Autocorrelation analysis for cryo-EM with sparsity constraints: Improved sample complexity and projection-based algorithms

The number of noisy images required for molecular reconstruction in single-particle cryo-electron microscopy (cryo-EM) is governed by the autocorrelations of the observed, randomly-oriented, noisy projection images. In this work, we consider the effect of imposing sparsity priors on the molecule. We use techniques from signal processing, optimization, and applied algebraic geometry to obtain new theoretical and computational contributions for this challenging non-linear inverse problem with sparsity constraints. We prove that molecular structures modeled as sums of Gaussians are uniquely determined by the second-order autocorrelation of their projection images, implying that the sample complexity is proportional to the square of the variance of the noise. This theory improves upon the non-sparse case, where the third-order autocorrelation is required for uniformly-oriented particle images and the sample complexity scales with the cube of the noise variance. Furthermore, we build a computational framework to reconstruct molecular structures which are sparse in the wavelet basis. This method combines the sparse representation for the molecule with projection-based techniques used for phase retrieval in X-ray crystallography.Comment: 31 pages, 5 figures, 1 movi

Manifold Rewiring for Unlabeled Imaging

Geometric data analysis relies on graphs that are either given as input or inferred from data. These graphs are often treated as "correct" when solving downstream tasks such as graph signal denoising. But real-world graphs are known to contain missing and spurious links. Similarly, graphs inferred from noisy data will be perturbed. We thus define and study the problem of graph denoising, as opposed to graph signal denoising, and propose an approach based on link-prediction graph neural networks. We focus in particular on neighborhood graphs over point clouds sampled from low-dimensional manifolds, such as those arising in imaging inverse problems and exploratory data analysis. We illustrate our graph denoising framework on regular synthetic graphs and then apply it to single-particle cryo-EM where the measurements are corrupted by very high levels of noise. Due to this degradation, the initial graph is contaminated by noise, leading to missing or spurious edges. We show that our proposed graph denoising algorithm improves the state-of-the-art performance of multi-frequency vector diffusion maps

3D unknown view tomography via rotation invariants

In this paper, we study the problem of reconstructing a 3D point source model from a set of 2D projections at unknown view angles. Our method obviates the need to recover the projection angles by extracting a set of rotation-invariant features from the noisy projection data. From the features, we reconstruct the density map through a constrained nonconvex optimization. We show that the features have geometric interpretations in the form of radial and pairwise distances of the model. We further perform an ablation study to examine the effect of various parameters on the quality of the estimated features from the projection data. Our results showcase the potential of the proposed method in reconstructing point source models in various noise regimes

The sample complexity of sparse multi-reference alignment and single-particle cryo-electron microscopy

Multi-reference alignment (MRA) is the problem of recovering a signal from its multiple noisy copies, each acted upon by a random group element. MRA is mainly motivated by single-particle cryo-electron microscopy (cryo-EM) that has recently joined X-ray crystallography as one of the two leading technologies to reconstruct biological molecular structures. Previous papers have shown that in the high noise regime, the sample complexity of MRA and cryo-EM is $n=\omega(\sigma^{2d})$, where $n$ is the number of observations, $\sigma^2$ is the variance of the noise, and $d$ is the lowest-order moment of the observations that uniquely determines the signal. In particular, it was shown that in many cases, $d=3$ for generic signals, and thus the sample complexity is $n=\omega(\sigma^6)$. In this paper, we analyze the second moment of the MRA and cryo-EM models. First, we show that in both models the second moment determines the signal up to a set of unitary matrices, whose dimension is governed by the decomposition of the space of signals into irreducible representations of the group. Second, we derive sparsity conditions under which a signal can be recovered from the second moment, implying sample complexity of $n=\omega(\sigma^4)$. Notably, we show that the sample complexity of cryo-EM is $n=\omega(\sigma^4)$ if at most one third of the coefficients representing the molecular structure are non-zero; this bound is near-optimal. The analysis is based on tools from representation theory and algebraic geometry. We also derive bounds on recovering a sparse signal from its power spectrum, which is the main computational problem of X-ray crystallography