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J Struct Biol. Author manuscript; available in PMC 2013 April 1.

Published in final edited form as:

Published online 2012 May 18. doi: 10.1016/j.jsb.2012.05.005

PMCID: PMC3498508

NIHMSID: NIHMS416313

The publisher's final edited version of this article is available at J Struct Biol

See other articles in PMC that cite the published article.

The high noise level found in single-particle electron cryo-microscopy (cryo-EM) image data presents a special challenge for three-dimensional (3D) reconstruction of the imaged molecules. The spectral signal-to-noise ratio (SSNR) and related Fourier shell correlation (FSC) functions are commonly used to assess and mitigate the noise-generated error in the reconstruction. Calculation of the SSNR and FSC usually includes the noise in the solvent region surrounding the particle and therefore does not accurately reflect the signal in the particle density itself. Here we show that the SSNR in a reconstructed 3D particle map is linearly proportional to the fractional volume occupied by the particle. Using this relationship, we devise a novel filter (the “single-particle Wiener filter”) to minimize the error in a reconstructed particle map, if the particle volume is known. Moreover, we show how to approximate this filter even when the volume of the particle is not known, by optimizing the signal within a representative interior region of the particle. We show that the new filter improves on previously proposed error-reduction schemes, including the conventional Wiener filter as well as figure-of-merit weighting, and quantify the relationship between all of these methods by theoretical analysis as well as numeric evaluation of both simulated and experimentally collected data. The single-particle Wiener filter is applicable across a broad range of existing 3D reconstruction techniques, but is particularly well suited to the Fourier inversion method, leading to an efficient and accurate implementation.

In electron cryo-microscopy (cryo-EM), as in X-ray crystallography, an important goal of the data processing is to minimize the effects of noise in a density map. In recent years, cryo-EM has matured into a tool capable of providing near-atomic-resolution reconstructions of non-crystalline (single particle) biomolecules (Grigorieff and Harrison, 2011), thus bypassing certain limitations of X-ray crystallography (for example, the requirement that the target molecule be grown into a crystal) and NMR spectroscopy (which is limited to highly-concentrated, relatively low-molecular mass samples). Key advances that led to this breakthrough include the development of better electron optical systems, as well as improvements in image processing methodology for three-dimensional (3D) reconstructions of the resulting electron micrographs.

In a high-resolution cryo-EM experiment there will typically be ~10^{4} - 10^{6} images of the target molecule, each of which suffers from high noise levels, and is corrupted by a contrast transfer function (CTF) of the microscope. After determining the orientations and positions of each molecule in the images, a reconstruction algorithm merges the images into a 3D density representing the molecule. A large body of literature exists on various aspects of the reconstruction step (Penczek, 2010), but due to its importance it remains the subject of ongoing investigation.

In this work we address the reconstruction step; specifically, we seek a method to estimate a so-called ‘optimal’ map, where the mean-squared error compared to the ideal, unknown noise-free reference volume is minimized. Several studies have addressed this problem using different formalisms. At least two studies have reported implementations of the Wiener filter applied to the problem of 3D reconstruction of single-particle cryo-EM data (Zhang et al., 2008a; Scheres, 2012). The underlying assumption (either implicit or explicit) in these studies was that this filter should minimize the mean-squared error in the resulting 3D map, with respect to the signal present in the image data. Similarly, a so-called ‘figure-of-merit’ (FOM) filtering scheme was proposed as a post-processing step intended to generate a ‘best map’ (i.e., lowest mean-squared error) given the data (Rosenthal and Henderson, 2003). The error remaining in a map when subjected to such filter schemes has not been carefully scrutinized in these reports, thus leaving the essential premise of the filters (error reduction) untested. Moreover, we have recently demonstrated that, in order to minimize error in averages of aligned two-dimensional (2D) images, the bulk solvent surrounding the particle must be adequately accounted for through the addition of a scale factor. This resulted in a modification to the Wiener filter which we called the ‘single-particle Wiener filter’ (SPW filter).

Here we extend our previous results with the SPW filter to the more involved problem of 3D reconstruction. We test the various assumptions of our theory by applying the resulting SPW filter to synthetic and experimentally acquired test data sets. We find that the resulting algorithm is generally applicable to reconstruction problems with single particles, and quantitatively minimizes the error within the particle density map in cases where neither the conventional Wiener filter nor the FOM filter is as effective. Our algorithm is the first adaptation of the Wiener filter to specifically address problems caused by the presence of bulk solvent surrounding the particle. We demonstrate that this approach leads to better real-space and Fourier space fidelity for reconstructed maps using a highly efficient Fourier inversion framework. The SPW filter described here has been implemented in the single particle software FREALIGN (Grigorieff, 2007) starting with version 8.10.

The Wiener filter (Wiener, 1949) has been applied to both 2D and 3D cryo-EM image processing problems, with the goal of optimally combining noisy images into a “best” representation of the noise-free object being imaged (Saxton, 1978; Ludtke et al., 2001; Zhang et al., 2008a; Scheres, 2012). If one can obtain an estimate of the signal-to-noise ratio (SNR) of the Fourier-space representation of the data, the Wiener filter will suppress the noise in poorly measured parts of the Fourier space in order to obtain better agreement with the noise-free signal. However, the utility of the Wiener filter is compromised in single-particle imaging applications by an ambiguity in the definition of the SNR: as noted in Sindelar and Grigorieff (2011), the SNR of a particular imaged particle can be made arbitrarily low just by increasing the field of view to include more noise in the surrounding solvent area. Thus, the behavior of the Wiener filter depends on the selected image size, for a given particle, and in general tends to give over-filtered results for images of single particles (Sindelar and Grigorieff, 2011).

The above deficiency can be linked to the observation that the Wiener filter is only guaranteed to be optimal for stationary processes, where the expected mean and variance of the target function does not vary under translation (Van Trees, 2001). In fact, the target function in the currently considered case, a 3D density map of a single particle, is highly non-stationary: the mean and variance of the density inside a particle will always be substantially different than the mean and variance in the solvent region. We therefore seek a modified filter that better captures the properties of single particles.

To a first approximation, a large number of randomly oriented images will contribute a variable number of Fourier space measurements ${F}_{i,hkl}^{2\mathrm{D}}$ (i=1, 2, ... *n _{hkl}*) to each discrete point

$${F}^{\mathrm{W}}\left({\mathbf{s}}_{hkl}\right)=\mathrm{DFT}\left\{{\rho}^{\mathrm{W}}\left(\mathbf{r}\right)\right\}=\frac{\sum _{i=1}^{{n}_{hkl}}{\left\{{\mathrm{CTF}}_{i,hkl}\right\}}^{}{F}_{i,hkl}^{2\mathrm{D}}\sum _{i=1}^{{n}_{hkl}}{{\mathrm{CTF}}_{i,hkl}2}^{+}}{}$$

(1)

where CTF* _{i,hkl}* are the previously estimated CTF values of the microscope for the given Fourier space measurement, accounting for the image defocus level, astigmatism, etc.

We now derive a modification to Eq. (1) that addresses the special properties of single particles. Following our approach for the case of aligned 2D images (Sindelar and Grigorieff, 2011), we define a 3D binary enveloping function, env_{3D}(**r**), outside of which the target particle density is known to be zero. We then seek the new set of linear coefficients to the measurements ${F}_{i}^{\text{samp}}$ that yield a real-space map where the error is specifically minimized inside the envelope. Applying a set of assumptions that are expected to be reasonable for single-particle cryo-EM data sets (for example, that the data set is sufficiently large to yield a well-localized particle map), it is straightforward to adapt the previously presented 2D SPW filter to its 3D analog (Appendix A). After including a “gridding” formalism to account for the fact that, in the 3D case, most Fourier space measurements do not fall exactly on the discrete grid points **s*** _{hkl}* (see Appendix B) we arrive at the following expressions for the 3D SPW filter:

$${F}^{\mathrm{SPW}}\left({\mathbf{s}}_{hkl}\right)=\frac{\sum _{i=1}^{{n}_{hkl}}{\left\{{\mathrm{CTF}}_{i,hkl}\right\}}^{}{F}_{i,hkl}^{2\mathrm{D}}\sum _{i=1}^{{n}_{hkl}}{{\mathrm{CTF}}_{i,hkl}2}^{+}}{}$$

(2)

with

$$\mathrm{PSSNR}\left(s\right)=\frac{1}{{f}_{\text{particle}}}\mathrm{SSNR}\left(s\right)$$

(3)

and

$${f}_{\text{particle}}={{\text{env}}_{3\mathrm{D}}^{2}\left(\mathbf{r}\right)\mathbf{r}V}_{}$$

(4)

(note that *f*_{particle} refers to a fraction of a 3D volume whereas in Sindelar and Grigorieff (2011) it referred to a fraction of a 2D image). Here and in the following, PSSNR and SSNR are functions of radial spatial frequency *s*. They approximate SNR values found at grid points **s*** _{hkl}* by averaging over all values in a resolution shell. It is important to note here that Eqs. (2) - (4), as is the case with the equivalent 2D SPW expressions, can be applied in the absence of any specific knowledge about the shape of the envelope function. Instead, all that is required is the mean squared value of the envelope function,

In order to implement Eq. (2) it becomes necessary to obtain an accurate estimate of the SSNR of the raw data images. The SSNR can be most accurately obtained from a ‘masked’ FSC calculated from two volumes each containing half the data (Harauz and Van Heel, 1986) where solvent noise surrounding the particle is suppressed with a soft-edged mask function env_{mask}. The term env_{mask} differs from env_{3D} as it usually has a simpler shape such as a sphere and therefore contains substantially more volume than the actual particle volume. As shown by Sindelar and Grigorieff (2011), for a set of aligned 2D images,

$$\mathrm{SSNR}\left(s\right)\approx {f}_{\text{mask}2\mathrm{D}}\frac{{n}_{S}}{\sum _{h,k\mathrm{S}\left(s\right)}}$$

(5)

where the FRC is the 2D analog of the FSC, formed by comparing two independently averaged image data sets (Harauz and Van Heel, 1986), S(*s*) is a resolution shell centered around radial spatial frequency *s*, and *n _{S}* is the number of Fourier space pixels contained within S.

For a 3D data set, the corresponding result is (see Appendix C):

$$\mathrm{SSNR}\left(s\right)\approx {f}_{\text{mask}}\frac{{n}_{S}}{\sum _{{\mathbf{s}}_{hkl}\mathrm{S}\left(s\right)}}$$

(6)

where ${f}_{\text{mask}}={{\text{env}}_{\text{mask}}^{2}\left(\mathbf{r}\right)\mathbf{r}V}_{}$ is the mean-squared value of the soft-edged mask function evaluated over the 3D (real-space) reconstruction volume. This expression estimates the SSNR found in the raw data images, including the noise found in the solvent region, and thus may be combined with Eq. (3) to obtain the PSSNR (assuming knowledge of *f*_{particle}; see below). Here and in the following we make the assumption that the SSNR does not vary significantly between images and therefore, an average SSNR for the entire data set can be assumed. In the Discussion, we will consider the case of variable SSNR in a data set.

The PSSNR term in the denominator in Eq. (2) systematically down-weights structure factors *F*^{SPW} where the number of measurements is not sufficient to overcome the measurement noise. *F*^{SPW} thus represents an optimal estimate of the true structure factors (in the least squares sense and ignoring gridding-related artifacts), and its calculation requires incorporation of the SSNR found in the 2D image data during calculation of the final reconstruction. An alternative scheme has been described that uses a filter based on an FOM (Rosenthal and Henderson, 2003). Unlike the Wiener filter and its SPW derivative described above, the FOM filter is not incorporated directly into the 3D reconstruction step, and is instead applied in a post-processing step after the reconstruction has been calculated. To compare these filtering methods, we relate *F*^{SPW} to unfiltered gridded reconstruction using Eqs. (2) and (A2.6):

$${F}^{\mathrm{SPW}}\left({\mathbf{s}}_{hkl}\right)=\frac{\sum _{i=1}^{{n}_{hkl}}{{\mathrm{CTF}}_{i,hkl}2}^{}\sum _{i=1}^{{n}_{hkl}}{{\mathrm{CTF}}_{i,hkl}2}^{+}{F}^{\mathrm{LSQ}}\left({\mathbf{s}}_{hkl}\right)}{}$$

(7)

where we have left out the small ε term from Eq. (A2.6), which is expected to have a negligible effect on this expression. The above expression represents a voxel-by-voxel correction to the unfiltered reconstruction *F*^{LSQ}. We now substitute our estimate of the SSNR as a function of the masked FSC given in Eq. (6):

$${F}^{\mathrm{SPW}}\left({\mathbf{s}}_{hkl}\right)=\frac{\sum _{i=1}^{{n}_{hkl}}{{\mathrm{CTF}}_{i,hkl}2}^{}\sum _{i=1}^{{n}_{hkl}}{{\mathrm{CTF}}_{i,hkl}2}^{+}}{}$$

(8)

The above expression still requires knowledge of the individual CTF terms in the 2D image data. To further simplify this expression, we now assume that the filter is approximately constant within a given resolution shell. This requires that the sum of squared CTF values (this can be considered as the effective number of Fourier-space measurements) is similar for all structure factors within the resolution shell. This condition will be met when (1) a sufficiently large number of images have been collected, such that every point in Fourier space is measured many times by a spread of defocus values, and (2) there are no strongly preferred orientations in the data set (note that the presence of astigmatism in the images would not affect our analysis, under the above conditions). The expected value of this filter is then

$$\begin{array}{c}\hfill {F}^{\mathrm{SPW}}\left({\mathbf{s}}_{hkl}\right)={\frac{\sum _{i=1}^{{n}_{hkl}}{{\mathrm{CTF}}_{i,hkl}2}^{}\sum _{i=1}^{{n}_{hkl}}{{\mathrm{CTF}}_{i,hkl}2}^{+}\hfill & \approx {\frac{2{\mathrm{FSC}}_{\text{mask}}\left({s}_{hkl}\right)}{2{\mathrm{FSC}}_{\text{mask}}\left({s}_{hkl}\right)+{f}_{\text{particle}}/{f}_{\text{mask}}(1-{\mathrm{FSC}}_{\text{mask}}\left({s}_{hkl}\right))}{F}^{\mathrm{LSQ}}\left({\mathbf{s}}_{hkl}\right)\mathrm{S}\left({s}_{hkl}\right)}_{}\hfill}{}}_{}\hfill \end{array}$$

(9)

where the brackets ${\mathrm{S}\left({s}_{hkl}\right)}_{}$ denote the average value for all possible instances of the noise in resolution shell S(*s _{hkl}*). The above expression is expected, upon application to a non-filtered 3D reconstruction, to optimally filter the density map to reduce noise.

Eq. (9) describes how to obtain an approximation to the SPW algorithm (Eq. (2)), by defining a post-processing filter to be applied to the unfiltered reconstruction (*F*^{LSQ}). This result may be compared with the FOM filter described by Rosenthal and Henderson (2003), which is written in our terminology as:

$${F}^{\mathrm{FOM}}\left(s\right){\mathrm{C}}_{\text{ref}}\left(s\right)F\left(s\right)=\sqrt{\frac{2{\mathrm{FSC}}_{\text{mask}}\left(s\right)}{{\mathrm{FSC}}_{\text{mask}}\left(s\right)+1}}F\left(s\right)$$

(10)

In contrast, we see that in the limit of *f*_{particle} = *f*_{mask}
Eq. (9) reduces to:

$${F}^{\mathrm{SPW}}\left(s\right)\approx \frac{2{\mathrm{FSC}}_{\text{mask}}\left(s\right)}{{\mathrm{FSC}}_{\text{mask}}\left(s\right)+1}F\left(s\right)={\mathrm{C}}_{\text{ref}}^{2}\left(s\right)F\left(s\right)$$

(11)

Note that while Rosenthal and Henderson applied masks to their reconstructed volumes prior to calculating the FSC, they did not explicitly consider the effects of masking in their expressions for C_{ref}.

The above results indicate that successful application of the SPW filter requires an accurate estimate for *f*_{particle}. However, *f*_{particle} is defined by the shape of the solvent envelope of the particle, which is frequently challenging to obtain in experimental applications. Here we present a strategy for estimating *f*_{particle} using only information available from the input images. We begin with the property that the SPW filter minimizes the expected error within the particle region, compared with a noise-free reference volume. We further note that the SPW filter minimizes the reconstruction error everywhere in the particle simultaneously. In other words, if a chosen value of *f*_{particle} minimizes the error in any given region within the solvent envelope, the error should also be minimized at all other regions within the envelope as well, assuming equal quality of the map in all regions. Thus, one may restrict the above error evaluation to a small mask located within a “core” region of the particle, which is straightforward to establish even when the solvent boundary is indistinct (see below). If the noise-free reference volume is available, it is therefore possible to estimate *f*_{particle} by systematically varying this quantity during application of the SPW filter. The best estimate of *f*_{particle} will be the value that minimizes the error in the “core” region of the filtered reconstruction, with respect to the reference volume. In this way, *f*_{particle} can be estimated without knowledge of the precise shape of the particle envelope.

In experimental studies, the noise-free reference volume remains unknown, requiring further modification to the above strategy. It is straightforward to show, however, that because the SPW filter minimizes the error with respect to the noise-free reference volume, this filter also minimizes the error with respect to a noisy reference volume (so long as the added noise is random, and the reference is otherwise unfiltered). For any given experimental data set, moreover, a noisy reference volume is readily obtained by gridded Fourier inversion (Eq. (A2.6)). This observation implies that we may use an experimentally obtained reference volume in the above estimation procedure for *f*_{particle}, rather than using a noise-free reference, and still obtain the same result.

We thus arrive at the following scheme for estimating *f*_{particle}: From one half of the data we calculate an unfiltered, noisy estimate, *F*^{LSQ}, of the reconstruction (Eq. (A2.6)). The second half of the data is used to calculate a filtered estimate, *F*^{SPW} in which we now allow *f*_{particle} to vary (Eq. (2)). We subsequently perform a series of reconstructions using values of *f*_{particle} that range from 0 to 1, comparing the real-space cross-correlation coefficient (*CCC*) between core regions of the filtered half-data density map and the unfiltered, noisier half-data map. We then choose the value of *f*_{particle} that optimizes the core region *CCC*. This value of *f*_{particle} is thus expected to give a SPW filter whose output minimizes the error in the particle region. Moreover, this value of *f*_{particle} is expected to correspond (at least approximately) to the fraction of the reconstructed volume that is occupied by particle density (this property will be tested below). Thus, the procedure just described will yield an approximation of the SPW filter using only the images and Euler angles that are standard input in any 3D reconstruction algorithm.

- Insert projections into Fourier volume via box convolution (equivalent to nearest-neighbor interpolation if box dimension is 1 × 1 × 1 in voxel units):
- Calculate sum in numerator of Eq. (2), stored on a per-voxel basis.
- Calculate sum in denominator of Eq. (2), also on a per-voxel basis (this and the preceding step are identical to the previously published FREALIGN implementation).
- Gather separate numerator, denominator tallies for two half-data-set reconstructions, for FSC computation.

- Perform Fourier inversion (Eq. (A2.6)) to obtain both half-data-set reconstructions, and compute the FSC between the two maps (using a smoothed mask where
*f*_{mask}is conservatively chosen to significantly exceed the volume of the particle) to obtain a lower bound on the reconstruction resolution. - Estimate the whole-image SSNR from the masked FSC, by Eq. (6).
- Select a ‘core region’ of the density by low-pass-filtering the reconstruction several times lower than the resolution lower bound computed in the last step, and defining the binary envelope to enclose a small fraction (i.e. ~10%) of the filtered reconstruction density.
- Perform a series of reconstructions using the second half data set, according to the formula:where$${F}^{\mathrm{SPW}}\left({\mathbf{s}}_{hkl}\right)=\frac{\sum _{i=1}^{{n}_{hkl}}{\left\{{\mathrm{CTF}}_{i,hkl}\right\}}^{}{F}_{i,hkl}^{2\mathrm{D}}\sum _{i=1}^{{n}_{hkl}}{{\mathrm{CTF}}_{i,hkl}2}^{+}}{}$$
*f*, representing the unknown quantity*f*_{particle}, is varied between 0 and 1. - Estimate
*f*_{particle}as the value of*f*that maximizes the real-space*CCC*between core regions of the first (unmodified) half-data set reconstruction and the filtered reconstructions generated in step E.

If the SPW filter is implemented with a post-processing filter rather than as an integrated reconstruction algorithm, any reconstruction algorithm may be used and fewer steps are necessary:

- Obtain unfiltered half-data-set reconstructions and compute the masked FSC and full-data-set reconstructions.
- Define a ‘core region’ of the density, as described in step D in the integrated SPW procedure.
- Apply a series of filters to the second half-data-set reconstruction, using the following form of the SPW post-processing filter:where$${F}^{\mathrm{SPW}}\left({\mathbf{s}}_{hkl}\right)=\frac{{\mathrm{FSC}}_{\text{mask}}\left({s}_{hkl}\right)}{{\mathrm{FSC}}_{\text{mask}}\left({s}_{hkl}\right)+f/{f}_{\text{mask}}(1-{\mathrm{FSC}}_{\text{mask}}\left({s}_{hkl}\right))}F\left({\mathbf{s}}_{hkl}\right)$$
*f*, representing the unknown quantity*f*_{particle}, is varied between 0 and 1. Note that this post-processing filter has been modified from Eq. (9) in order to take into account the reduced signal-to-noise ratio found in a reconstruction made with half the data, compared with a full-data-set reconstruction. - Estimate
*f*_{particle}as the value of*f*that maximizes the real-space*CCC*between core regions of the first half-data set reconstruction and the filtered reconstructions generated in step C. - Apply the post-processing SPW filter (Eq. (9)), using the value for
*f*_{particle}estimated in part D, to the unfiltered full-data-set reconstruction step A.

Eq. (6) predicts that using the FSC to estimate the SSNR for a reconstructed particle map will yield a result that is inversely proportional to the fraction of solvent that is included in the FSC comparison. We tested this prediction using a synthetic data set composed of noisy projection images of a small (~35kD) protein molecule (crystal structure of the kinesin motor domain, PDB ID 1MKJ), from randomly sampled viewing orientations (Fig. 1A, B). Special care was taken to avoid interpolation artifacts during the projection process (see Methods), thus allowing the SSNR characteristic of the projection images to be precisely established *a priori* (Fig. 1C). Images were divided into two equal sets and subjected to gridded Fourier inversion (Eq. (A2.6)) using the exact (known) Euler angles of the projections in order to compute a pair of 3D reconstructions from each set, and a third reconstruction for the combined full image set. We then performed FSC comparisons of the resulting reconstructions, after multiplying the maps with a solvent mask. Three different mask sizes were used: a tight binary mask (Fig. 2E), generated from the reconstruction itself by the method of Wang (Wang, 1985) with parameters chosen such that the mask volume was ~2x the particle volume (see Methods); a looser mask (Fig. 2F), generated from the former mask by applying a cosine edge smoothing function (mask volume was ~5x the particle volume); and a smoothed spherical mask (Fig. 2G) where the radius matched the maximum linear dimension of the particle map (net mask volume was ~10x the particle volume).

The results of these FSC calculations (Fig. 3; note that FSC values are scaled into estimates of C_{ref} using Eq. (9)) illustrate that the application of masking yields substantially different results, due to the varying amount of solvent noise eliminated by the masking. However, we can resolve this discrepancy by defining a quantity PSSNR_{final} that places the SSNR of the reconstruction onto an absolute scale, applying the same logic that was used to derive Eq. (6):

$${\mathrm{PSSNR}}_{\text{final}}\left(s\right)\approx \frac{{f}_{\text{mask}}}{{f}_{\text{particle}}}\frac{2{\mathrm{FSC}}_{\text{mask}}\left(s\right)}{(1-{\mathrm{FSC}}_{\text{mask}}\left(s\right))}$$

(12)

Following application of Eq. (12), the scaled C_{ref} estimates converge to approximately the same value throughout most of spatial frequency range (Fig. 3A, B), indicating that Eq. (12) yields a consistent resolution measure.

We cross-validated these estimates by separately computing the Fourier shell correlation between the noise-free reference volume and a masked full-dataset reconstruction (we refer to this latter function as C_{ref}, following the convention of Rosenthal and Henderson (2003). The resulting C_{ref} curve was scaled to form an estimate of PSSNR_{final} by combining Eqs. (6), (9) and (12), and is also shown in Fig. 3A-C. The estimates for PSSNR_{final} generated from this latter approach are in excellent agreement with the FSC-generated estimates. As with the FSC calculations, the C_{ref} calculations showed smaller fluctuations (indicating higher fidelity) as tighter masking was applied (results not shown). Thus, while tight masking is desirable to reduce the random error in the PSSNR_{final} estimates, our results demonstrate that the mask size may be expanded as necessary (for example, to avoid mask-related artifacts in the FSC computation; see Discussion) without introducing systematic under-estimation of the reconstruction resolution, so long as the values are adjusted by Eq. (6).

Using the known SSNR characteristic of the synthetic images, we then derived an upper bound for the expected value of PSSNR_{final} for an idealized Fourier inversion algorithm (assuming no reconstruction artifacts):

$${\mathrm{PSSNR}}_{\text{ideal}}\left(s\right)=1/{f}_{\text{particle}}\frac{\sum _{{\mathbf{s}}_{hkl}\mathrm{S}\left(s\right)}}{}$$

(13)

This limiting function is defined purely by the signal and noise characteristics of the data images, together with the number of images taken, imaging geometry, CTF conditions, and microscope parameters; all of these values are precisely known for the synthetic data set used here. As shown in Fig. 3C, the values for PSSNR_{ideal} are in excellent agreement with the C_{ref} function derived from the conventional Fourier inversion reconstruction. The estimated SSNR showed higher fluctuations about the known value in the lowest-resolution shells (corresponding to resolutions lower than 10 Å), due to the combination of poor statistics (fewer voxels per shell) and small CTF values at these spatial frequencies, leading to higher noise variance. These errors, however, did not strongly affect the performance of the SPW filter (see below) because of the high overall SSNR of the final reconstruction at low resolution. The estimated PSSNR also showed a tendency to under-estimate the known SSNR values at resolutions higher than 3 Å, likely due to incomplete sampling of Fourier transform by the data. Again, however, these errors did not significantly affect the performance of the SPW filter because these errors occurred at spatial frequencies beyond the nominal resolution of the reconstruction (C_{ref} < 0.5, Fig. 3C). Thus, the agreement between these three different SSNR estimation methods (FSC-derived, C_{ref}-derived, and ‘ideal’) indicate that our expressions are self-consistent and quantitative, under the given (simulated) imaging conditions.

We estimated the SSNR in our data set by applying Eq. (6), using the soft-edged mask in Fig. 2F (*f*_{mask} = 0.101); we then back-calculated an estimate of PSSNR for the original data images by applying Eq. (3). We note that this back-calculation formula is based on the assumption of a perfect, artifact-free Fourier inversion algorithm, which our tests indicated was approximately valid (see above). As shown in Fig. 3D, the resulting estimates for the image SSNR were in excellent agreement with the known, pre-defined SSNR characteristic of the synthetic images used in these tests, although minor deviations below the known value are visible at the highest spatial frequencies.

To test the validity of the Wiener filter when applied within a 3D Fourier inversion scheme, we performed a series of 3D reconstructions using the synthetic data images from Fig. 1 as inputs, and employing the known SSNR characteristic of the images for the Wiener filter. As shown in Fig. 4A-B, applying the Wiener filter within a Fourier inversion scheme filters away high-resolution noise from the resulting 3D reconstruction, improving the real-space agreement with the noise-free 3D reference map. To further test the validity of the Wiener filter within the approximations inherent in our gridded reconstruction algorithm, we systematically perturbed the SSNR term in the denominator of Eq. (1) above and below its true value in order to test whether the mean-squared error was properly minimized with respect to the reference volume. This test is mathematically equivalent to applying Eqs. (2) - (4) using values of *f*_{particle} scaled above and below 1, which is how the results are presented here (Fig. 5A, inset). These calculations show that, as expected, the error is minimized near *f*_{particle}=1, although the peak is relatively broad. This perturbation experiment thus indicates that incorporating the Wiener filter into a Fourier inversion reconstruction scheme approximately minimizes the mean-squared error of the full 3D reconstruction volume with respect to the filtering parameters.

Fig. 4B also shows that the 3D density map that results from the Wiener filter reconstruction appears to be strongly over-filtered, especially when compared with the output of the SPW reconstruction methods (see below). This over-filtering results from the Wiener filter's sensitivity to the noise in the solvent region, such that the larger the solvent region, the lower the measured SSNR and hence the greater the over-filtering effect ((Sindelar and Grigorieff, 2011); see Eq. (3) above).

The above drawback in the Wiener filter can be corrected by re-defining the reconstruction problem to neglect the reconstruction error that occurs within the solvent region, and instead to minimize the error within the particle envelope only. The resulting SPW filter (Eq. (2)) is predicted to minimize the mean-squared error within an arbitrarily shaped enveloping function characterized by a fractional volume *f*_{particle}, so long as the envelope fully encloses the particle. We note that the mask function itself is not a required input to the SPW filter; instead, *f*_{particle} is the only additional input required (with respect to the conventional Wiener filter).

To test the performance of the SPW filter within the Fourier inversion scheme, we applied both the integrated as well as the post-processing SPW filters to our synthetic image data set. The resulting density maps (Figs. 4C-D) were visibly improved relative to the unfiltered or Wiener filtered maps. We tested the SPW filtered maps by real-space cross-correlation comparison with the noise-free reference volume, confining the comparison within either (1) a relatively tight binary mask (envelope mask in Fig. 2E), generated from a moderately filtered reconstruction (see Methods); or (2) a large spherical binary mask having a diameter slightly larger than the longest particle dimension. We then systematically perturbed *f*_{particle} throughout the range from 0 to 3.0 and computed the masked *CCC* where the comparison was restricted to the defined envelope region. As predicted (Fig. 5A), the SPW filter reduced the error within both envelopes, for values of *f*_{particle} close to the exactly computed value $\left({\text{env}}_{3\mathrm{D}}^{2}\right)$ for these envelopes. For the tight mask, *f*_{particle} was estimated as 0.06 vs. the known value of 0.051; for the spherical mask the estimated value was 0.19, compared to the known value of *f*_{particle} = 0.223.

In contrast, the whole-volume *CCC* for the map produced by the SPW filter was not minimized as a function of the SSNR function, and indeed was substantially lower than the whole-volume *CCC* yielded by the conventional Wiener filter (data not shown). Thus, *CCC* comparisons indicate that the SPW filter optimizes the error within the particle envelope, but that this improvement is accomplished at the expense of increased noise in the solvent region. The increased noise in the solvent region, however, is readily removed by multiplying the reconstruction with the binary particle envelope, yielding a highest-quality map where the error has been completely eliminated from the solvent region and minimized within the particle envelope.

These results demonstrate that our modified Wiener filter expression specifically tunes the noise suppression in the particle volume defined by *f*_{particle}. It follows that *f*_{particle} should be made as small as possible, while still corresponding to an envelope that fully encloses the particle, in order to completely minimize the error within the particle region. Below we evaluate our scheme for empirically determining such a value of *f*_{particle} even in the absence of precise description of the particle shape.

To assess the Fourier-space signal of the SPW reconstruction scheme compared with other reconstruction methods, we computed masked Fourier shell correlation functions comparing the reconstructions with the noise-free reference map. The resulting C_{ref} curve was increased across the entire spatial frequency range, relative to the corresponding result for the equivalent unfiltered reconstruction (Fig. 5B), although the gains were relatively minor. For comparison, we also evaluated several other published reconstruction schemes with the identical synthetic data set (Fig. 5C), including back-projection with phase flipping CTF correction (Frank et al., 1996) and an iterative algebraic method also combined with phase-flipping (Sorzano et al., 2004). These reconstructions yielded C_{ref} curves similar or lower than our unfiltered, gridded reconstruction, but falling below the SPW values (Fig. 5C).

The basis for our method of estimating *f*_{particle} is to find the filter function that maximizes the agreement in a representative “core” region of two half-data set reconstructions (see Theory). To generate a “core” mask containing only particle density, we applied a 30 Å low-pass filter to the initial, unfiltered, gridded reconstruction, then selected a threshold value to define a mask limited to a subset of the protein interior (Fig. 2H; mask volume was ~20% of the protein envelope volume). We generated a series of half-data set reconstructions using our synthetic data set, applying the integrated SPW filter to one half-data set reconstruction (Eq. (2)) but scaling the *f*_{particle} term systematically from 0 to 1. The second half-data set reconstruction was generated using the gridded Fourier inversion algorithm without the SPW filter (Eq. (A2.6)). The SSNR of the data was estimated via. Eq. (6). As shown in Fig. 5D, maximizing the *CCC* between the “core” density of the two half-data-set maps (defined by the central ~20% of the kinesin protein envelope, see Methods) led to the assignment of *f*_{particle} = ~0.022. A similar result was seen for the post-processing version of the SPW filter (Fig. 5D). For comparison, the volume contained by the molecular surface defined by the atomic model, which captures the solvent envelope of a high-resolution structure (see Methods), was 0.023. Thus, the simple scheme described here produces an estimate for *f*_{particle} that closely agrees with the “true” value expected from basic principles.

Similarly accurate estimates of *f*_{particle} were obtained with both the integrated and the post-processing forms of the SPW filter, although the *CCC* values were slightly lower in the case of the post-processing filter (Fig. 5D). We also experimented with different “core” mask choices by using the molecular surface itself, or subfragments thereof (Fig. 2A-D), for the core mask in the *f*_{particle} estimation procedure; these latter experiments (Fig. 5D, upper dashed curves) indicated that the results of the estimation procedure were relatively insensitive to the choice of core region.

We tested our filter expressions on a set of papillomavirus images that were used to obtain a near-atomic resolution 3D map (Wolf et al., 2010). We used the FREALIGN software (Grigorieff, 2007) to duplicate the methods of Wolf et al., generating a full-data-set gridded (unfiltered) reconstruction (Fig. 6A) and two half-data set reconstructions output by the program for the purpose of computing the FSC function (icosahedral averaging was performed, but no other averaging was done). We then applied our estimation scheme for *f*_{particle}, varying *f*_{particle} until we observed the maximum real-space correlation (Fig. 6C) between a non-filtered gridded reconstruction (half-data set #1), and the post-filtered SPW map (half-data set #2), restricting the comparison to small core regions within the protein interior (Fig. 6B). For FSC computations, we duplicated the mask parameters of Wolf et al. resulting in a mask in the form of hollow sphere (*f*_{mask} ≈ 0.26). This strategy yielded an estimated value for *f*_{particle} of 0.075 (Fig. 6C). To visualize this value of *f*_{particle}, we rendered an isosurface of the low-pass-filtered virus reconstruction, adjusting the threshold until the enclosed volume was equivalent to *f*_{particle}. As can be seen in Fig. 6E and 6F, this isosurface tightly encloses the volume occupied by the virus capsid proteins, indicating that our methods find a reasonable approximation to *f*_{particle} in this case. We also compared the actual filter function values of the FOM scheme vs. our SPW post-processing filter (Fig. 6D); remarkably, the filter function originally obtained by Wolf et al. using the FOM scheme (solid curve) nearly coincides with the post-processing SPW filter function values (lower dashed curve). Thus, for this particular instance the FOM filter closely matches the SPW post-processing filter, at least for the chosen masking parameters.

We have used a new theoretical framework to derive a least-squares solution to the single-particle 3D reconstruction problem, specifically accounting for the presence of a noisy solvent region of uniform density. Key to our analysis was the observation that the SSNR of an image or volume of a single particle is linearly related to the fractional area/volume occupied by the particle (Sindelar and Grigorieff, 2011) – a result that enabled us to quantify the effects of masking on FSC calculations, hence permitting much more accurate SSNR estimation. We find that the resulting SPW reconstruction algorithm is closely related to the Wiener filter, from which it was derived. We also find that the SPW method is closely related to an FOM weighting scheme proposed by Rosenthal and Henderson (2003). However, our analysis demonstrates that the SPW method improves on these two earlier methods. Moreover, our theoretical treatment connects the earlier methods to each other, and explains why they fail to produce optimal results under certain circumstances.

The least-squares method we have implemented here, as embodied by Eq. (2) (and which we previously described for the treatment of aligned 2D images (Sindelar and Grigorieff, 2011)), differs from the classically defined Wiener filter (Saxton, 1978) in a subtle but important way. In the SPW method, we have introduced the assumption that the density of interest occupies only a fraction of the reconstructed map, which is otherwise occupied by a uniform background value. When this assumption is applied to the problem of 2D or 3D averaging, an approximate least-squares solution results whose form (Eq. (2)) is nearly identical to the Wiener filter, but where the SSNR term is scaled by the inverse of the fractional particle volume, *f*_{particle} (Sindelar and Grigorieff, 2011). Given the approximately linear relationship found between image area/volume and the SSNR (Sindelar and Grigorieff, 2011), it is tempting to identify the scaled SSNR function, 1/*f*_{particle}SSNR(*s*), as the signal-to-noise ratio “inside the particle region”. While this identification is appealing intuitively, it is not strictly correct because the scaled SSNR function contains low-frequency terms that describe the overall shape of the particle, not only its interior. Thus, the PSSNR term in the denominator of Eq. (2) does not correspond to the signal-to-noise ratio of an actual image (or volume), indicating that the SPW filter is distinct from a true Wiener filter. As we have shown, *f*_{particle} tends to diverge quite far from unity in typical single-particle applications (for example, 0.075 in the papillomavirus data set considered here), leading to substantially different behavior of the SPW filter compared with the Wiener filter.

Many sources of error can degrade the quality of a 3D reconstruction. Not only does error arise due to noise in the images themselves, but also due to errors in the orientation and translation parameters that have been assigned to the images during the course of structure refinement. Artifacts and errors in the 3D reconstruction algorithm itself will reduce the quality of the final map.

Importantly, the method we have described for estimating the SSNR of the data images, Eq. (6), does not distinguish between these various error sources. Because Eq. (6) is a measure of the consistency between two separate data sets after image processing is completed, this formula therefore yields a composite description of most or all sources of signal attenuation and noise. This feature of Eq. (6) is particularly advantageous in the process of single-particle orientation and translation refinement, because misalignment of images is a major source of signal attenuation (and hence resolution degradation) during single particle structure refinement. Eq. (6) will automatically measure a lower SSNR when images are misaligned. Thus, the SPW filter will behave more aggressively with poorly aligned images, and will do so in a way to “optimize” whatever signal does emerge after summing the current image alignment. Our approach, which parallels the Bayesian approach of Scheres (2012), contrasts with other Wiener filter methods (for example, Ludtke et al. (2001)) where the SSNR is estimated via separate measurements of the signal strength and noise strength, derived from the sample itself (Ludtke et al., 2001). This latter approach may lead to suboptimal behavior of the Wiener filter due to the presence of other, undetected error sources during refinement/reconstruction. On the other hand, our SSNR estimation approach, similar to that of Scheres (2012), is expected to filter away noise in the map due to alignment errors; this could in principle lead to faster and more accurate convergence of alignment parameters during 3D structure refinement.

Key to the successful application of the SPW reconstruction scheme is knowledge of both the image SSNR characteristics as well as the fractional particle volume, *f*_{particle}. We have shown how a combination of masking and FSC computation (Eq. (6)) allows the composite SSNR of the input images to be estimated with high accuracy. Perhaps more surprising was our finding that *f*_{particle} can be estimated via a real-space comparison of two half-data-set reconstructions (Fig. 5D), essentially in the absence of any knowledge of the particle/solvent boundary. We note that the accuracy of the estimate for *f*_{particle} depends on a number of factors, including the availability of an accurate estimate of the image SSNR (for example, by Eq. (6)). Indeed, some underestimation of the ground-truth image SSNR by Eq. (6) is apparent in Fig. 3D at higher spatial frequencies. A favorable aspect of our estimation scheme for *f*_{particle}, however, is that it inherently seeks the value which best optimizes the filter performance (as judged by the measured error between FSC half-data-set reconstructions). Thus, one expects *f*_{particle} to be underestimated for the data set in Fig. 3, in order to compensate for the underestimation of the SSNR. Consistent with this prediction, our methods report a value for *f*_{particle} that falls slightly below the molecular volume of the particle (Fig. 5D). Thus, within our formalism the *f*_{particle} term will function to at least partially compensate for errors in the determination of the image SSNR (insofar as correction is possible by a scalar factor), in order to better approximate the ‘perfect’ SPW filter.

We note that a potential problem occurs when the FSC computation is affected by over-refinement which can artificially increase the FSC (Stewart and Grigorieff, 2004). The increased FSC will increase the estimated SSNR (Eq. (6)) while also artificially increasing the real-space *CCC*. However, we would argue that once over-refinement has occurred, it is no longer possible to distinguish ‘real’ signal from artifactual signal due to noise correlations. The SPW defines ‘signal’ as the information that is consistently present between two half-data-set reconstructions and optimally represents this information in a least-squares sense, whether it is ‘real’ or artifact. As such, however, the SPW filter can itself be used to reduce the possibility of over-refinement, by suppressing noise in maps produced at intermediate stages during iterative single-particle parameter refinement. A related approach has recently been explored by Scheres (2012) with promising results (see below). In addition, the SPW filter can serve as a tool for the user to diagnose the presence of over-refinement, if the map has reached a resolution where recognizable features such as secondary structure or chain traces would be evident. If the resolution of the refinement has reached 8 Å, for example, alpha helices and beta sheets should be evident in the SPW-filtered map.

Zhang et al. (Zhang et al., 2008a) incorporated a Wiener filter into their nearest-neighbor Fourier inversion reconstruction algorithm, thus yielding an algorithm very similar to ours but lacking the *f*_{particle} term. Thus, although Zhang et al. do not give a detailed analysis of the effects of noise in their reconstruction algorithm, the expectation based on our analysis is that their implementation would produce strongly over-filtered maps. More recently, Scheres presented a 3D reconstruction scheme (Scheres, 2012) within a Bayesian formalism, yielding an algorithm very similar to the filters of Zhang et al. and in the current work. In Scheres’ method, the term corresponding to SSNR is multiplied by an adjustable coefficient *T*, which was arbitrarily set to 4. Thus, *T* corresponds to 1/*f*_{particle} in our formalism, and so yields a scheme that is expected to yield an approximate least-squares solution for the case of a particle that occupies ¼ of the reconstruction volume. While Scheres does not supply a detailed analysis of the reconstruction error as we have done, the multiplication by *T* would lead to substantially less over-filtering than the method of Zhang et al., although the implied value of 0.25 for *f*_{particle} nevertheless seems too high for many (if not most) cryo-EM images that are analyzed. Importantly, Scheres selected *T* not on the basis of minimizing the error found in the reconstructed map, but rather on a more indirect measure – *T* was selected so as to minimize the degree of noise bias that occurred during the course of a refinement loop. In the absence of more sophisticated schemes for minimizing noise bias, over-filtering the reference volume is expected to reduce noise bias during map refinement (Stewart and Grigorieff, 2004), so that *T* = 4 is probably a reasonable choice for this purpose unless the particle occupies an exceptionally large fraction of the map volume.

Rosenthal and Henderson (2003) observed that the error in a reconstructed 3D map is reduced when the structure factors are scaled by the FSC curve (*C*_{ref}, or ‘figure-of-merit’) that would correspond to a comparison between the initially reconstructed map and the true but unknown, noise-free reference volume. In connecting our SPW filter to the weighting scheme of Rosenthal and Henderson, we identified a potentially significant correction to their formula. As seen by comparing Eqs. (10) and (11), in the limit of a particle that entirely fills the reconstruction volume (*f*_{particle}=1) our post-processing filter expression converges to the square of theirs. In non-limiting cases where *f*_{particle} < 1, the correction factor we derive is more complicated, but easily quantified (Fig. 7). Remarkably, we find that the FOM scheme yields filter values fairly close to our corrected expression when *f*_{particle} ≈ 0.33. While this value of *f*_{particle} is unrealistic for typical cryo-EM particles (as noted above), Rosenthal and Henderson compute the weighting factor using masked FSC calculations, which implicitly adds a correction factor of 1/*f*_{mask} to the SSNR estimate for the reconstruction (see Eq. (6)). Thus, when masked volumes are used to compute the FSC, the FOM weighting scheme reasonably approximates a least squares solution when the *ratio* of the particle volume to the mask volume (*f*_{particle} / *f*_{mask}) is ~0.33. This value is not unreasonable for typical soft-edged masks used in cryo-EM applications; for example, for the papillomavirus data set considered here (Wolf et al., 2010) we determined the particle/mask volume ratio to be ~0.3 (Fig. 6). Accordingly, the density map produced by SPW post-processing filter for this case was virtually indistinguishable from the FOM weighted map (results not shown).

One aspect of 3D reconstruction we have not explicitly considered here is the high variability in image quality that is usually inherent in a cryo-EM data set. Notably, our expression for estimating the SSNR of the image data (Eq. (6)) yields a single function that expresses the composite SSNR of the entire data set. In contrast, the experimental papillomavirus data set analyzed here contains particle images with significant variations in quality (Wolf et al., 2010). We addressed this variability using the identical methods as Wolf et al. (2010): within the FREALIGN refinement program, an exponential weighting function was applied to each particle Fourier transform (Grigorieff, 2007). While heuristic in nature, the FREALIGN weighting function adopts a similar mathematical form as the individual noise terms found in Wiener filter implementations where particle-to-particle variations in SSNR were explicitly accounted for (Ludtke et al., 2001; Scheres, 2012). We therefore anticipate that the SPW formalism could be expanded to include a formal treatment of variability in particle SSNR.

As a variant of the Fourier inversion method, the single-particle reconstruction scheme presented here is among the most computationally efficient. Furthermore, we have demonstrated that its accuracy (by FSC or real-space correlation criteria) exceeds that of other methods under carefully controlled testing conditions. Moreover, the theoretical relationships presented here clarify the relationship between particle size and error minimization, and are sufficiently general to be applied to other forms of image analysis.

A randomized set of viewing orientations was generated by first creating a set of 10000 quasi-uniformly spaced Euler angle triplets using the “VO EA” command from the SPIDER package (Frank et al., 1996). This set of 10000 Euler angles was then randomly sampled 1000 times to simulate 1000 random orientations of the particles. Projections were then generated using the resulting set of Euler angles. In order to avoid artifacts and/or signal loss at high resolutions due to interpolation, the following projection protocol was used. The atomic coordinates of 1MKJ were rotated in 3D space according to specified Euler angles, and subsequently used to generate a 3D Coulomb potential map (using CP FROM PDB from the SPIDER image processing package). We then formed a 2D projection image down the z-axis of the map coordinate system (using the PJ 3Q command from SPIDER). This projection protocol entirely avoids interpolation, and is thus predicted to maintain full signal strength all the way to the Nyquist frequency. This prediction is confirmed by a comparison of the average signal power in the projected images as a function of resolution (Fig. 1) to the average structure factors in the reference volume.

To ensure proper treatment of the simulated contrast transfer function (CTF) of the microscope, images were padded to a final size of 256 × 256 before convolving the noise-free projection images with the simulated CTF, thus allowing for information delocalization (Glaeser, 2007) to a distance of ~1 particle diameter = 96 Å from the boundary of the imaged particle. Each projection image was assigned a random defocus in an approximately uniform distribution between 0.5μm and 1.5μm. Other parameters for CTF simulation were: an accelerating voltage of 400kV, a spherical aberration constant of 4.1 (no CTF envelope function was modeled). Gaussian-distributed white noise images were generated using the MO function of SPIDER, and the noise images were scaled and added to the CTF-modulated molecular projections in order to produce a final signal-to-noise ratio (computed for the image size of 256 × 256) of 0.002.

Envelope mask volumes: For the synthetic data set, the known protein envelope mask volume was computed as the molecular surface of the 1MKJ coordinate set (Connolly, 1983), using a solvent radius parameter of 1.6. Experimental solvent mask volumes for FSC calculations were generated from the reconstructed maps similar to the method described in Grigorieff (2007), by applying a 14 Å low-pass filter to the maps and subsequently defining a binary envelope by selecting a density threshold such that the envelope contained a specified volume. The binary envelope was then smoothed by a cosine edge mask (edge distance was 14 Å).

The FREALIGN software (Grigorieff, 2007) was used for all 3D reconstructions, but was modified to separately save to disk the accumulated sum of CTF-multiplied image data (numerator term in Eq. (A2.6)), as well as the accumulated sum of CTF squared terms (denominator term in Eq. (A2.6)). No parameter refinement was done in FREALIGN; all input parameters were either set to default values (for the synthetic data set), or taken from the published refinement (for the virus data set (Wolf et al., 2010)). The intermediate data files from FREALIGN were then read into the Octave open-source numerical analysis package (http://www.gnu.org/software/octave/doc/interpreter/), where subsequent analysis was completed.

We gratefully acknowledge the reviewers for providing extremely helpful and thorough comments and suggestions, and we thank Hemant Tagare for sharing his insights on the limitations of the Wiener filter. N.G. was supported by NIH Grant P01 GM- 623580.

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