Estimating the tails of the spectrum of the Hessian of the log-likelihood for \textit{ab-initio} single-particle reconstruction in electron cryomicroscopy

Aaditya V. Rangan; Kexin Zhang; Nikolaus Grigorieff; Pilar Cossio; Wai-Shing Tang

REVIEW 3 major objections 5 minor 53 references

Estimating the tails of the spectrum of the Hessian of the log-likelihood for \textit{ab-initio} single-particle reconstruction in electron cryomicroscopy

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A cryo-EM reconstruction can now say where it is uncertain: in the low-noise limit its log-likelihood collapses to a single 3D integral, and the smallest eigenvalues of the resulting second-derivative matrix mark the deformations the data…

desk verdict Cryo-EM uncertainty via Hessian soft modes; the derivation is clean and the idea is useful, but the gap between the surrogate likelihood and the real one at empirical noise levels keeps the central claim provisional. read the letter →

arxiv 2411.13263 v2 pith:YLHTKMKM submitted 2024-11-20 q-bio.QM

Aaditya V. Rangan , Wai-Shing Tang , Pilar Cossio , Kexin Zhang , Nikolaus Grigorieff This is my paper · ORCID

classification q-bio.QM MSC 65F1565R3092C55

keywords cryo-electronmicroscopysingle-particlereconstructionHessianspectrumsoftestmodeslog-likelihoodLanczositerationuncertaintyquantificationviewing-angledistribution

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

At stake is whether a cryo-EM reconstruction can tell you where it is wrong. This paper argues that it can, locally and cheaply: when the image noise is small, or after marginalizing over the noise level, the log-likelihood of an image stack against a reconstructed volume collapses from a sum of two-dimensional projection integrals into a single three-dimensional integral over frequency space. With that form, the Hessian (second-derivative matrix) of the log-likelihood can be applied to any perturbation quickly enough for Lanczos iteration, yielding the smallest eigenvalues and their eigenvectors—the 'softest modes' of the reconstruction problem. These modes encode the volumetric deformations and the accompanying per-image alignment shifts that the data most readily confuses with the current map, so soft-mode amplitude marks which regions of a map are trustworthy or unreliable, and the associated eigenvalues give a local measure of Fisher information. If the claim holds, cryo-EM sensitivity analysis gains a quantitative complement to Fourier-shell correlation, with consequences for data collection, validation, and free-energy estimation.

What carries the argument

The central object is the Hessian $H$ of the scaled negative-log-likelihood $L$ in the reduced model where each image's optimal viewing angle is implicitly determined by the volume. The key move is to hold two representations of $L$ side by side: the template-wise 2D sum, which makes per-image alignment derivatives easy, and the volumetric 3D integral, which makes volume derivatives diagonal in Fourier space. The moments $M_{20} = \sum_j |\mathrm{CTF}_j(k)|^2$, $M_{11} = \sum_j \mathrm{CTF}_j(k)\hat T(-\delta_j,k)\hat A_j(k)$, and $M_{02} = \sum_j |\hat A_j(k)|^2$ over the back-propagated shells carry all of the data; $H\cdot \Delta\hat F$ is built from the blocks $H_{FF}, H_{F\tau}, H_{\tau F}, H_{\tau\tau}$ with the implicit alignment response $\Delta \tau_j$ computed to first or second order from Euler-angle expansions of the template. Lanczos iteration (an iterative method for the extreme eigenvalues of a symmetric matrix) then targets the smallest eigenvalues while the three rigid-rotation directions are projected away, and the same machinery yields the Rayleigh quotient $R(s) = \partial_s \hat F^\dagger H(s)\partial_s \hat F$ for paths in volume space.

What would settle it

On a small single-particle problem where the exact Gaussian likelihood can still be evaluated pixel-by-pixel, compute the posterior covariance by direct numerical integration or by extensive sampling at a realistic signal-to-noise ratio, and compare its leading eigenvectors with the soft modes obtained from the 3D-integral Hessian. A large subspace angle between the two, or a mismatch between the inverse Hessian eigenvalues and the sampled variances along those directions, would show that the low-temperature approximation breaks down precisely where uncertainty quantification is needed.

Watch

Extended reading notes

Core claim

The paper's central discovery is that the Hessian of the low-temperature log-likelihood has a usable spectral structure. It rewrites $L = -\hat{\sigma}^2 \log P$ as $\frac{1}{2}\int \bigl(M_{20}(\kappa)|\hat F(\kappa)|^2 - 2\Re(M_{11}^\dagger(\kappa)\hat F(\kappa)) + M_{02}(\kappa)\bigr)\,d\kappa/|\kappa|$, where $M_{20}, M_{11}, M_{02}$ are accumulated Fourier-space moments of the CTF-weighted, alignment-corrected image data. From the template-wise 2D form the alignment block of the Hessian is block-diagonal per image and the volumetric block is diagonal in Fourier space; the cross-blocks $H_{F\tau}$ and $H_{\tau F}$ are evaluated through the 3D form, using back-propagated monopole and dipole impulses. When each image's optimal alignment is treated implicitly as a function of the volume, the Hessian of the reduced likelihood is assembled from these blocks, a first- and second-order expansion of the implicit alignment shift gives the correction, Lanczos iteration extracts the extremal spectrum after rigid rotations are projected out, and the small eigenvalues are the soft modes. On TRPV1 with realistic image pools the softest modes localize uncertainty to the bottom of the map under polar-biased views, shrink dramatically for the empirical viewing-angle distribution, move with viewing-angle geometry for other molecules, and stiffen as defocus increases; along an RNA conformational path the overlap of the path tangent with the soft subspace flags locations where free-energy estimates will be poorly resolved.

Load-bearing premise

The whole calculation is done on a simplified likelihood: the low-noise limit, or a noise-marginalized form in which $\log \ell$ is replaced by an affine fit $a\ell + b$ from a single dataset. The soft modes describe the true posterior uncertainty only if that approximation faithfully represents real, noisy cryo-EM data; if it does not, they describe an approximation instead.

Editorial extensions

If this is right

The softest eigenvalue converts into a hypothesis-testing bound: a deformation along the softest mode can be scaled until the likelihood ratio drops by a chosen factor (for example 20, as in the paper's figures), bounding the largest volumetric change that a standard test could not reject, together with its linked alignment shifts.
Soft-mode amplitude per map region acts as a per-voxel trust map: regions with large soft-mode component are the least constrained by the image pool, which is exactly the information needed when interpreting flexible or low-confidence domains.
Changing the image pool shifts the soft subspace in predictable ways: polar-biased viewing angles soften the caps, equatorial bias softens a complementary set, and broader viewing-angle coverage raises the smallest eigenvalues, quantifying which additional viewing angles are needed.
Increasing defocus at fixed resolution raises the softest eigenvalues and shrinks the soft subspace, so imaging conditions can be ranked by information content rather than by CTF appearance alone.
Along a conformational path, locations where the path tangent has small Rayleigh quotient $R(s)$ with the Hessian are exactly where neighbouring conformations are projectionally confusable; a collective variable whose tangent follows stiff modes avoids this source of poor conditioning in free-energy estimates.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural extension the paper leaves implicit is to use the soft subspace as a local covariance model: the inverse Hessian eigenvalues provide per-direction variances, so a per-voxel error map could be produced at the cost of a few Lanczos iterations and compared against bootstrap or split-map FSC uncertainty.
The Hessian spectrum suggests an experimental-design rule that the authors only gesture at: new images should be chosen to enlarge the smallest eigenvalues of the Hessian (for instance by adding viewing angles or defocus values that couple to the soft modes), rather than only increasing the total image count.
The constructed degeneracies point to a practical diagnostic threshold: if the softest eigenvalues are of order the noise variance divided by the number of images, the local likelihood surface is nearly flat; flagging such reconstructions could catch the spurious heterogeneity the paper demonstrates, since the fake sub-volumes show enhanced softness in exactly the regions where they were under-con
For the noise-marginalized variant, the paper's validity check is the scatter around a single empirical affine fit; monitoring that scatter on each new dataset would give a cheap, dataset-specific test of whether the computed modes still describe the true posterior uncertainty.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The new thing here is concrete: a way to write the cryo-EM log-likelihood as a 3D volumetric integral (Eq. 77) so that Hessian-vector products become cheap, and then to use Lanczos iteration to pull out the low end of the spectrum. That combination is not in the cited literature, and it is a genuinely useful tool for local uncertainty analysis. The derivation is careful and mostly self-contained; the case studies show the expected physics (polar-concentrated views make the poles soft, defocus changes the soft subspace, and so on). The ill-posedness examples in the appendix are also good, and the paper is honest about its own limits, including the explicit statement in Section 19 that no efficient general solution to the exact smallest-eigenvalue problem is known.

The soft spot is the one the stress-test flags: the Hessian whose tails you compute is the Hessian of a low-temperature surrogate objective, not the exact Gaussian posterior, and the paper never checks how much those differ at realistic SNR. The noise-marginalized version in Section 23 replaces the exact marginal with an affine fit to one lamellae dataset (Fig. 26). That is a thin basis for the central claim that the computed soft modes describe the true reconstruction uncertainty. The quadratic approximation is also extrapolated to finite perturbations without checking higher-order terms. These are validation gaps rather than obvious errors; the derivation itself holds together. But the paper as written would benefit from a direct comparison of the approximate and exact Hessians on a small, computationally tractable case, or at least a sensitivity check across noise levels. No code is provided, and there is no quantitative comparison against FSC or bootstrap, which the authors themselves position as complementary.

Who is this for? People building cryo-EM methods and possibly structural biologists who want a principled way to think about where a map is locally reliable. A serious referee should engage with it, but the referee should push hard on the validation question, because the uncertainty claim is only as good as the surrogate. I would not desk-reject this, but I would send it back for real work on that front.

Referee Report

3 major / 5 minor

Summary. The paper develops a computational strategy for estimating the smallest eigenvalues and associated eigenvectors (the 'softest modes') of the Hessian of the log-likelihood in ab-initio single-particle cryo-EM reconstruction. The central device is a rewriting of the low-temperature negative log-likelihood as a three-dimensional volumetric integral (Eqs. 54, 73, 77), which makes Hessian-vector products tractable and permits iterative Lanczos estimation of the spectral tail. The authors argue that the softest modes quantify which volumetric deformations and image-alignment perturbations are most confusable with a given reconstruction, and they use this to identify trustworthy and untrustworthy regions of maps. The main text presents case studies on TRPV1 (EMPIAR-10005), on synthetic viewing-angle distributions, on defocus variation, and on free-energy paths; the appendix derives the low-temperature and noise-marginalized representations, the multipole treatment of implicit alignments, and several ill-posedness examples.

Significance. If the central approximation is valid at realistic noise levels, this is a useful and potentially important contribution to cryo-EM uncertainty quantification. The derivation from the Gaussian likelihood to the volumetric form is self-contained, elegant, and largely parameter-free in its low-temperature version; the Hessian-vector-product formulation and the Lanczos procedure are natural and computationally plausible. The synthetic-image case studies are well controlled and illustrate that the method can recover intuitive effects of viewing-angle coverage, defocus, and noise. The paper also connects the approach to free-energy estimation and multi-particle ill-posedness, broadening its potential impact. However, the method's practical value depends on whether the computed soft modes describe the Hessian of the actual cryo-EM log-likelihood at realistic signal-to-noise ratios, and this is not tested in the manuscript. The noise-marginalized generalization also relies on an empirical affine approximation fitted to a single dataset, which needs independent validation.

major comments (3)

[Sections 13, 20; Eqs. (54), (73), (75), (77)] The Hessian whose soft modes are estimated is the Hessian of the low-temperature surrogate objective, not of the exact Gaussian posterior marginalized over alignments. Equation (54) and the subsequent volumetric forms (73), (77) are obtained by replacing the alignment integral in Eq. (75) with a saddle point at the optimal alignments; at realistic noise the exact Hessian of Eq. (75) contains additional second-order contributions arising from the width of the alignment posterior. Section 19 explicitly states that the exact smallest-eigenvalue problem has no known efficient solution, so this approximation is not a minor technicality. The manuscript uses empirical noise levels, CTFs, and finite image numbers in the case studies, but provides no comparison between the approximate Hessian and the exact Hessian of Eq. (75) (or the noise-marginalized version) at those noise levels. I recommend adding a small-scale numerical comparison—for example, a system with few images and low resolution where the exact Hessian can be formed by brute-force integration over a discretized alignment grid—and reporting the resulting eigenvalue differences. Without such a check, the claimed link between the computed soft modes and the actual uncertainty of the reconstruction is not established.
[Section 23, Fig. 26, Eq. (176)] The noise-marginalized version of the method rests on an empirical affine approximation log(ℓ) ≈ aℓ + b, with a ≈ 0.537 and b ≈ −0.378, fitted to a single FIB-milled lamellae dataset. The text claims these constants are 'fixed and independent of the image-index j', but Figure 26 shows variations of the same underlying micrograph set under different added noise levels; it does not show independent image pools, different specimens, or different imaging conditions. Since Eq. (176) and the subsequent volumetric representation depend on this approximation being universal, the manuscript needs either a derivation of the affine relation from the noise-marginalized integral or a demonstration across multiple independent datasets, together with a sensitivity analysis of the resulting soft modes to plausible variations in a and b. Without this, the noise-marginalized scenario—which the abstract identifies as a practical route for realistic data—remains unvalidated.
[Section 3.1 and Section 18] The case studies validate internal consistency rather than predictive accuracy. The synthetic images in Section 3.1 are generated from the same volume that is later treated as the optimal reconstruction, and the text explicitly assumes this volume is a global optimum. Real ab-initio reconstructions often converge to local minima or contain systematic errors, and the Hessian and its soft modes will generally be evaluated at such imperfect reconstructions. The manuscript does not test whether the softest-mode structure is stable with respect to moderate reconstruction error or whether it correctly predicts localization of errors when the reconstruction is not the truth. A numerical experiment where the input volume is perturbed away from the global optimum, or where a known suboptimal local minimum is used, would strengthen the practical relevance of the trustworthiness interpretation.

minor comments (5)

[Section 3.1] In the first paragraph of Section 3.1, 'quallity' should be 'quality'.
[Section 20] In the sentence introducing the 3-dimensional representation, 'emphazised' should be 'emphasized'; similarly, 'trival' in Section 20.1 should be 'trivial'.
[Discussion] In the membrane-protein example, 'lammelae' should be 'lamellae'.
[Fig. 17] The figure shows Rayleigh quotients for the softest seven modes at three defocus values, but no error bars, iteration counts, or convergence criteria are reported; a brief description of how many Lanczos iterations were used and how convergence was assessed would be helpful.
[Section 23.1, Eq. (174)] After offset marginalization, the text states images and templates can be assumed centered; it would be clearer to state explicitly whether the Hessian calculation is performed on the centered data and whether this changes the definition of M11 in Eq. (80).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Hessian derivation is self-contained, and the fitted or approximate elements are explicitly labeled as approximations rather than predictions.

full rationale

The paper's central derivation chain is mathematically self-contained. The log-likelihood is first written as a product of per-image likelihoods (Eq. 47) and then, in the low-temperature limit, as a sum of 2D template integrals (Eqs. 54 and 90); Sections 16-17 re-express the same sum as a 3D volumetric integral (Eqs. 73 and 77) with moments M20, M11 and M02 defined directly from the data (Eqs. 79-81). This is an algebraic rearrangement of the same objective, not a fit, and the Hessian blocks in Section 20 are derivatives of that objective. The implicit alignment response in Section 21 is derived from the first-order stationarity condition (Eq. 104), giving the reduced Hessian via the implicit function theorem, so the soft modes are not assumed but computed. The synthetic-image case studies generate data from the same volume, but Section 18 explicitly frames this as a consistency construction (Eq. 82) and a validation design, not as evidence for the method's conclusions. The only fitted constants are the affine noise-marginalization parameters a and b from Fig. 26, used in Eq. 176; these are fit to a separate FIB-lamellae dataset and are stated to be an approximation to log(l), not used to force the spectral results, and the paper does not present them as predictions. The self-citations [42, 44, 47, 48] provide numerical algorithms and context, but the core volumetric-likelihood and Hessian derivation does not rely on them. The skeptic's concern that the computed Hessian is that of a low-temperature or noise-marginalized surrogate, rather than the exact marginal likelihood at realistic SNR, is a correctness and approximation risk, not a circularity, and no equation reduces to its own input by construction.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

No new physical entities are introduced. The central claim rests on standard Fourier/mathematical identities, the Gaussian noise model, and the low-temperature or noise-marginalized approximation of the likelihood, plus an empirical affine fit for noise marginalization. The free parameters are case-study choices (noise level, image count, resolution) and the affine constants.

free parameters (5)

temperature/noise level sigma-hat = estimated from EMPIAR-10005, value not stated
Case studies set the noise level from the empirical image stack; the magnitude of softest eigenvalues and the displayed perturbation scales depend on this choice.
number of images J = 1024 in most case studies; user-chosen in synthetic pools
The Hessian spectrum is a function of image-pool size; J is chosen by hand and affects Rayleigh quotient magnitudes.
resolution cutoff 2*pi*K = 48 or 32
The reconstructed volume and Hessian are computed at low resolution; spectral estimates are resolution-dependent.
noise-marginalized affine constants a and b = a ~ 0.537, b ~ -0.378
Fig 26 fits log(l) ~ a*l + b on one FIB-lamellae dataset; the volumetric-likelihood reduction in the noise-marginalized limit assumes this affine relation holds across image pools.
mollifier design parameters (beta_north, beta_south, D_min, eps_pole) = chosen by precomputation, values not given
Section 16.1 requires user-specified trade-offs for the back-propagation mollifier, affecting numerical accuracy of Hessian-vector products.

assumptions (7)

standard math Fourier slice theorem and Plancherel/Parseval identities hold for the image and volume representations used.
Sections 11 and 17 rely on the Fourier slice theorem to connect 2D template integrals to 3D volumetric integrals; this is a standard mathematical result.
standard math Spherical-harmonic and Fourier-Bessel expansions, with Wigner-d rotation, allow efficient projection and slicing operations.
Sections 6, 10, and 11 use these basis expansions to compute rotations and projections of volumes and images.
domain assumption Images are modeled as signal plus independent Gaussian pixel noise, with no structural noise.
Section 7 explicitly assumes iid Gaussian noise in real space, yielding a Gaussian likelihood in Fourier space; structural noise is excluded.
domain assumption The reconstructed volume and optimal alignments are at a local minimum of the log-likelihood, and perturbations are small enough for a quadratic (Hessian) expansion.
Section 2 (Eq. 5) and Section 19 assume a critical point and focus on infinitesimal perturbations, though case studies display finite perturbations.
domain assumption The low-temperature limit (sigma-hat to zero) or the noise-marginalized large-dof limit applies to the data.
Section 13 derives the low-temperature form of the likelihood; Section 23 derives the noise-marginalized form, both foundational for the volumetric representation.
ad hoc to paper The empirical affine relation log(l) ~ a*l + b holds across datasets, with constants independent of the image pool.
Fig 26 fits this relation to one experimental dataset; the paper uses this to claim noise-marginalization is practical in many scenarios, but the generality is not established.
domain assumption Main derivation assumes zero translations (delta = 0); displacements are described as an extension.
Section 12.1 states 'we will typically assume delta = 0', and the main Hessian calculations suppress image-specific translations; real data may require including them.

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Cite this review

Pith. "Pith review of Estimating the tails of the spectrum of the Hessian of the log-likelihood for \textit{ab-initio} single-particle reconstruction in electron cryomicroscopy." pith.science (2026). https://pith.science/paper/YLHTKMKM

@misc{pith2026241113263,
  author       = {Pith},
  title        = {Pith review of: Estimating the tails of the spectrum of the Hessian of the log-likelihood for \textitab-initio single-particle reconstruction in electron cryomicroscopy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YLHTKMKM}},
  note         = {Machine review of arXiv:2411.13263}
}

read the original abstract

Electron cryomicroscopy (cryo-EM) is a technique in structural biology used to reconstruct accurate volumetric maps of molecules. One step of the cryo-EM pipeline involves solving an inverse-problem. This inverse-problem, referred to as \textit{ab-initio} single-particle reconstruction, takes as input a collection of 2d-images -- each a projection of a molecule from an unknown viewing-angle -- and attempts to reconstruct the 3d-volume representing the underlying molecular density. Most methods for solving this inverse-problem search for a solution which optimizes a posterior likelihood of generating the observed image-data, given the reconstructed volume. Within this framework, it is natural to study the Hessian of the log-likelihood: the eigenvectors and eigenvalues of the Hessian determine how the likelihood changes with respect to perturbations in the solution, and can give insight into the sensitivity of the solution to aspects of the input. In this paper we describe a simple strategy for estimating the smallest eigenvalues and eigenvectors (i.e., the `softest modes') of the Hessian of the log-likelihood for the \textit{ab-initio} single-particle reconstruction problem. This strategy involves rewriting the log-likelihood as a 3d-integral. This interpretation holds in the low-noise limit, as well as in many practical scenarios which allow for noise-marginalization. Once we have estimated the softest modes, we can use them to perform many kinds of sensitivity analysis. For example, we can determine which parts of the reconstructed volume are trustworthy, and which are unreliable, and how this unreliability might depend on the data-set and the imaging parameters. We believe that this kind of analysis can be used alongside more traditional strategies for sensitivity analysis, as well as in other applications, such as free-energy estimation.

Figures

Figures reproduced from arXiv: 2411.13263 by the authors.

**Figure 2.** In this figure we show a collection of τ -specific alignment-perturbations {∆τ (τ )} (left) along with a corresponding volumetric-perturbation δF (right). The arrows on the left indicate the direction of the alignment-perturbations for each (β, α), with ∆γ(τ ) = 0. Larger/darker arrows indicate larger values of ∆τ (τ ), with the arrow-area proportional to |∆τ | (and dark purple arrows corresponding to the largest v… view at source ↗

**Figure 3.** This figure is similar to Fig 2. In this figure we show a collection of [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (44 more)

**Figure 4.** Figure 4: This figure illustrates level surfaces of [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: This figure is similar to Fig 4. This time the level-surfaces of [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: This figure is similar to Fig 3, except that we show the second softest observed eigenvector of [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: This figure is similar to Fig 5, except this time we choose ∆ [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

**Figure 8.** Figure 8: This figure is similar to Fig 1, except this time we illustrate the empirical viewing-angle distribution for the [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]

**Figure 9.** Figure 9: This figure is similar to Fig 2. In this figure we show a collection of [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]

**Figure 10.** Figure 10: This figure is similar to Figs 5 and 7, except this time we choose ∆ [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]

**Figure 11.** Figure 11: This figure is similar to Fig 10. Here we show the softest mode associated with the ISWINCP molecule at [PITH_FULL_IMAGE:figures/full_fig_p012_11.png]

**Figure 12.** Figure 12: This figure is similar to Fig 10. Here we show the softest mode associated with the ISWINCP molecule [PITH_FULL_IMAGE:figures/full_fig_p012_12.png]

**Figure 13.** Figure 13: This figure is similar to Fig 10. Here we show the softest mode associated with the MlaFEDB molecule [PITH_FULL_IMAGE:figures/full_fig_p013_13.png]

**Figure 14.** Figure 14: This figure is similar to Fig 10. Here we show the softest mode associated with the MlaFEDB molecule at [PITH_FULL_IMAGE:figures/full_fig_p013_14.png]

**Figure 15.** Figure 15: This figure is similar to Fig 10. Here we show the softest mode associated with the MlaFEDB molecule [PITH_FULL_IMAGE:figures/full_fig_p014_15.png]

**Figure 17.** Figure 17: Note that, as the defocus increases, the softest eigenvalues of the Hessian increase, corresponding to the increase in [PITH_FULL_IMAGE:figures/full_fig_p014_17.png]

**Figure 16.** Figure 16: In this figure we illustrate three different collections of CTF-functions, corresponding to defocus-values of [PITH_FULL_IMAGE:figures/full_fig_p015_16.png]

**Figure 17.** Figure 17: Here we show the rayleigh-quotients associated with the softest 7 modes for each defocus-value. Note that, [PITH_FULL_IMAGE:figures/full_fig_p015_17.png]

**Figure 18.** Figure 18: This figure is similar to Fig 9. In this case we show the perturbation associated with the softest mode the [PITH_FULL_IMAGE:figures/full_fig_p016_18.png]

**Figure 19.** Figure 19: This figure is similar to Fig 18. In this case we show the perturbation associated with the softest mode the [PITH_FULL_IMAGE:figures/full_fig_p016_19.png]

**Figure 20.** Figure 20: This figure is similar to Fig 18. In this case we show the perturbation associated with the softest mode the [PITH_FULL_IMAGE:figures/full_fig_p016_20.png]

**Figure 21.** Figure 21: This figure is similar to Fig 10. In this case we show the perturbation associated with the softest mode at a [PITH_FULL_IMAGE:figures/full_fig_p017_21.png]

**Figure 22.** Figure 22: This family of conformations corresponds to a particular path of interest in volume-space, and can be parametrized [PITH_FULL_IMAGE:figures/full_fig_p017_22.png]

**Figure 22.** Figure 22: In this figure we illustrate the rayleigh-quotient [PITH_FULL_IMAGE:figures/full_fig_p018_22.png]

**Figure 23.** Figure 23: In this figure we illustrate the properties of the hessian [PITH_FULL_IMAGE:figures/full_fig_p034_23.png]

**Figure 24.** Figure 24: In this figure we illustrate some properties of a latitudinal-perturbation Θ( [PITH_FULL_IMAGE:figures/full_fig_p040_24.png]

**Figure 25.** Figure 25: In this figure we illustrate some properties of a longitudinal-perturbation Φ( [PITH_FULL_IMAGE:figures/full_fig_p042_25.png]

**Figure 26.** Figure 26: In this figure we illustrate values of ℓ(Abj | Sbopt j , µ) taken from micrographs of FIB-milled yeast lamellae as described in [46] (see [PITH_FULL_IMAGE:figures/full_fig_p046_26.png]

**Figure 27.** Figure 27: On the left we show an idealized viewing-angle-distribution [PITH_FULL_IMAGE:figures/full_fig_p047_27.png]

**Figure 28.** Figure 28: In this figure we show a collection of τ -specific alignment-perturbations {∆τ (τ )} (left) along with a corresponding volumetric-perturbation δF (right). The arrows on the left indicate the direction of the alignment-perturbations for each (β, α), with ∆γ(τ ) = 0. L…

**Figure 29.** Figure 29: In this figure we show one of the perturbed volumes [PITH_FULL_IMAGE:figures/full_fig_p048_29.png]

**Figure 30.** Figure 30: This figure has the same format as Fig 29, except that we illustrate [PITH_FULL_IMAGE:figures/full_fig_p048_30.png]

**Figure 31.** Figure 31: This figure illustrates an even more extreme volumetric perturbation that, again, has the same collection of [PITH_FULL_IMAGE:figures/full_fig_p049_31.png]

**Figure 32.** Figure 32: This figure illustrates an even more extreme volumetric perturbation that, again, has the same collection of [PITH_FULL_IMAGE:figures/full_fig_p049_32.png]

**Figure 33.** Figure 33: This figure illustrates an even more extreme volumetric perturbation that, again, has the same collection of [PITH_FULL_IMAGE:figures/full_fig_p050_33.png]

**Figure 34.** Figure 34: This figure illustrates an even more extreme volumetric perturbation that, again, has the same collection [PITH_FULL_IMAGE:figures/full_fig_p050_34.png]

**Figure 35.** Figure 35: In this figure we illustrate the frequency-shells [PITH_FULL_IMAGE:figures/full_fig_p051_35.png]

**Figure 36.** Figure 36: In this figure we illustrate level-sets of the volumes produced by convolving the frequency-shells shown in Fig [PITH_FULL_IMAGE:figures/full_fig_p051_36.png]

**Figure 37.** Figure 37: This figure is similar to Fig 35. This time the frequency-shells adopt more than one value in the polar-regions. b [PITH_FULL_IMAGE:figures/full_fig_p052_37.png]

**Figure 38.** Figure 38: In this figure we illustrate level-sets of the volumes produced by convolving the frequency-shells shown in Fig [PITH_FULL_IMAGE:figures/full_fig_p053_38.png]

**Figure 39.** Figure 39: In this figure we illustrate the viewing-angles corresponding to [PITH_FULL_IMAGE:figures/full_fig_p055_39.png]

**Figure 40.** Figure 40: In this figure we illustrate three level-sets of volumes [PITH_FULL_IMAGE:figures/full_fig_p055_40.png]

**Figure 41.** Figure 41: This is similar to Fig 40, except for a different level-surface. [PITH_FULL_IMAGE:figures/full_fig_p056_41.png]

**Figure 42.** Figure 42: This is similar to Fig 40, except for a different level-surface. [PITH_FULL_IMAGE:figures/full_fig_p056_42.png]

**Figure 43.** Figure 43: This is similar to Fig 40, except for a different level-surface. [PITH_FULL_IMAGE:figures/full_fig_p056_43.png]

**Figure 44.** Figure 44: This is similar to Fig 41. We show the original [PITH_FULL_IMAGE:figures/full_fig_p057_44.png]

**Figure 45.** Figure 45: This is similar to Fig 44. This time the alternative volume on the right is fit to a different subset of 64 noisy [PITH_FULL_IMAGE:figures/full_fig_p057_45.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

A. B. Goncharov. Methods of integral geometry and finding the relative orientation of identical particles arbitrarily arranged in a plane from their projections onto a straight line. Dokl. Phys., 32:173, 1987. 57

work page 1987

[2] [2]

van Heel

M. van Heel. Angular reconstitution: A posteriori assignment of projection directions for 3D reconstruction. Ultramicroscopy, 21:111–123, 1987

work page 1987

[3] [3]

A. B. Goncharov and M. S. Gelfand. Determination of mutual orientation of identical particles from their projections by the moments method. Ultramicroscopy, 25:317–328, 1988

work page 1988

[4] [4]

Jonic, C.O

S. Jonic, C.O. Sorzano, P. Thevenaz, C. El-Bez, S. De Carlo, and M. Unser. Spline-based image-to-volume registration for three-dimensional electron microscopy. Ultramicroscopy, 103(4):303–317, 2005

work page 2005

[5] [5]

Penczek, and Thomas Walz

Yifan Cheng, Nikolaus Grigorieff, Pawel A. Penczek, and Thomas Walz. A primer to single-particle cryo-electron microscopy. Cell, 161:439–449, 2015

work page 2015

[6] [6]

Nogales and S.H

E. Nogales and S.H. Scheres. Cryo-EM: a unique tool for the visualization of macromolecular complexity. Mol. Cell., 58:677–689, 2015

work page 2015

[7] [7]

Cryogenic electron microscopy and single-particle analysis

Dominika Elmlund and Hans Elmlund. Cryogenic electron microscopy and single-particle analysis. Annu. Rev. Biochem., 84:499–517, 2015

work page 2015

[8] [8]

Murata and M

K. Murata and M. Wolf. Cryo-electron microscopy for structural analysis of dynamic biological macromolecules. Biochim. Biophys. Acta Gen. Subj., 1862(2):324–334, 2017

work page 2017