Abstract
Electron tomography is widely used for nanoscale determination of 3D structures in many areas of science. Determining the 3D structure of a sample from electron tomography involves three major steps: acquisition of sequence of 2D projection images of the sample with the electron microscope, alignment of the images to a common coordinate system, and 3D reconstruction and segmentation of the sample from the aligned image data. The resolution of the 3D reconstruction is directly influenced by the accuracy of the alignment, and therefore, it is crucial to have a robust and dependable alignment method. In this paper, we develop a new alignment method which avoids the use of markers and instead traces the computed paths of many identifiable ‘local’ centerofmass points as the sample is rotated. Compared with traditional correlation schemes, the alignment method presented here is resistant to cumulative error observed from correlation techniques, has very rigorous mathematical justification, and is very robust since many points and paths are used, all of which inevitably improves the quality of the reconstruction and confidence in the scientific results.
Background
Electron tomography has been a powerful tool in determining 3D structures and characterization of nanoparticles in the biological, medical, and materials sciences [13]. The method is carried out by acquiring a series of 2D projection images of an object and then using these 2D projections to reconstruct the 3D object. Using the transmission electron microscope, these projections are collected at a number of different orientations, typically by tilting the sample about a fixed tilt axis [4], while other dual axis tilting schemes also exist [5]. A demonstration of the projection scheme is shown for a 2D object in Figure 1. We will focus only on the case of a single fixed tilt axis in this paper, although our methods can easily be translated to dual axis schemes.
Ideally, between two consecutive projections acquired at nearby tilts of the sample, one would observe only a small rotation of the projected image. However, due to unavoidable mechanical limitations, significant translation shifts are present. Therefore, the projections must be aligned into a common coordinate system to be properly interpreted. Once the projections are aligned, they can then be merged to approximate the 3D structure of the sample. The alignment is a crucial part of the process, for the resolution of the reconstructed 3D structures are limited to the accuracy in the alignment. In this paper, we demonstrate a new mathematically justified method for the alignment based on the apparent motion of the center of mass of many 2D crosssections of the sample.
Over the years, many traditional alignment techniques have been developed by the biological sciences [6]. The most commonly practiced are correlation techniques, feature tracking, and fiducial marker tracking. Correlation techniques are performed by selecting one of the projections as a reference image and aligning each pair neighboring images by selecting the crosscorrelation peak between the images for the shift [7]. This method has been proven useful but can yield poor results, as small cumulative errors may result in a serious drift of the sample [8]. As we will show, crosscorrelation will not recover the correct alignment even for noisefree data subjected to random shifts. The current work finds a solution without this deficiency.
Fiducial marker tracking is done by decorating the sample with small highdensity particles that create high contrast in the projection images [912]. Individual markers are then identified in all projections. The alignment is determined based on tracking of the path of each marker through the projections. This method can be very accurate but requires a lot of manual interaction to properly locate and center the markers. The main drawback of marker tracking is that the markers will be present in the reconstruction and must be removed for accurate characterization of the sample. Since the markers are of such high density, the reconstruction of the markers will inevitably mix with the reconstruction, making the task of removal nontrivial and possibly inaccurate.
Feature tracking uses regions of high contrast or intensity as fiducial markers [13,14]. It requires the identification of suitable regions of high contrast that remain visible throughout the tilt series.
Others have begun to perform alignment techniques based on a refinement approach [6]. After a coarse alignment from crosscorrelation, one proceeds in computing an initial 3D reconstruction. This 3D reconstruction is then reprojected and compared with the original projections. A new alignment arises from aligning the reprojected reconstruction with the original projections, and this process is iterated until convergence is met. In our experience with this method, the reconstruction always satisfies the projections, even if they’re misaligned, so that insignificant refinement occurs from updating.
Most recently, Scott et al. [15] introduced a technique based on the observation that as the sample is tilted about a fixed axis, the center of mass of the sample will spin in a circle, and if the center of mass is on the tilt axis, then it remains fixed. In this way, it was suggested to shift each projection so that the center of mass in each projection is fixed on a point and taking the line through this point parallel to the axis of rotation as the tilt axis. We believe this is not always applicable and can yield poor results in many settings. First, it requires a tilt series in which the total projected volume is fixed for each projection. However, in most practical settings, some mass will move in and out of the projection range as the sample is tilted, which will then significantly affect the location of the center of mass within the projection along both axes of the projection images. This transition of mass must be accounted for, as this transition will be along the edges of the projections, far from the center, and will thus weigh heavily on the calculated center of mass. Figure 1 demonstrates this transition of mass, with the small ball located on the left edge of the object that has only been projected at certain angles. An additional drawback is that using only the single center of mass point in each projection removes the use of any local structure of the projections as criteria for alignment.
In this paper, we give an alignment method that makes more detailed use of the path of the projected center of mass along many crosssections of the object, perpendicular to the axis of rotation. In an ideal experiment, points on the sample move in circular trajectories. We define a viable path as the projection of such a circular orbit. By simple calculation, we derive an equation which describes all such viable paths of the projected centers of mass, as opposed to the one trivial path of a single point. From here, we show how one can determine a shift for each projection so that the center of mass of all crosssections perpendicular to the axis of rotation nearly follows a viable path. In this way, since all crosssections are considered in our alignment method, we will be able to avoid problems involved with error in the calculated centers of mass due to transition of volume in and out of the projections, and we maintain local analysis of the projections as means for the alignment. Additionally, our model aligns the projections based on the rotation about a chosen axis, so that manual interaction for determining the positioning of the tilt axis is avoided. In general, our method can be considered more statistically accurate, and we will show that it provides very dependable alignment and definitively improves the resolution of the reconstruction.
Methods
Notation
The 3D density function for reconstruction will be denoted f(x,y,z)=f(x,(y z)), with (y z) a 2D row vector. The data generated are the projections of f in the zaxis, about rotations around the xaxis. A rotation of f through θ about the xaxis can be written as:
A projection about the rotation θ is then defined as:
We note that for each fixed x=x _{0}, P _{ θ }(f)(x _{0},y) only contains information from f(x _{0},y,z), and therefore, many of the alignment and reconstruction processes can be considered as 2D rather than 3D. Therefore, for convenience, we will sometimes denote:
In practice, we are given the unaligned data; therefore, we will regularly refer to the misaligned projections, denoted by \( {\overset{\sim }{P}}_{\theta }(f) \). We define these projections as:
where the coordinates (x _{ θ },y _{ θ }) are the shifts to be determined for the alignment. Similarly, we will denote:
where in this instance the shift x _{ θ } is not included. We do not include it, for determining the shifts x _{ θ } is a much more trivial task, so that most of our work here focuses on determining y _{ θ } after the xaxis alignment is completed.
We will denote the total mass about a crosssection x by \( {M}_x=\underset{{\mathbb{R}}^2}{\int }{f}_x\left(y,z\right)\kern1em dy\kern1em dz \). Then, the coordinates for the center of mass of a crosssection are denoted as:
We will denote the center of mass of a projected crosssection of f by:
We take the conventional L _{ p } norm (denoted by ∥·∥_{ p }) of a function, say g, defined over ℝ ^{n} to be:
Similarly, for a vector x∈ℝ ^{n}, we take the ℓ _{ p } norm (denoted ∥·∥_{ p }) to be:
Theoretical model
In practice, we are given the set of misaligned angular projections:
Typically, the number of projections, k, can be from 50 to 200, with maximum tilts of ± 70°. The domain is of course limited, but for theoretical purposes, we will assume that the domain for y is all of ℝ. The problem is then to approximate the set of shifts \( \left({x}_{\theta_i},{y}_{\theta_i}\right) \) for alignment, so that \( {\left\{{\overset{\sim }{P}}_{\theta_i}(f)\left(x,y\right)\right\}}_{i=1}^k \) correspond to the aligned projections \( {\left\{{P}_{\theta_i}(f)\left(x,y\right)\right\}}_{i=1}^k \). Determining the shifts for the xaxis is much simpler, since the xaxis is the axis of rotation. We simply observe that the total mass in each crosssection should remain fixed, so that:
Based on this simple observation, one should be able to approximate all shifts \( {x}_{\theta_i} \) based on a ‘conservation of mass’ approach. We design a ‘global’ alignment method for determining these shifts, by taking \( {x}_{\theta_i} \) to be the shift which minimizes the difference between the observed mass in each crosssection of \( {\overset{\sim }{P}}_{\theta_i}(f)\left(x{x}_{\theta_i},y\right) \) and the average mass of all projections in each crosssection. More precisely, we let:
Of course, the averaged term, \( \frac{1}{k}\sum_{l=1}^k\left(\underset{\mathbb{R}}{\int }{\overset{\sim }{P}}_{\theta_l}\right.\left.(f)\left(x,y\right)\kern1em dy\right) \), is subject to error since the projections are not yet aligned, so the determination of each \( {x}_{\theta_i} \) is iterated a few times until there is no change. The number of iterations will depend on just how large the offset of the projections are, but we have typically observed no change in each \( {x}_{\theta_i} \) after just two iterations. A demonstration of this xaxis alignment is given in Figure 2.
One could also perform a similar ‘local’ method, by comparing the consecutive projections to each other instead of the average. This approach is subject to cumulative error in the alignment similar to crosscorrelation; therefore, we avoid this approach.
From here forth, we will now assume that the \( {x}_{\theta_i} \) have been accurately determined, and consider each crosssection. For alignment along the yaxis, we again want to make use of physical properties. It has been noted, as f _{ x }(y,z) is rotated about the origin, the center of mass \( \left({c}_x^y,{c}_x^z\right) \) will spin in a circle around the origin. It is not immediately clear, however, how this property can be observed within the projections and used for alignment. Computing the center of mass of a projected slice, we obtain:
where we applied the substitution \( \left(\alpha \kern1em \beta \right):=\left(y\kern1em z\right){Q}_{\theta_i} \). This tells us that the center of mass of each projected crosssection should follow the path given by:
This equation gives us a local relationship between the relative positioning of all of the projections to use for the alignment. As discussed earlier, in [15], it was simply noted that if the center of mass is located at the origin on the tilt axis, then it does not move under rotations about that axis. This observation can be made through similar computations where the integrand is first taken over x, and then, the center of mass is computed for the total sum of the crosssections, that is:
where c ^{y} and c ^{z} here denote the centerofmass coordinates along the y and zaxes, respectively, independent of x, and M denotes the total mass of f. Therefore, it is suggested to shift each projection so that \( {t}^{\theta_i}=0 \) for all i, so that c ^{y}=c ^{z}=0. While this approach is theoretically sound in an ideal setting, summing over x immediately removes any consideration of local behavior of the projections of f. As we will show, in many settings, this simplification can be a major drawback.
Therefore, our approach is to determine a sequence of shifts so that for each crosssection there exists some deterministic center of mass \( \left({c}_x^y,{c}_x^z\right) \) so that Equation 3 is nearly satisfied. With this in mind, let us denote:
We note that from the acquired projection data we can compute both Θ and t _{ x }. Now from Equation 3, if our alignment is good, then for each crosssection x, there should exist some c _{ x } so that Θ c _{ x }≈t _{ x }. Therefore, in order to yield a good alignment, we would like to determine:
so that there exist some c _{ x } satisfying:
or equivalently:
In practice, we will have some finite number of crosssections, say x _{ j }, for j=1,2,…n. Then, we would like solve the minimization problem:
Now we can compute the minimization over c _{ x } directly. Given Θ and t _{ x }, the least square solution \( {c}_x^{\ast } \), to \( \parallel \varTheta {c}_x\left({t}_x+{y}_{\varTheta}\right){\parallel}_2^2\kern0.3em \) :
can simply be found by differentiation so that:
Solving these equations, the solution can be found to be:
where Θ ^{+} denotes the pseudoinverse of Θ, given by Θ ^{+}=(Θ ^{T} Θ)^{−1} Θ. It should be noted that Θ ^{T} Θ is a 2×2 matrix with entries:
which is clearly invertible and without any notable computational cost.
Then, our minimization in Equation 5 becomes:
If we let:
then the minimization problem in Equation 6 is equivalent to solving a standard least squares problem:
Practical implementation
The major consideration that we have ignored so far in the theoretical development but will handle in this section is that certainly the domain for y for \( {\overset{\sim }{P}}_{\theta_i}\left({f}_x\right)(y) \) is finite, say [−m,m]. As before with x, for all practical purposes, we will now additionally consider the yaxis to be discrete, and for each projection \( {P}_{\theta_i}(f)\left(x,y\right) \), the domain is given as:
We chose the indexing for y symmetrically for convenience in the centerofmass computations so that the center of the projections is along the modeled axis of rotation at y=0. Computing \( {t}_x^{\theta_i} \) now becomes:
The first issue is that M _{ x } may vary through the tilt series for each crosssection; in particular, since the domain for y is limited, there may be some observable mass moving in and out of the field of view after rotation and projection, as we demonstrated in Figure 1. This is again why it’s important that we choose the alignment to be considered over many projected crosssections.
To handle this issue, we multiply \( {\overset{\sim }{P}}_{\theta_i}(f)\left(x,y\right) \) by a window function, \( {\omega}_{\theta_i}\left(x,y\right) \), in the computation of \( {t}_x^{\theta_i} \) in order to alleviate some of this transition of mass in and out of the frame. The window function allows for the balance of the total mass within each projection. We choose our window functions to satisfy the following properties:

(i)
\( 0\le {\omega}_{\theta_i}\left(x,y\right)\le 1; \)

(ii)
\( M=\sum_{x=1}^n\sum_{y=m}^m{P}_{\theta_i}(f)\left(x,y\right){\omega}_{\theta_i}\left(x,y\right) \), for i=1,2,…,k;

(iii)
\( {\omega}_{\theta_i}\left(x,y\right)\le {\omega}_{\theta_i}\left(x,y+1\right)\kern1em \mathrm{if}\kern1em y<0, \)
\( {\omega}_{\theta_i}\left(x,y\right)\ge {\omega}_{\theta_i}\left(x,y+1\right)\kern1em \mathrm{if}\kern1em y\ge 0; \)

(iiii)
\( {\omega}_{\theta_i}\left(x,y\right)={\omega}_{\theta_i}\left(x+1,y\right) \), for x=1,2,…,n−1.
The first property simply emphasizes that multiplication of \( {\overset{\sim }{P}}_{\theta_i}(f) \) by \( {\omega}_{\theta_i} \) reweighs the projection values in order to dampen the introduction of new mass in to the frames. The second property then tells us that this dampening of the values of \( {P}_{\theta_i}(f) \) by multiplication of \( {\omega}_{\theta_i} \) yields the same total mass in each projection. Finally, properties (iii) and (iiii) describe how this dampening should be done. Property (iii) says that the window function decreases as the function moves away from the yaxis. This is because new mass would be introduced along the edge of the plane of view, so that we dampen these values more significantly. Property (iiii) is an additional property to help us better characterize \( {\omega}_{\theta_i} \) in a simple manner and simply says that we place the same weight for each crosssection x. One could remove property (iiii) and change property (ii) so that instead the mass M _{ x } is fixed for each crosssection of each projection. This could potentially cause bias in the alignment of the crosssections, especially ones with considerable noise, and it would require much greater computational time to determine a window for each crosssection of each projection.
After the windowing function is determined, we then compute the center of mass for each projected crosssection \( {t}_{x_j} \), for j=1,2,…,n as:
and solve a variant of Equation 6. The variation is that we only choose to minimize only a subset of the crosssections, say T⊂{1,2,…,n}. This subset is chosen so that the selected crosssections have a significant quantity of mass in each projection so that introduction of new mass along the edges has considerably less effect on the center of mass of this projected crosssection area. In addition, we only choose those in which the observable total mass within that crosssection varies little throughout all projections, to again avoid the crosssections with large transition of mass.
More precisely, we pick the crosssections in which the ratio of the average observed mass through the projections to the variance of the mass in the projections is above some specified tolerance. This tolerance can be chosen based upon quality of the data. Finally, the minimization for determining the shifts becomes:
which can again be converted into a standard least squares minimization problem as done in Equation 7. We summarize the method with the simple schematic shown in Figure 3.
Reconstruction method
After the alignment, for the reconstruction, we use a compressed sensing approach by total variation (TV) minimization [16]. These methods have recently been gaining popularity for electron tomographic reconstructions [1719]. In order to briefly describe the method, let us denote the 3D reconstructed approximation of f by \( g={\left\{{g}_{x,y,z}\right\}}_{x,y,z=1}^N \), where for simplicity we now let our discrete 3D domain be:
Most reconstruction methods are then designed so that numerical reprojection of g agrees with the experimental projections \( {P}_{\theta_i}(f) \), for i=1,2,…,k. In particular, reconstruction techniques typically minimize the distance between the projections of g and the experimental projections, sometimes called the projection error. This projection error can be expressed as:
However, simple minimization of the projection error does not necessarily produce optimal results in the presence of noise. Therefore, methods, such as TV minimization, additionally apply regularization conditions on the reconstruction. In the case that our sample consists of homogeneous materials and relatively smooth surfaces, compressivesensing theory allows us to assume that the reconstruction should have a small total variation norm, given by:
With this in mind, we would like for Equation 9 to be relatively small, while also applying a penalty on ∥g∥_{ TV } for noise reduction, so that our method solves:
Results and discussion
We will give the results for experimental and simulation data. We compare the reconstructions from alignment using crosscorrelation and our centerofmass technique, while also demonstrating the advantage of using many slices for the centerofmass alignment, as opposed to just one centerofmass calculation.
Experimental results
For the experimental data, we have an alumina particle sitting on a holey carbon grid. The sample was prepared by grinding the alumina spheres into powder. A suspension of the powder is prepared in ethanol and sonicated for 5 min. The suspension was then added dropwise over the lacey carbon film supported on 200 mesh Cu TEM grids (Structure Probe, Inc., West Chester, PA, USA) and dried at room temperature. The sample is analyzed using the FEI Titan 80300 Scanning Transmission Electron Microscope equipped with a sphericalaberration probecorrector (CEOS GmbH, Heidelberg, Germany) operating at 200 kV. The images were collected using the highangle annular detector with the camera length of 195 mm and at 80,000 X magnification. The acquisition time was set to 15 s over an image area of 1024 X 1024 pixels resulting in a pixel size of 0.2411 nm. The tilt series is collected using linear tilt scheme continuously from 70° to +70°with tilt increments of 2°. Dynamic STEM focus function is used to compensate for change in focus across the image. The projection of the sample at 30°degrees is shown in Figure 2, and the aligned projections are shown in a video in Additional file 1.
Total variation minimization is valid for this data set, as the alumina particle and the carbon grid are known to be uniform in density. In addition, regularization of the reconstruction with TV minimization is critical to the quality of the results due to the lowdose sampling conditions necessary for acquisition of the projections due to beam sensitivity of the material. The reconstructed images from crosscorrelation and our alignment methods are shown in Figure 4. While the overall particle morphologies are similar, the reconstruction resulting from our alignment displays much more uniform densities and clearer particle structures. This will result in more confident segmentation and characterization of the reconstructed particle, which is crucial to the interpretations of the experiment. In the 3D images (visualized using tomviz software [20]), the overall structures appear similar. However, less rigid particle structure is recovered with the crosscorrelation alignment, as the red glow around the particle demonstrates blurring from the main particle structure to a lower gray level represented by red in the colormap. In Figure 5, we plotted the centers of mass, t _{ x }, for two crosssections. Plotted together with t _{ x } are least squares solutions of the center of mass, \( \left({c}_x^y,{c}_x^z\right) \), based Equation 3 given the computed t _{ x }. It is evident that our method finds a nearly viable path for the motion of the center of mass, as we set out to do. On the other hand, the alignment from crosscorrelation clearly fails to do so, resulting in lowresolution reconstructions.
In Figure 6, additional results are given using the alignment method described in [15]. Again the 3D visual comparison of the reconstructions show that our alignment has produced a more rigid structure, as there is less red glow from the main particle but less significant than the results from crosscorrelation. Similarly, the images in Figure 6c,d,e,f of the 2D crosssections show a more rigid structure and less noisy artifacts due to misalignment. The plots in Figure 6 give a quantitative comparison of the alignment approaches. In Figure 6g,h, the location of the global projected center of mass along the yaxis is shown for the two methods. The plot in Figure 6g shows the only consideration for the originally proposed centerofmass alignment, as the center of mass in the projections along the yaxis is shifted to the tilt axis. With pixelation of the images, there is still a small negligible distance (less than half a pixel) between the center of mass in each projection and the tilt axis. The location of this center of mass resulting from our approach is shown in Figure 6h and does not necessarily follow a viable path, because we choose a different minimization and allow our approach to avoid problematic crosssections. In Figure 6i,j, the path of the projected center of mass is shown for a single crosssection for the two alignment methods, where, for this crosssection, our methods demonstrate a viable path and the approach based on the single global center of mass does not. Inevitably, our method produces better reconstruction results, demonstrating that a more sophisticated alignment approach should be taken for dependable results as we have done, taking into account not one single data point but rather all crosssections as unique data points. The resulting segmentation of the alumina particle is shown in 3D in a video in the Additional file 2.
Simulation results
As a numerical test, we reconstructed simulated data by projecting a discrete 3D volume with binary intensities at the same tilt angles as the experimental data: a maximum tilt range of ± 70 °in 2°angle increments. We align the projection images according to the various alignment methods, and each realigned set of projections is reconstructed again using TV minimization. The results from the simulations are shown in Figure 7. The total projected volume shows little variation depending on the tilt angle, with the exception of a small mass appearing in the projection range at hightilt angles. This is indicated in the projection images shown in Figure 7a,b, where, in Figure 7a, the bundle of mass is located towards the upper right of the projection image, and in Figure 7b, this bundle of mass has nearly moved completely out of the projecting range. With the special example we have here, this small transition of mass will significantly affect the results of an alignment approach such as in [15]. This is very clear from the resulting blurry reconstruction in Figure 7e that does not resemble a binary reconstruction. In addition, it can be seen in Figure 7d that even in this noisefree simulation crosscorrelation also produces very poor results simply because the model is not appropriate. In Figure 7c, it is seen that our centerofmass approach still yields optimal results displaying a near binary reconstruction image that almost completely resembles the original phantom not presented in the figure. The adaptability of our method to choose only the appropriate crosssections with little variability of mass is clearly advantageous as demonstrated in these simulations.
Conclusions
Our method has a sound physical basis: the movement of the center of mass in each crosssection. By selecting shifts for individual tiltseries images that globally lead to physically plausible motions for the centers of mass of many crosssections, our method effectively utilizes the assumption that the sample object is rigid to improve the alignment and the resolution of the final reconstruction. We have shown that conventional alignment procedures, which shift the global center of mass to the origin, may not produce physically plausible motions in other crosssections. We have generalized these methods in a computationally feasible manner that can be easily be incorporated into electron tomography workflows. We have demonstrated the significance of such consistency between crosssections and the effectiveness of the presented method by improving the resolution of 3D reconstructions of simulated and actual data.
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Acknowledgements
The authors would like to thank Dr. Ilke Arslan for her helpful discussions. This research was supported in part by NSF grant DMS 1222390. It was also funded by the Laboratory Directed Research and Development program at Pacific Northwest National Laboratory, under contract DEAC0576RL01830.
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Authors’ contributions
TS derived the alignment methods and algorithms. TS and MP analyzed the technical issues of the methods and algorithms. PB assisted in the analysis of the methods and supervised the research. CA generated the tomography data and analyzed the quality of the reconstructions. TS created the simulated tomography data. TS performed the alignment and reconstruction algorithms and performed the analysis. TS drafted the manuscript. TS and MP revised the manuscript, and all authors discussed it. All authors read and approved the final manuscript.
Additional files
Additional file 1
Video that shows the sequence of aligned projection images of the alumina particle using the method proposed in this paper.
Additional file 2
Video that shows the reconstructed alumina particle in 3D.
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Sanders, T., Prange, M., Akatay, C. et al. Physically motivated global alignment method for electron tomography. Adv Struct Chem Imag 1, 4 (2015). https://doi.org/10.1186/s4067901500057
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DOI: https://doi.org/10.1186/s4067901500057