal spread out andaway from the real signal into the surrounding, causing a dilutionof the actual signal relative to the background. Thus there is ahigher chance to lose a weak signal altogether in the FBP case.

ExamplesIn this section we compare reconstructions obtained by FBP andELT. We start out with simulated two-dimensional data to illus-trate the properties of the ELT operator. Next, we move on to realET data.

Simulated Two-Dimensional Data. This experiment illustrates thefundamental properties of ELT. First, we compare ELT and FBPreconstructions of a single disk (the top two pictures in Fig. 2).Then, we compare ELT and FBP reconstructions for the samedisk plus two objects outside the ROI (the bottom two picturesin Fig. 2). This will show the effects of locality of the ELT opera-tor. The ROI is a disk of radius two units and the phantom in thetop pictures in Fig. 2 consists of one disk of radius 0.5 units cen-tered at the origin (see Fig. 2 for the data geometry). Parameters

21844 www.pnas.org / cgi / doi / 10.1073 / pnas.0906391106 Quinto et al.

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Fig. 2. Reconstructions of simulated data for one disk of radius 0.5 in a ROIthat is a disk of radius 2 from FBP (Left) and ELT (Right). In the top two pic-tures, there is nothing outside the ROI. In the bottom two pictures, there aretwo disks of radius 0.5 centered at (0, ±2.6) outside the region. The maxi-mum tilt angle is 60◦ from the z axis (vertical) and 120 tilts are used with 200projections per tilt. Note that the ELT reconstruction on the right highlightsboundaries and is less affected by objects outside the region than the FBPreconstruction.

are chosen to be essentially the same as the ones we used in ourexperiments with real data.

The FBP reconstruction in the top left picture of Fig. 2 showsvery dark triangular troughs to the left and to the right of the diskthat could mask details near the disk. Although the triangles arevisible in the ELT reconstruction in the bottom right picture inFig. 2, they are less pronounced than in the FBP reconstruction.

In the bottom two reconstructions in Fig. 2 we see the effects ofshadowing from far away as well as nearby densities. The recon-structions in those bottom pictures include two disks outside theROI of radius 0.5 centered at (2.6, 0) and (−2.6, 0) and density25 times that of the origin centered disk. These disks affect boththe ELT and FBP reconstructions since they affect data throughthe ROI. At the left and right boundaries of both reconstructionsare triangular troughs from these added disks, but the trianglesare much darker in the FBP reconstruction than the ELT recon-struction. To only have small troughs is important because deeptroughs might mask smaller nearby objects. Furthermore, thelambda boundaries of the center disk are better defined. Takentogether this leads to less disturbances in the structures themselvesin the ELT density.

Real ET data. We tested the behavior of ELT versus FBP on a realin vitro biological sample of tobacco mosaic virus (TMV) suppliedby J. Butler (MRC laboratory for molecular biology, Cambridge,UK). The rationale for choosing this sample was that the struc-ture of the TMV is well-known. The FBP algorithm used on thisdata is the optimized version used at the Department of Cell andMolecular Biology at the Karolinska Institute.

The TMV is made up of a helical cylinder with an outer diame-ter of ≈18 nm and an inner cylindrical hole along the helical axiswith a diameter of ≈4 nm (24, 25). The size and regularity of theTMV deems it a suitable sample for a simple comparison– even atthe rather low resolution inherent in this particular experiment at≈7.5 nm resolution. The contrast between the protein (and RNA)

mass and the uniform cylindrical nature of the hole are enough toenable the visualization of the hole despite it being smaller thanthe resolution of 7.5 nm.

Specimen preparation. The concentration of TMV particles was3 mg/mL and the colloidal gold markers added for alignmenthad a diameter of ≈10 nm. The gold markers were coated withBSA (Amersham AuroProbe EM protein A G10) and washedto remove unbound BSA. The specimen was placed on carbon-coated Quantifoil R2/2 grids that were glow discharged on bothsides. A Vitrobot was used for the vitrification process at 100%humidity with a 2 s blotting time. After vitrification, we obtaineda specimen consisting of TMV particles embedded in a slab ofvitrified aqueous buffer with a thickness slightly less than 115 nm.

TEM imaging protocol. The imaging was done at Sidec AB witha FEI Tecnai Polara with a FEG at 300 kV. The detector wasa Gatan UltraScan 1000 with a CCD of 14 μm pixel size giving2,048×2,048 pixels per image and a sensitivity of about five countsper electron. The magnification was calibrated to 19,830× giving apixel size after magnification of 0.5757 nm. The low-dose tilt serieswas collected at 10-μm underfocus following the single-axis tiltingscheme with 65 low-dose images collected at every second degree.The total dose used for the low-dose tilt series was 1230 e−/nm2,which gives an average dose of 18.92 e−/nm2 (or 6.27 e− per pixel)in one image in the low-dose tilt series. After the low-dose tiltseries had been recorded, a post-image was collected at consid-erable higher dose at 0◦ tilt-angle, but still at 10-μm underfocus,enabling a better view of the imaged area of the specimen.

TEM images of the TMV sample are shown in Fig. 3. The200×200 pixel square region insert in Fig. 3 is overcasting an area

Fig. 3. TEM images of an in vitro TMV specimen. The large region is theentire 2,048 × 2,048 pixel high-dose TEM image taken at the 0◦ tilt anglewith a 10-μm underfocus and recorded after the tilt series has been collected(post-image). The small 200 × 200 pixel highlighted square slightly to theleft shows the projection of the ROI onto the post-image. The reconstruc-tions of the ROI are shown in Figs. 4 and 5. The upper right-hand square is a4× enlargement of the previously mentioned small square. Finally, the lowerright-hand square is the projection of the ROI onto the low-dose image fromthe tilt series corresponding to the 0◦ tilt angle. This is one of the 65 TEMimages that constitute the single axis tilt series. The 10-nm gold particles arevisible as black spots surrounded by a whitish “halo” caused by the largeunderfocus.

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slightly to the left in the post-image. This square region shows theprojection of the ROI that is reconstructed. It contains two verti-cal and five horizontal TMV particles crossing the ROI and thatwe expect to see in the ensuing 3D reconstructions. The reasonfor choosing the ROI in such an area is to see how the ELT andFBP reconstructions are compromised in a crowded region. Inaddition we have an enlarged version of the ROI insert at the topright and lower right. The lower right shows the same area fromthe low-dose not-tilted specimen with 200 e−/nm2. The whole Fig.3 image spans 2,048 × 2,048 pixels with 0.5757 nm pixel size, thuscovering 1,179 nm in each direction.

Data pre-processing. The images in the low-dose tilt series werealigned using the colloidal gold markers. The average alignmenterror over all the tilts was <0.75 nm (i.e. about 1.3 pixels). From thealigned images we then extracted a data set which in turn servedas input data for the 3D reconstructions of the ROI. The ROI hasa size of 200 × 200 × 200 voxels and it was calculated from animage support of 300 × 300 pixels from the aligned images in thelow-dose tilt series.

Reconstruction protocol. Crowther’s criterion applied to this datadictates using a reconstruction kernel with a half-width of 4nm in FBP. In order to compensate for the loss in resolutiondue to missing data, following ref. 26, the kernel half-width isincreased by ≈50%. Furthermore, the low signal-to-noise-ratio indata motivates further increase in the kernel half-width to 7.5 nm.

For ELT, the half-widths of the convolution kernel representingD2

S in Eq. 4 (the derivative kernel) and the convolution represent-ing the averaging over slices (slice averaging kernel) are related tothe signal-to-noise-ratio in the data. This in turn is related to thesensitivity of the detector, the contrast in the specimen, andthe dose. In general one can use larger half-widths to smooththe reconstruction more when the signal-to-noise ratio in thedata decreases. To prescribe a specific protocol for choosing thehalf-width would require extensive testing on a large number ofspecimens. However, it turns out that the half-widths can be cho-sen close to the above value of 7.5 nm used for the FBP. We canhowever account for the anisotropy in resolution caused by lim-ited data and we settle for a half-width of 6 nm for the derivativekernel and 7 nm for the slice averaging kernel.

The TEM data of TMV is represented in Figs. 3. Note that itis not easy to see the centrally placed hole even in the high-dosepost-image shown in Fig. 3. This is due partly to induced beamdamages and partly due to the large defocus-induced extra con-trast shadowing the fainter low density hole. The hole is howevervisible in the 3D reconstructions.

Fig. 4 shows a planar cross-section of the FBP and ELT recon-structions of the TMV. In this figure, the emphasis is put on thetwo TMV particles that can be seen passing vertically throughthe ROI insert in Fig. 3. The two virions pass on top of the five

Fig. 4. Displays of a 1.15-nm-thick slice through the central part of thevirions in the FBP (Left) and ELT (Right) reconstructions. Emphasis is put onthe two TMV particles that pass vertically through the ROI insert in Fig. 3.Although the cylindrical hole is clear in both FBP and the ELT tomograms,there is less clutter in the ELT reconstruction. The maximum tilt angle is 60◦.

others, and in Fig. 4 we show a 1.15-nm-thick slice through thetomograms showing the central part of the virions. The diameterof the TMV particle can be estimated to be just below 18 nm forboth virions (the side of the square image is 200 pixel, or 115 nm),and the hole between 4.5 and 5 nm. These values agree with thepublished structural parameters for TMV (24, 25) and are usedas benchmarks to justify our choices of half-widths described inthe reconstruction protocol above. Although the cylindrical hole isvisible in both FBP (left) and the ELT reconstructions, the bound-aries are more prominent in ELT. Furthermore, it is clear in Fig.4 that the FBP density is more cluttered than the ELT one, andthat e.g. the virion is more connected in the ELT density.

In Fig. 5 we show a 200×200×160 voxel sub-region of the ROIreconstructed by FBP and ELT. The sub-region is viewed alongthe beam direction having the tilt-axis horizontal. The thresholdsused for contouring the two tomograms have each been selectedso that the structural parameters (outer diameter and diameter ofthe inner cylindrical hole) related to the virions in the tomogramsagree with the published ones. Still, it’s clear that the ELT tomo-gram is easier to interpret due to the obviously lower clutter-level.The virions are also more disconnected in the FBP tomogramwhen this threshold is used for contouring. If we decrease thethreshold level, the background clutter makes the analysis diffi-cult. One can also see the increased clutter in FBP versus ELTwhen one rotates the 3D reconstructions even though the FBPreconstructions have been optimized. If the FBP reconstructionsare smoothed more, boundaries fade; ELT can smooth more andstill show boundaries since boundaries are emphasized in ELT.

Application of Theorem 1 and ConclusionsThe principle in Theorem 1 is that boundaries tangent to lines inthe data set are easier to reconstruct than boundaries not tangentto any line in the data set. This is illustrated in each of our recon-structions above. In the simulated reconstructions, the angularrange of tilts is ±60◦ from the z axis. Therefore, according to thisprinciple, the visible boundaries will be those on the left and rightsides of the circles, and these are exactly the boundaries that aremost clearly visible in both reconstructions in Fig. 2. Neither FBPnor ELT show the other, invisible, boundaries particularly well.

One can see from each of the reconstructions how the ELTreconstructions highlights the visible boundaries compared withthe FBP reconstructions. For example, the thin cylinder wall ofthe TMV is more emphasized in the ELT reconstruction in Fig. 4than in the FBP.

In summary, our reconstructions show that ELT emphasizesvisible boundaries, even with noisy data. This allows one to

Fig. 5. Volumetric displays of the FBP (Left) and ELT (Right) reconstructionsof TMV. Shown is a 200 × 200 × 160 voxel sub-region of the FBP and ELTreconstruction of the ROI. The sub-region is viewed along the beam direc-tion having the tilt axis horizontal. The thresholds used for contouring thetwo tomograms have each been selected so that the outer diameter of thevirions agree with the known value. It’s clear that the ELT tomogram is easierto interpret due to the obviously lower clutter-level and that the virions aremore disconnected in the FBP tomogram.

21846 www.pnas.org / cgi / doi / 10.1073 / pnas.0906391106 Quinto et al.

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smooth more to get rid of clutter. ELT is local so objects out-side the ROI affect reconstructions less than in FBP. Finally,ELT is straightforward to adapt to many data acquisitiongeometries (1, 23).

ACKNOWLEDGMENTS. We thank Sara Sandin and Lars-Göran Öfverstedt foracquiring the TMV tilt-series data and Wenner Gren Stiftelserna and Sidec,Stockholm, for support that has greatly aided this research. E.T.Q. was sup-ported by National Science Foundation Grants 0456858 and 0908015 and aTufts University Faculty Research Award.

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