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Two Datasets Are Better Than One: Method of Double Moments for 3-D Reconstruction in Cryo-EM

Joe Kileel, Oscar Mickelin, Amit Singer, Sheng Xu

TL;DR

This work addresses 3-D molecular reconstruction in cryo-EM under unknown and anisotropic orientations by introducing the method of double moments (MoDM), which fuses two datasets with distinct orientation distributions and relies on only second-order statistics. The authors establish a uniqueness result showing that the first- and second-order moments jointly identify the structure and a low-pass form of the non-uniform rotation distribution, and they develop a convex-relaxation-based algorithm with alternating refinement to recover bandlimited structures efficiently. Numerical experiments on real cryo-EM datasets demonstrate robust reconstructions up to prescribed bandlimits, with improvements over single-dataset or higher-order-moment approaches. The work highlights the potential of data fusion in computational imaging, offering practical benefits for identifiability, computational efficiency, and applicability to other modalities such as XFEL data.

Abstract

Cryo-electron microscopy (cryo-EM) is a powerful imaging technique for reconstructing three-dimensional molecular structures from noisy tomographic projection images of randomly oriented particles. We introduce a new data fusion framework, termed the method of double moments (MoDM), which reconstructs molecular structures from two instances of the second-order moment of projection images obtained under distinct orientation distributions: one uniform, the other non-uniform and unknown. We prove that these moments generically uniquely determine the underlying structure, up to a global rotation and reflection, and we develop a convex-relaxation-based algorithm that achieves accurate recovery using only second-order statistics. Our results demonstrate the advantage of collecting and modeling multiple datasets under different experimental conditions, illustrating that leveraging dataset diversity can substantially enhance reconstruction quality in computational imaging tasks.

Two Datasets Are Better Than One: Method of Double Moments for 3-D Reconstruction in Cryo-EM

TL;DR

This work addresses 3-D molecular reconstruction in cryo-EM under unknown and anisotropic orientations by introducing the method of double moments (MoDM), which fuses two datasets with distinct orientation distributions and relies on only second-order statistics. The authors establish a uniqueness result showing that the first- and second-order moments jointly identify the structure and a low-pass form of the non-uniform rotation distribution, and they develop a convex-relaxation-based algorithm with alternating refinement to recover bandlimited structures efficiently. Numerical experiments on real cryo-EM datasets demonstrate robust reconstructions up to prescribed bandlimits, with improvements over single-dataset or higher-order-moment approaches. The work highlights the potential of data fusion in computational imaging, offering practical benefits for identifiability, computational efficiency, and applicability to other modalities such as XFEL data.

Abstract

Cryo-electron microscopy (cryo-EM) is a powerful imaging technique for reconstructing three-dimensional molecular structures from noisy tomographic projection images of randomly oriented particles. We introduce a new data fusion framework, termed the method of double moments (MoDM), which reconstructs molecular structures from two instances of the second-order moment of projection images obtained under distinct orientation distributions: one uniform, the other non-uniform and unknown. We prove that these moments generically uniquely determine the underlying structure, up to a global rotation and reflection, and we develop a convex-relaxation-based algorithm that achieves accurate recovery using only second-order statistics. Our results demonstrate the advantage of collecting and modeling multiple datasets under different experimental conditions, illustrating that leveraging dataset diversity can substantially enhance reconstruction quality in computational imaging tasks.

Paper Structure

This paper contains 35 sections, 7 theorems, 141 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. Image formation model
  4. Existing methods
  5. Maximum likelihood-based approaches.
  6. Method of moments.
  7. Contributions
  8. Preliminaries
  9. Basis representation of structure
  10. Basis representation of rotation distribution
  11. Analytic expressions for moments
  12. Formal problem statement
  13. Algorithm: Method of Double Moments
  14. Step 1: Kam’s method via uniform sample moment
  15. Step 2: Formulate optimization with the non-uniform sample moment
  16. ...and 20 more sections

Key Result

Proposition 2.2

The second-order population moment eq:def_pop_moment for a bandlimited structure of the form eq:expand_phi_hat_sph_bessel with a rotation distribution of the form eq:def_rho_in_plane_uniform can be written as In particular, $\mathcal{B}^n_{\ell,\ell'}$ satisfies the Hermitian property $\mathcal{B}^n_{\ell,\ell'} = (\mathcal{B}^n_{\ell',\ell} )^{\mathsf{H}}$. Also, $\mathcal{B}^n_{\ell,\ell'}=0$ f

Figures (3)

  • Figure 2.1: Illustration of the basis expansion of \ref{['eq:expand_phi_hat_sph_bessel']}, using $L = 10,20,30$ from leftmost to second-most right figure, and the rightmost figure as the ground-truth.
  • Figure 2.2: Illustration of the basis expansion of \ref{['eq:def_rho_in_plane_uniform']} as function of $\theta(R)$ and $\varphi(R)$, using $P = 3,5,10$ from leftmost to second-most right figure, and the rightmost figure as the ground-truth.
  • Figure 3.1: Result of running Algorithm \ref{['alg:modm']} on the dataset EMD-2660 wong2014cryo from the online electron microscopy data bank lawson2016emdatabank. (a) FSC curves for the reconstructins, for different values of bandlimit $L$ and image size $M$. (b) Reconstructions (top) and ground truth structures (bottom).

Theorems & Definitions (17)

  • Remark 2.1
  • Proposition 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.7
  • Theorem 4.1
  • Remark 4.2
  • Lemma 4.3
  • Remark 4.4
  • Lemma 4.5
  • ...and 7 more