# Orthogonal Matrix Retrieval with Spatial Consensus for 3D Unknown-View Tomography

arXiv: https://arxiv.org/abs/2207.02985 · HTML: https://mlsanity.com/paper/2207.02985/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 47 references that are in the map.
- First posted on arXiv 6 Jul 2022.

## Builds on (most cited first, 15 of 15)

- [Implicit Regularization in Matrix Factorization](https://mlsanity.com/paper/1705.09280/) (2017)
- [Designing RNA Secondary Structures is Hard](https://mlsanity.com/paper/1710.11513/) (2017)
- [DeepPicker: a Deep Learning Approach for Fully Automated Particle Picking in Cryo-EM](https://mlsanity.com/paper/1605.01838/) (2016)
- [Estimation under group actions: recovering orbits from invariants](https://mlsanity.com/paper/1712.10163/) (2017)
- [Fast Steerable Principal Component Analysis](https://mlsanity.com/paper/1412.0781/) (2014)
- [Denoising and Covariance Estimation of Single Particle Cryo-EM Images](https://mlsanity.com/paper/1602.06632/) (2016)
- [A deep convolutional neural network approach to single-particle recognition in cryo-electron microscopy](https://mlsanity.com/paper/1605.05543/) (2016)
- [Method of moments for 3-D single particle ab initio modeling with non-uniform distribution of viewing angles](https://mlsanity.com/paper/1907.05377/) (2019)
- [Centering noisy images with application to cryo-EM](https://mlsanity.com/paper/2009.04810/) (2020)
- [3D ab initio modeling in cryo-EM by autocorrelation analysis](https://mlsanity.com/paper/1710.08076/) (2017)
- [Reconstructing Point Sets from Distance Distributions](https://mlsanity.com/paper/1804.02465/) (2018)
- [Geometric Invariants for Sparse Unknown View Tomography](https://mlsanity.com/paper/1811.09940/) (2018)
- [3D unknown view tomography via rotation invariants](https://mlsanity.com/paper/2003.10107/) (2020)
- [Orthogonal Matrix Retrieval in Cryo-Electron Microscopy](https://mlsanity.com/paper/1412.0494/) (2014)
- [Anisotropic twicing for single particle reconstruction using autocorrelation analysis](https://mlsanity.com/paper/1704.07969/) (2017)

## Built on by (most cited first, 2 of 2)

- [Solving Quadratic Systems with Full-Rank Matrices Using Sparse or Generative Priors](https://mlsanity.com/paper/2309.09032/) (2023)
- [Two Datasets Are Better Than One: Method of Double Moments for 3-D Reconstruction in Cryo-EM](https://mlsanity.com/paper/2511.07438/) (2025)

## Abstract (arXiv)

Unknown-view tomography (UVT) reconstructs a 3D density map from its 2D projections at unknown, random orientations. A line of work starting with Kam (1980) employs the method of moments (MoM) with rotation-invariant Fourier features to solve UVT in the frequency domain, assuming that the orientations are uniformly distributed. This line of work includes the recent orthogonal matrix retrieval (OMR) approaches based on matrix factorization, which, while elegant, either require side information about the density that is not available, or fail to be sufficiently robust. For OMR to break free from those restrictions, we propose to jointly recover the density map and the orthogonal matrices by requiring that they be mutually consistent. We regularize the resulting non-convex optimization problem by a denoised reference projection and a nonnegativity constraint. This is enabled by the new closed-form expressions for spatial autocorrelation features. Further, we design an easy-to-compute initial density map which effectively mitigates the non-convexity of the reconstruction problem. Experimental results show that the proposed OMR with spatial consensus is more robust and performs significantly better than the previous state-of-the-art OMR approach in the typical low-SNR scenario of 3D UVT.