Orbit recovery under the rigid motions group

TL;DR

This work analyzes orbit recovery under the rigid-motion group SE(n) by connecting it to the well-understood SO(n) rotational problem via an explicit algebraic reduction: the d-th order SO(n) moments can be recovered from the (d+2)-order SE(n) autocorrelations. This transfer enables sharp high-noise sample-complexity bounds for SE(n) and the multi-target detection model, showing lower bounds scaling as ω(σ^{2d}) and upper bounds scaling as ω(σ^{2d+4}). The authors provide a provable end-to-end pipeline for SE(n) orbit recovery in 2D and 3D, recovering SO(n) moments from SE(n) autocorrelations and performing frequency-marching inversions to reconstruct volumes, with validation on end-to-end 2D/3D reconstructions. They also connect the theory to cryo-ET/cryo-EM, deriving implications for detection-free structure recovery under extreme noise and outlining open questions for non-well-separated and heterogeneous MTD settings. Collectively, the paper establishes a rigorous framework for invariant-based orbit recovery under rigid motions and demonstrates a concrete pathway toward practical 3D molecular reconstruction from highly noisy data.

Abstract

We study the orbit recovery problem under the rigid-motion group SE(n), where the objective is to reconstruct an unknown signal from multiple noisy observations subjected to unknown rotations and translations. This problem is fundamental in signal processing, computer vision, and structural biology. Our main theoretical contribution is bounding the sample complexity of this problem. We show that if the d-th order moment under the rotation group SO(n) uniquely determines the signal orbit, then orbit recovery under SE(n) is achievable with $N\gtrsim σ^{2d+4}$ samples as the noise variance $σ^2 \to \infty$. The key technical insight is that the d-th order SO(n) moments can be explicitly recovered from (d+2)-order SE(n) autocorrelations, enabling us to transfer known results from the rotation-only setting to the rigid-motion case. We further harness this result to derive a matching bound to the sample complexity of the multi-target detection model that serves as an abstract framework for electron-microscopy-based technologies in structural biology, such as single-particle cryo-electron microscopy (cryo-EM) and cryo-electron tomography (cryo-ET). Beyond theory, we present a provable computational pipeline for rigid-motion orbit recovery in three dimensions. Starting from rigid-motion autocorrelations, we extract the SO(3) moments and demonstrate successful reconstruction of a 3-D macromolecular structure. Importantly, this algorithmic approach is valid at any noise level, suggesting that even very small macromolecules, long believed to be inaccessible using structural biology electron-microscopy-based technologies, may, in principle, be reconstructed given sufficient data.

Figures (8)

• Figure 1: Overview of the central problems studied in this work.(a) A primary motivation for this work arises from single-particle cryo-electron microscopy (cryo-EM), where the goal is to reconstruct 3D molecular structures from noisy 2D projection images bendory2020single. The raw measurements, known as micrographs, contain many randomly oriented copies of the target molecule embedded in high levels of noise. Each particle instance is given by $f_i = \Pi(g_i \cdot f)$, where $g_i$ is a random 3D rotation and $\Pi$ is a tomographic projection operator. At low SNR, individual particles cannot be reliably detected, but direct structure recovery from micrographs may still be possible bendory2023toward. (b) The orbit recovery problem under the special orthogonal group $\mathsf{SO}(n)$ (Problem \ref{['prob:orbitRecoverySOn']}), where each observation is a noisy measurement of a randomly rotated signal (e.g., Einstein image), studied across varying signal-to-noise ratio (SNR) regimes. (c) Orbit recovery under the rigid motion group $\mathsf{SE}(n)$ (Problem \ref{['prob:orbitRecoverySEn']}), where each signal undergoes an unknown rotation and translation. This setting introduces additional uncertainty due to unknown spatial locations. (d) Signal recovery from MTD observations (Problem \ref{['prob:orbitRecoveryMTD']}), where multiple rotated instances of a signal $f$ are embedded at unknown positions in a noisy measurement. At high SNR, particle instances can be detected, aligned, and averaged. However, at low SNR, detecting the particle locations becomes infeasible. In this challenging regime, addressed in this work, the goal is to characterize the sample complexity required for successful signal recovery despite the presence of both rotational and positional uncertainty, in high levels of noise.
• Figure 2: Properties of antipodal points.(a) Applying the same translation to both antipodal points in the support of the signal results in their product being zero, since at least one of the translated antipodal points lies outside the support of the signal $f$. (b) The integral of the product of antipodal points over the support of the signal is invariant under the group action (see Lemma \ref{['lem:invarianceOfAntipodalProdSum']}).
• Figure 3: Comparing $\mathsf{SO}(2)$ moments and $\mathsf{SE}(2)$ autocorrelations.(a) $\mathsf{SO}(2)$ moments: fix $d=3$ locations in the image and average the product of their intensities over all rotations of the underlying signal (e.g., the Einstein image). (b) $\mathsf{SE}(2)$ autocorrelation: Choose five fixed relative offsets $\bm{\tau}_0,\ldots,\bm{\tau}_4$ around a window center. Each row shows a different rotation of the signal. Each column corresponds to a different sliding-window center (green dot). For every (row, column) pair, sample the five locations at the red offsets $\{\bm{x}_{\text{center}}+\bm{\tau}_j\}_{j=0}^4$ in the transformed image and multiply their intensities. Some products are zero when one or more samples fall outside the object, while others are nonzero when the pattern overlaps the object. Averaging these per-window products across all centers and rigid–motion realizations yields $A_f^{(5)}(\bm{\tau}_0,\bm{\tau}_1,\bm{\tau}_2,\bm{\tau}_3,\bm{\tau}_4)$.
• Figure 4: Extracting $\mathsf{SO}(2)$ moments from $\mathsf{SE}(2)$ autocorrelations.Left panel: Fifth-order autocorrelation with an antipodal pair. Fix $d=3$ internal points $\bm{\eta}_1,\bm{\eta}_2,\bm{\eta}_3$ inside the support of $f$ (Theorem \ref{['thm:reductionFromAutocorrelationToTensorMoment']}). Each row shows a different rigid-motion instance $g_\ell\in\mathsf{SE}(2)$ in which the entire image is rotated and translated. For each transformed image, integrate the product of the intensities at these three fixed points together with the intensities at all pairs of (nearly) antipodal locations on a circle (Definition \ref{['eqn:nearlyAntipodalPoints']}), sweeping over all angles. This yields a directional slice of the $(d+2)$-order autocorrelation $A^{(d+2)}_{f,\rho}$ (the numerator of \ref{['eqn:mainTheoremExtraction']}). Right panel: Second-order autocorrelation of antipodal points. For each transformed image, integrate over all antipodal pairs on the boundary, as in Assumption \ref{['assum:nonVanishingSupport']} (the denominator of \ref{['eqn:mainTheoremExtraction']}). Extraction. The $d$-th order $\mathsf{SO}(2)$ moment $M^{(d)}_{f,\rho}$ is obtained by letting the antipodal pair approach the boundary and normalizing by the second–order autocorrelation, per \ref{['eqn:mainTheoremExtraction']}.
• Figure 5: Asymptotic equivalence between empirical and ensemble mean autocorrelations. This figure illustrates the convergence of the empirical autocorrelation (Definition \ref{['eqn:autoCorrealtionMoments']}) to the ensemble mean autocorrelation (Definition \ref{['eqn:ensembeleRV']}) as the number of samples $N \to \infty$, as formalized in Proposition \ref{['thm:prop0']}. In the ensemble model, each observation is drawn either from the signal distribution with probability $\gamma$ or from pure noise with probability $1 - \gamma$. The empirical autocorrelation thus asymptotically matches the expected autocorrelation under this probabilistic mixture.

Theorems & Definitions (53)

• Theorem 1.1: Informal: Extracting ${d}$-th order rotational moments from ${(d+2)}$-order rigid motion autocorrelations
• Corollary 1.2
• Theorem 1.3: Informal: Sample complexity of the orbit recovery problem under $\mathsf{SE}(n)$ and of the MTD model
• Definition 2.2: The Special Euclidean group $\mathsf{SE}(n)$
• Remark 2.4
• Lemma 2.8: Rotational invariance of antipodal correlation under group action
• Definition 3.1: Population moments under $\mathsf{SO}(n)$
• Definition 3.2: Population autocorrelations under $\mathsf{SE}(n)$
• Lemma 3.4
• Proposition 3.5: Invariance of the autocorrelation under the rigid motion group