Correction of Missing-Wedge Artifacts in Filamentous Tomograms by Template-Based Constrained Deconvolution

Abstract

Cryo-electron tomography maps often exhibit considerable noise and anisotropic resolution, due to the low-dose requirements and the missing wedge in Fourier space. These spurious features are visually unappealing and, more importantly, prevent an automated segmentation of geometric shapes, requiring a subjective and labor-intensive manual tracing. We developed a novel computational strategy for objectively denoising and correcting missing-wedge artifacts in homogeneous specimen areas of tomograms, where it is assumed that a template repeats itself across the volume under consideration, as happens in the case of filaments. In our deconvolution approach, we use a template and a map of corresponding template locations, allowing us to compensate for the information lost in the missing wedge. We applied the method to tomograms of actin-filament bundles of inner-ear stereocilia, which are critical for the senses of hearing and balance. In addition, we demonstrate that our method can be used for cell membrane detection.

Graphical Abstract

INTRODUCTION

Cryo-electron tomography (cryo-ET) of unstained frozen-hydrated samples is a widely used technique for imaging supramolecular complexes in organelles, cells, and tissues in their near-native state.¹ It consists of tilting the sample in the electron microscope to obtain a series of views of the specimen in known orientations. A disadvantage of cryo-ET when used for biological specimens is that radiation damage limits the number of images that can be obtained with this approach. This limitation of the electron dose results in typically high noise levels. Furthermore, the tilting range of the electron microscope is almost always limited, resulting in missing views (“missing wedge” in the case of single-axis tilting) and, consequently, anisotropic resolution in the direction of the electron beam (z coordinate), showing a reduction of 50% to 66% relative to the resolution in the x-y plane.²

The cryo-ET field is currently in a state of rapid development and innovation. At the recent cryo-ET conference in Les Diablerets, Switzerland,³ several speakers (Briegel, Medalia, Li, Dietrich, Pigino) presented 3D reconstructions that achieved resolutions better than 1 nm. This was achieved with a technique called subtomogram averaging, which improves signal-to-noise ratios in isolated particles extracted from the tomogram.⁴ In some cases, the secondary structure was visible, which enabled the flexible refinement of a fitted atomic structure; but in many ways, the interpretation of cryo-ET maps is still much more challenging than single-particle cryo-EM (featured in other articles of this special issue), due to high noise and missing wedge artifacts in the 3D density. In this work we focus on the relatively low-resolution problem of segmentation and missing wedge correction of cryo-ET maps, but significant modeling challenges also remain in other areas of interest to chemists, such as the flexible refinement and atomic modeling of tomograms.

Previous approaches by us and other groups have addressed the problem of tracing filaments in electron tomograms by “template convolution,” i.e., maximizing the local cross-correlation with a single shape template as a function of the template’s direction at each voxel position.^5–8 These methods take the 3D tomogram, with noise and artifacts present, at face value and attempt to fit shape features into the (imperfect) experimental map. Noise artifacts are smoothed over by averaging voxel intensities across the template footprint; also, the template shape can be deformed to mimic the missing wedge to some degree.^5–7

As has been pointed out by Weber et al.⁷ and confirmed by us on tomograms of microtubules (Hampton, Frank, Rusu, and Wriggers, unpublished) and on actin-filament bundles of inner-ear stereocilia (below), template convolution encounters difficulties in cases of densely packed structures, i.e., when the map resolution is comparable to, or worse than, the spacing between filaments due to fusion of nearby density features. High noise levels typical in cryo-ET further exacerbate this issue. The effect of such imprecise tracing can be likened to a stylus of a phonograph record player that is skipping laterally between groove indentations on a record due to dirt or an eroded groove wall. Similar to nudging the stylus to reset a broken vinyl record, Weber et al. proposed a human intervention, where the software user would manually guide the tracing; however, only a limited number of “skips” can be corrected in this way by human experts.⁷

Here we describe a new algorithmic approach that fully automates the tracing problem for “hard” cases that exhibit high noise and the problematic missing wedge-induced fusion of filamentous features. Instead of taking the (imperfect) experimental map as a gold standard, our new approach seeks the most likely model–informed by the template–that explains the observed data, including the noise and missing wedge artifacts. Specifically, we are seeking a new map U whose convolution with the template and the “resolution kernel” (inverse Fourier transform of the data-covered region in Fourier space) best reproduces the given experimental map (except for the noise). By considering the physical meaning of such convolution–namely placing copies of the template at locations prescribed by the map U–we see that the values of U must all be non-negative (without imposing non-negativity, we would simply reproduce exactly the given experimental map with all its imperfections). Through the use of this prior knowledge, we can then obtain U by solving a least-squares optimization problem under linear constraints (see Methods).

While we restrict this paper to the problem of tracing filaments using cylindrical templates, the long-term goal of this work is to extend the analysis to the multitude of shapes exhibited by cellular components. Also, our mathematical framework of missing-wedge correction could be of interest to other developers in cryo-ET, specifically in subtomogram averaging, where the missing wedge can limit the alignment of structural features in real space.

METHODS

Modeling Assumptions.

For the present work, we restricted our approach to the case of tomograms, in which the direction of the pattern can be assumed to change relatively slowly across the map (Figure 1). In this case we can divide the map into subregions in which the shape patterns (e.g., filaments) exhibit a predominant orientation. As we show later, this assumption still allows significant flexibility in the directions, even as high as about 25° deviation relative to the average direction within the region, and so it is not a strict requirement. As a rule of thumb, the size of the subregions should be chosen such that the vast majority of filaments still falls within those 25° of the average orientation.

Figure 1.

General schematic overview of the spatial subdecomposition used in the implementation. (a) The tomogram (2D cross section shown) is partitioned into equal-sized 3D subregions, within which the filaments (or the relevant repeating units) can be expected to have nearly constant direction. The subregion size is chosen empirically, taking into account the computational cost of the resulting quadratic programming problem and also the curvature of the filaments (higher curvature requires smaller volumes). (b) The partitions are enlarged by 50% of the size of the shape kernel S to create overlap between them (to avoid boundary effects after the subregion tracings are combined later). The shape kernel is represented as a Gaussian function, whose longitudinal full width at half-maximum (fwhm) is the template length, and whose transversal fwhm is the known thickness of the filament. For the actin filaments below we used a shape kernel length of 50 voxels and a filament thickness of 7 voxels (1 voxel = 0.947 nm). When choosing the length of the kernel, there is a trade-off between sensitivity to noise and the ability to detect actual changes along the filament.⁵ Our empirical result of 47 nm is in close agreement with a previous report⁵ of an optimal length of the cylindrical template of about 42 nm. The final size of the enlarged subregions was 74 × 195 × 77 voxels for the actin filaments below. (c) Focusing on one of the regions, the shape kernel S is shown centered at a generic location on the map, marked by a red dot. (d) The solution of the constrained-deconvolution equations yields a map U in each subvolume that indicates the locations (depicted by red dots that are more closely spaced than the longitudinal fwhm) where copies of S are to be placed (along with the corresponding scalings) to recover the “true map” T (free of noise and missing-wedge artifacts), according to eq 1. Then all the local U and T maps are merged, eliminating half of the overlap from each side (so they attach without overlapping). Finally, the tracing is performed on the global U and T maps (see Methods for details).

Focusing, then, on one of these subregions, the template or “shape kernel” S (Figure 1c) is assumed to have the same direction at every point. Thus, our main assumption is that within each of the regions, the “true map” T can formally be expressed as a convolution of U and S

$$T=U^{\ast }S$$ (1)

that is

$$T(p)=\sum _{q}U(q)\cdot S(p-q)$$ (2)

where the summation runs over all voxels q of the map.

Usually, for filament detection and tracing, the template is taken as a simple cylindrical mask of certain length and diameter.^5,6 In our approach, however, it is more appropriate to use a smooth function, which (for mathematical convenience) is a Gaussian function, and this is why we term it “shape kernel” here rather than template. This allows a proper formulation of the model as a kernel-based expansion given by the above equation.

Our second assumption is that the true map T is related to the observed experimental map Φ by the image-formation model

$$T^{\ast }R+\text{ noise }=\Phi$$ (3)

where R is the resolution kernel, i.e., the point-spread function of the final reconstruction (Figure 2). The Fourier transform of R is the mask in Fourier space defining the data-covered region. In the case of single-axis tilt geometry, which we focus on in this paper, this region is a cylinder with a wedge excluded (Figure 2). Other tilt geometries are easily accommodated by using the mask corresponding to their data coverage.

Figure 2.

Resolution kernel. (a) The missing wedge in Fourier space is the region where data is not available due to the limited tilt angles of the specimen (about the y axis). This is given by the angle Θ < 180°. Likewise, data is not available outside of the cylinder shown. These conditions limit the resolution of the reconstruction and make it worse along the z axis (missing-wedge direction). (b) Inverse Fourier transform of the data-covered region, depicted as an isolevel surface. This is the effective point-spread function of the tomogram, which we call here resolution kernel. This enters the image-formation model according to eq 3.

Overview of the Constrained Deconvolution Problem.

By combining eqs 1 and 3, we get the following for the location map U:

$$U^{\ast }W+\text{ noise }=\Phi ,\text{\hspace{1em} where }W=S^{\ast }R$$ (4)

This is a constrained deconvolution problem for U, which can be recast as a least-squares problem

$$\sum _p{\left[\left(U^{\ast }W\right)(p)-\Phi (p)\right]}^2\to \text{min}$$ (5)

subject to the constraints U(p) ≥ 0 for all voxels p. This leads to the following quadratic programming problem

$$\{\begin{array}{l}{u}^{T}Hu-2u^{T}g\to \text{min}\hfill \\ u\ge 0\hfill \end{array}$$ (6)

where the matrix H and the vector g involve the kernel W, and the vector g additionally involves the map Φ. The solution vector u is a one-dimensional version of the map U, which will in general have relatively few voxels where it is nonzero—i.e., it is a discrete map. Once we have solved these equations for U, the “true map” T is obtained via eq 1. The maps T and U can then be used to carry out the tracing of the filaments, as described below.

A similar constrained-deconvolution approach was proposed years ago in the context of biopharmaceutics,⁹ although without the smoothing component S, which is essential in our case to reduce the effect of noise. This extra factor S can be seen as a change of variables: instead of using T directly in an equation such as T * R + N = Φ subject to T ≥ 0, we use eq 4 to solve for U under U ≥ 0 and then change the variables back through T = U * S. This provides a built-in mechanism to effectively denoise the data in an adaptive manner.

Detailed Mathematical Derivation of the Constrained Deconvolution Equations.

This section may be skipped by applied readers. Here we write the equations in the general case where a variable direction of the shape kernel S is allowed. Let us denote this dependence by S_y. Then, instead of the special main assumption T = U * S (eq 1), we have the following general one:

$$T(x)=\sum _{y}U(y)\cdot S_y(x-y)$$ (7)

The image-formation model (eq 3) remains unchanged. Writing the convolution explicitly results in the following

$$\left(T^{\ast }R\right)(z)=\sum _{x}T(x)\cdot R(z-x)=\sum _x\left[\sum _{y}U(y)\cdot S_y(x-y)\right]\cdot R(z-x)$$ (8)

$$=\sum _{y}U(y)\cdot \sum _x{S}_y(x-y)\cdot R(z-x)=\sum _{y}U(y)\cdot W_y(z-x)$$ (9)

where

$$W_y(z)=\sum _{x^{\prime }}{S}_y\left(x^{\prime }\right)\cdot R\left(z-x^{\prime }\right),\text{\hspace{1em} i.e., }{W}_y=S_y\ast R$$ (10)

Hence, the general least-squares equations for U are

$$\{\begin{array}{l}\sum _z{\left[\sum _{y}U(y)\cdot W_y(z-y)-\Phi (z)\right]}^2\to \text{min}\hfill \\ \text{ subject to }U(y)\ge 0\text{ for all }y\hfill \end{array}$$ (11)

Note that if the shape kernel direction is the same for all points in the map, then the W kernel is as well: W = S * R, as in eq 4, and we obtain eq 5 as a special case of eq 11.

If the tomogram is of sufficient quality to allow the determination of the shape kernel direction at each point beforehand, as conducted in previous works,^5,6 then our approach would not be needed. In harder cases, such as the stereocilia data (Figure 3), attempts of prior determination of the filament direction at each point have failed (see the Introduction). Our approach is imperative for this type of tomographic data.

Figure 3.

Side and top views of the two stereocilia tomograms analyzed in this paper. The yellow regions, located in the central part of each tomogram, show in more detail the artifacts that we address: noise, missing wedge, and imbalance of Fourier spot intensities. (a) The shaft region is mostly ordered, with filaments running nearly parallel to the y axis in an approximately hexagonal arrangement. (b) The taper region is less organized, with filaments exhibiting more pronounced twists and turns. The bands, clearly visible in the top views of both tomograms, are a result of the Fourier spot imbalance, presumably due to variations in defocus at different tilt angles.

The quadratic form is expanded as follows:

$$\sum _z\left[{\left(\sum _{y}U(y)\cdot W_y(z-y)\right)}^2-2\cdot \Phi (z)\cdot \sum _{y}U(y)\cdot W_y(z-y)\right]\to \text{min}$$ (12)

If N = number of voxels in the map, and the voxels are enumerated as {x_i}_i=1,…,N, we define the deconvolution matrix A by

$$A_{ij}=W_{x_j}\left(x_i-x_j\right)$$ (13)

Let us also use the notation U_i = U(x_i) and Φ_i = Φ(x_i). Then, our least-squares problem takes the form

$$\{\begin{array}{l}\sum _j{\left(\sum _i{U}_i{A}_{ji}\right)}^2-2\sum _j{\Phi }_j\sum _i{U}_i{A}_{ji}\to \text{min}\hfill \\ \text{ subject to }{U}_i\ge 0\text{ for all }i\hfill \end{array}$$ (14)

In order to write the equations in a matrix-vector notation, let us use the following column-array representation of the maps involved: u = (U₁,…,U_N)^T and f = (Φ₁,…,Φ_N)^T, where the superscript T denotes the transpose. Then ∑_iu_iA_ji = (Au)_j, and so

$$\sum _j{\left(\sum _i{u}_i{A}_{ji}\right)}^2=u^T{A}^{T}Au\text{\hspace{1em} and \hspace{1em}}\sum _j{\Phi }_j\sum _i{u}_i{A}_{ji}=u^T{A}^{T}f$$ (15)

Therefore, denoting H = A^TA and g = A^Tf, we arrive at our quadratic programming problem (eq 6).

These equations (eq 6) involve, in general, 3N unknowns: N coefficients u_i and 2N parameters describing the direction vector of the shape kernel at each of the N voxels. The dependence of these equations on u is quadratic, but the dependence on the direction vectors (on which H and g depend) is more complex. We envision the development of more sophisticated methods to solve such systems of equations under these general conditions. As mentioned before, for the purpose of the present paper, we assume that the whole tomogram can be partitioned into regions, within each of which the filament direction can be considered approximately constant. In this case, the matrix H and the vector g are constant within each region. Each subsystem then involves only N_k + 2 unknowns instead of 3N, where N_k is the number of voxels in region k. The various regions should overlap along their boundaries in order to avoid aliasing and boundary effects (Figure 1).

The Solution of the Quadratic Programming Problem.

We solved eq 6 iteratively within each of the regions. We started by using the initial direction of the shape kernel as the y axis. With this direction fixed, we solved eq 6 by means of the software package SNOPT.¹⁰ This step was wrapped inside a loop consisting of a Newton–Raphson scheme to minimize the objective function with respect to the direction vector of the shape kernel (that is, the remaining 2 variables, on which H and g depend). This process converged very quickly, in two or three iterations. In this manner we obtained the U maps for all the regions. Each of these maps was then cropped along its boundaries by half of the overlap width, yielding a global U map corresponding to the whole tomogram. The global T map was then computed by means of eq 7.

Filament Tracing Using the Global U and T Maps.

In principle, tracing could be done by just using T, but the availability of U allows for a much simpler and faster procedure.

After obtaining the global U and T maps, as described in the previous section, the voxels of the T map are scanned sequentially in order of increasing y coordinates (Figure 4). Each voxel is checked to determine if it has a local maximum of T on that particular x-z cross section. If so, a new trace starts from this voxel c₁. To obtain the next trace point, a local “orthogonal distance regression” is performed based on the U map: the shape kernel S is centered at c₁, and the voxels p where U(p) > 0 are weighted according to S(p – c₁) · U(p). These weights are used in the regression, which yields a line that best fits those points p in a local neighborhood of c₁. Denoting by c′ the point on the regression line closest to c₁, the next point c₂ of the trace is taken at a prescribed distance δ from c′. When a trace point is reached whose T value is less than a prescribed threshold, the trace ends. Every trace point determines a ball of “used” voxels around it, which serve to avoid repetitions of starting voxels and collisions with previously determined traces.

Figure 4.

Tracing of the map T by means of a local regression technique. Centering the shape kernel S at the previous (or first) trace point c₁, a local “orthogonal distance regression” line is computed based on nearby nonzero points p of the U map, using weights S(p – c₁)U(p). A new trace point c₂ is then taken on this line at a prescribed distance δ from the point on the line closest to c₁.

Balancing Fourier Spot Intensities.

In the shaft region of the tomogram, which exhibits a clear hexagonal pattern in the bundle, we noticed an imbalance of the Fourier peaks relative to the expected spectrum of a hexagonally symmetric structure (Figure 5). This seems to be due to a variation in the defocus of the microscope when the tilt angle changes. Hence, we used the hexagonally symmetric shaft portion to define an ad-hoc “balancing filter”, which equalizes the magnitudes of the weak spots (located nearer the missing wedge) to those of the corresponding strong spots (located at 60° from the weak ones). This filter F was introduced in the image-formation model thus

$$T^{\ast }{F}^{\ast }R+\text{ noise }=\Phi$$ (16)

Our method then proceeds as described above but with R replaced by R′ = F * R.

Figure 5.

Sometimes, an imbalance among the various Fourier spots becomes apparent, as in the tomogram of the shaft region of stereocilia with its nearly hexagonal packing of filaments. The expected power spectrum (seen down the filament direction) would have four spots of the same intensity (plus two inside the missing wedge), but the observed power spectrum shows one pair of spots with significantly weaker intensity (a). This results in the banded structure of the experimental tomogram (c). The result of applying the deconvolution approach directly shows remnants of these artifacts in the deconvolved map T (d). By scaling the weaker spots (i.e., those nearer the missing wedge) to match the intensity of the stronger spots (b), we obtain a better result where these artifacts in the deconvolved map are virtually eliminated (e). All tomograms are shown in the x-z plane.

RESULTS

Stereocilia are actin-bundle-filled membrane protrusions of the apical hair cell surface.¹¹ Within individual stereocilia, the precise 3D organization of the whole actin bundle has never been determined experimentally; instead, most of our knowledge is derived from transmission electron micrograph images of 70–100 nm ultrathin cross sections’ directions.¹² Such studies revealed that actin filaments generally appear to be spaced ≈12 nm apart in a hexagonal pattern in the shaft region, which constitutes the bulk of the stereocilium length.

Cryo-electron tomography data had been collected from unstained frozen-hydrated individual stereocilia.¹³ The 3D volumes contain over 300 filaments per stereocilium. So far, manual tracing has been the only way to trace the filaments in such challenging data sets, which is very laborious: it takes weeks for each data set. Manual tracing¹⁴ starts with hexagonal seed points and follows contiguous filaments all the way through the volume (see the SI).

Figure 3 shows side and top views of the tomograms mentioned above. Filaments are oriented in the y direction. The shaft region of the actin bundle consists of mostly parallel actin filaments in a nearly hexagonal arrangement, whereas the taper region is more disorganized and hence significantly more challenging for both manual and automated approaches. Besides the missing wedge artifacts, manifested as the complex point-spread function (Figure 2(b)), both tomograms exhibit an additional artifact, as banded patterns are clearly visible on the top views. These bands are caused by the imbalance of the Fourier spot intensities, presumably due to variations in the defocus for different tilt angles (Figure 5(a)). We addressed this problem by the use of a “balancing filter”, which restores the intensities of the weak spots (closer to the missing wedge), thereby equalizing them with the strong spots (Figure 5(b)). The effect of using this type of filter is exemplified in Figure 5(c)–(e), with (c) being the top view of the same region shown in Figure 3(a) and (d),(e) contrasting the results without the use of the filter (d) from those with it (e).

A preliminary attempt at semiautomatically tracing filaments in the shaft region has been proposed.⁸ A set of “seed points” is manually selected by the user at one of the end faces of the bundle, and then the algorithm would trace the filaments from those starting points. The success of this approach, however, was partial, since the missing wedge artifacts made it necessary to restrict the z coordinate in order to obtain reasonable traces. Given the more complex organization of the taper region,¹⁴ we were concerned that this approach may not work in the taper region of the bundle.

We applied our method to both the shaft and taper regions. The results from the shaft region revealed that the filaments exhibit slight nonlinear variations in the x and z coordinates when traced along y and match very well with the experimental map (Figure 6(a,b)).

Figure 6.

Results of our approach on the subregions shown in Figure 3, superposed on the original tomogram data. (a) and (b) Subregions of the shaft tomogram with ordered filaments traced earlier.¹⁴ In (a), the deconvolved map T is shown as a gray isocontour surface, while (b) shows the corresponding new traces (Figure 4). Some distortions appear near the boundaries, since these results are based only on the data in these particular subregions. (c) and (d) Subregions of the more challenging new taper tomogram. In (c), the deconvolved map T is shown as a gray isocontour surface, while (d) shows the corresponding traces. The boundary effects here are more pronounced and would prevent a correct tracing if started right at the edge in the y axis. Hence, the tracing excludes the portions of the map within half of the length of the shape kernel (the tracing of the global map is also done in this way). The variations in x-z make it more difficult to assess the quality of the tracing from the side view, but the top view reveals a good agreement especially in the parts that are more ordered, such as the upper half of the top view.

In the taper region (Figure 6(c,d)), it is harder to compare filament traces with the experimental map due to the curvature of filaments, making the judgment of the matching accuracy more difficult than in the shaft region (where the straight and aligned filaments allow the use of projection in the common direction for a visual assessment). Nonetheless, the traces generally follow the map density, as observed in the example cross sections (Figure 7).

Figure 7.

Tomogram of the taper region (a) and top views of the three indicated slabs (b–d). Each slab is 30 voxels thick. In all of them we can discern a very good agreement between the traces and the density. The bottom slab (d) is the most ordered, as can be seen by its roughly hexagonal packing of the filaments. In the middle and upper slabs, the filaments are less well organized, and one can see some instances of density without trace and some of trace without density. This is because filament traces shorter than the length of the shape kernel (50 voxels in our case) were excluded and also small gaps in density (i.e., density lower than the isocontour value used to display the map) belonging to actual filamentous densities that extend above and beneath the slab.

The SI shows a comparison of the current automated tracing with the earlier¹⁴ manual tracing of the same map. We also performed a point cloud distance analysis with CloudCompare¹⁵ over the entire map shown in Figure 7. The distance distribution between manual and automated traces has a peak at 1.1 nm or 1.2 voxels. This peak value is very similar to the best agreement of 1.3 voxels we obtained earlier with a hexagonal template fitting approach that was applicable only to the regular shaft region.⁸

DISCUSSION

We have described an automated computational strategy for denoising and correcting missing-wedge artifacts in cryo-ET tomograms containing filamentous structures. Our deconvolution approach can follow curved filaments and does not require regular packing⁸ to guide the tracing. Although we use templates in our modeling, unlike the earlier template convolution,^5–8 our new deconvolution seeks a model that best describes the experimentally observed data before performing the tracing. This enables the tracing of adjacent features that would otherwise be obscured by noise or the missing wedge effect. Such a reinterpretation of the experimental map is similar to a map regularization in cryo-ET proposed two decades ago by Skoglund et al.,^16,17 but instead of producing “the most featureless reconstruction” in the deconvolution,¹⁶ we inform the reinterpretation of the map with a known template shape.

Our new deconvolution has some advantages and limitations compared to the labor-intensive traditional manual tracing (see the SI). Manual tracing¹⁴ starts with hexagonal seed points and follows contiguous filaments all the way through the volume. In contrast, the automated traces can exhibit gaps due to the influence of density variations. (Gaps resulting from our approach mean the data is insufficient to determine traces in those areas.) Moreover, the automated tracing explains the entire density in the form of filaments; for instance, it renders membrane surfaces in the form of filamentous sheets (Figure 8). The results suggest that both manual and automated approaches have their own merits and complement each other. The automated approach is less contiguous and likely to have more false positives, whereas manual tracing might have some false negatives and might be too contiguous. At the limit of detectability in the current experimental tomogram we do not know the ground truth, but the 1.2 voxel peak of the point cloud distance distribution shows that both manual and automated tracing are detecting the correct transverse x-z locations of the y-oriented filaments.

Figure 8.

Side views of the traces in the membrane region of the taper sterocilia tomogram. The view in (a) corresponds to Figure 7(a). The view direction in (b) is perpendicular to (a). Membranes are detected as they are seen by the algorithm (as arrangements of 1D paths). This property can be helpful and complement other methods more specifically targeted at membrane detection.

Our approach can also be used to detect membranes if they have a roughly cylindrical shape in each region of the partition (Figure 8). The tracing is valid until membrane directions deviate by more than about 25° from the average filament direction, as the attenuation at those angles is higher than the prescribed detection threshold. This property of detecting membranes can be helpful and complement other methods more specifically targeted at membrane detection.

Deconvolution of the entire experimental map in a unified mathematical framework comes at the cost of needing to solve an extremely large system of equations. In the most general way, these equations include 3 unknowns for each voxel of the map: the coefficient of the template centered at that voxel, plus the two polar angles specifying the direction of the template at that voxel (see Methods). This typically means hundreds of millions of variables. Furthermore, the dependence on the polar angles is nonlinear. To make the implementation more efficient and to support parallel processing, we have therefore implemented a spatial subdecomposition that works for cases of tomograms in which embedded structures are well ordered, such as in loosely packed cytoskeletal filaments.

Future developments we are planning include the following:

• Increased ability to handle multiple and highly variable filament directions by adding two extra variables at each voxel (describing the template direction vector at the point). This will require the development of novel methods to solve the 3× larger (and nonlinear) system of equations.

• Extension to other tilting geometries, such as double-axis tilt and conical tilt, which involve a modification in the resolution kernel R.

• As a particular case of the above, the ability to detect membranes in arbitrary shapes and directions.

• The development of realistic simulated tomograms for objectively validating automated tracing approaches.

Finally, in cases where subtomogram averaging is not helpful to reduce missing-wedge artifacts due to the preferred orientation, our method could become an excellent complementary alternative, since it works best when the repeating subunits are nearly parallel to each other.

The documented source code of this project will be available for download.¹⁸