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Provable orbit recovery over SO(3) from the non-uniform second moment

Tamir Bendory, Dan Edidin, Josh Katz, Shay Kreymer, Nir Sharon

Abstract

We study the recovery of an unknown three-dimensional band-limited signal from multiple noisy observations that are randomly rotated by latent elements of SO(3), where the rotations are drawn from an unknown, non-uniform distribution. Because the rotations are unobserved, only the signal orbit under the rotation group can be recovered. We show that the signal orbit and the rotation distribution are jointly identifiable from the first and second moments. This yields an improved high-noise sample complexity that scales quadratically with the noise variance, rather than cubically as in the uniform-rotation case. We further develop a provable, computationally efficient reconstruction algorithm that recovers the 3-D signal by successively solving a sequence of well-conditioned linear systems. The algorithm is validated through extensive numerical experiments. Our results provide a principled and tractable framework for high-noise 3-D orbit recovery, with potential relevance to cryo-electron microscopy and cryo-electron tomography modeling, where molecules are observed in unknown orientations.

Provable orbit recovery over SO(3) from the non-uniform second moment

Abstract

We study the recovery of an unknown three-dimensional band-limited signal from multiple noisy observations that are randomly rotated by latent elements of SO(3), where the rotations are drawn from an unknown, non-uniform distribution. Because the rotations are unobserved, only the signal orbit under the rotation group can be recovered. We show that the signal orbit and the rotation distribution are jointly identifiable from the first and second moments. This yields an improved high-noise sample complexity that scales quadratically with the noise variance, rather than cubically as in the uniform-rotation case. We further develop a provable, computationally efficient reconstruction algorithm that recovers the 3-D signal by successively solving a sequence of well-conditioned linear systems. The algorithm is validated through extensive numerical experiments. Our results provide a principled and tractable framework for high-noise 3-D orbit recovery, with potential relevance to cryo-electron microscopy and cryo-electron tomography modeling, where molecules are observed in unknown orientations.
Paper Structure (28 sections, 9 theorems, 56 equations, 6 figures, 2 tables, 1 algorithm)

This paper contains 28 sections, 9 theorems, 56 equations, 6 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Harmonic analysis on compact groups
  4. Harmonic analysis over $\mathop{\mathrm{SO}}\nolimits(3)$
  5. Clebsch-Gordan coefficients
  6. In-plane uniform distributions
  7. Signal and distribution models
  8. A Fourier-theoretic perspective of $\mathop{\mathrm{SO}}\nolimits(3)$ moments
  9. Main results
  10. Main results
  11. Proof of Theorem \ref{['thm:main']}
  12. Base case.
  13. Induction step.
  14. Proof of Theorem \ref{['thm:inplane']}
  15. Base case.
  16. ...and 13 more sections

Key Result

Lemma 2.1

For a given $\ell$, let $a^\ell = (a^\ell_{-\ell}, \ldots, a^\ell_\ell) \in H_\ell$ denote a vector expanded in the spherical harmonic basis. For $|{\ell_2} - {\ell_3}| \le \ell_1 \le \ell_2 + \ell_3$, the projection $(a^{\ell_2} \otimes a^{\ell_3})_{\ell_1} \in H_{\ell_1}$ has components with respe where $-\ell_1\leq k_1\leq \ell_1$ and the sum is over $|k_2| \le \ell_2$ and $|k_3| \le \ell_3$.

Figures (6)

  • Figure 1: A schematic illustration of the frequency-marching recovery. Signal coefficients (blue, left) and distribution Fourier coefficients (orange, right) are recovered at progressively higher frequency bands. At each band $\ell \geq 2$, the distribution $\hat{\rho}(H_\ell)$ is first solved from the second moment using previously recovered signal coefficients (diagonal and dashed arrows), and then $X_\ell$ is recovered from the first moment by inverting $\hat{\rho}(H_\ell)$ (horizontal arrows). The numbers show the order of computation.
  • Figure 2: Volume reconstruction results from $n=50{,}000$ noisy measurements with $\text{SNR} = 1/2$, using expansion parameters $L = 13$ and $R = 8$ shells. (a) Plasmodium falciparum 80S ribosome reconstruction. (b) TRPV1 volume reconstruction. In both panels, from left to right: a representative noisy measurement; the reconstructed volume; and the ground truth volume expanded to $L$.
  • Figure 3: The average reconstruction error as a function of the SNR. The second-moment estimator is more robust in the high-noise regime ($\text{SNR} < 1$), while the third-moment estimator bendory2025orbit converges faster as the noise decreases.
  • Figure 4: The average reconstruction error as a function of the number of observations. The error decays at a rate of approximately $O(1/\sqrt{n})$ for both methods, with the second-moment estimator showing superior performance compared to the algorithm of bendory2025orbit.
  • Figure 5: The average reconstruction error as a function of the nonuniformity parameter $\eta$, for the second-moment and third-moment methods. Small values of $\eta$ (high non-uniformity) yield low errors for the second-moment method. As $\eta \to 2$ (approaching uniformity), the second-moment method's error increases, whereas the third-moment method remains stable.
  • ...and 1 more figures

Theorems & Definitions (15)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • Theorem 3.2
  • Corollary 3.3
  • Remark 3.4: The optimality of the results
  • Lemma 3.5
  • proof
  • Lemma 4.1
  • ...and 5 more