Super-resolution multi-reference alignment

Abstract

We study super-resolution multi-reference alignment, the problem of estimating a signal from many circularly shifted, down-sampled and noisy observations. We focus on the low SNR regime, and show that a signal in ${\mathbb{R}}^M$ is uniquely determined when the number $L$ of samples per observation is of the order of the square root of the signal’s length ($L=O(\sqrt{M})$). Phrased more informally, one can square the resolution. This result holds if the number of observations is proportional to $1/\textrm{SNR}^3$. In contrast, with fewer observations recovery is impossible even when the observations are not down-sampled ($L=M$). The analysis combines tools from statistical signal processing and invariant theory. We design an expectation-maximization algorithm and demonstrate that it can super-resolve the signal in challenging SNR regimes.

Multi-target detection with rotations

“…As such, (10) relates the third-order autocorrelation of the measurement A M to the invariant features S F and µ F . In practice, σ 2 and γµ F can be estimated from M : σ 2 can be estimated by the variance of the pixel values of M in the low SNR regime, while γµ F can be estimated by the empirical mean of M .…”

“…Future work includes extending Theorem 3.2 to the two-dimensional case, studying the high-dimensional regime where the size of the target image is large (see for example [29,12]), and exploring super-resolution limits in the MTD model [10]. Our ultimate goal is to complete the program outlined in [7] and devise a computational framework to recover a three-dimensional molecular structure directly from micrographs.…”

Denoiser-based projections for 2-D super-resolution multi-reference alignment

“…The SR-MRA model, first studied for 1-D signals [2], is a special case of the multi-reference alignment (MRA) model: the problem of estimating a signal from its noisy copies, each acted upon by a random element of some group, see for example [3], [4], [5], [6], [7], [8]. The MRA model is mainly motivated by the singleparticle cryo-electron microscopy (cryo-EM) technology: an increasingly popular technique to constitute 3-D molecular structures [9].…”

“…This section compares the performance of the projected method of moments of Algorithm (1) and the projected EM of Algorithm (2). We applied the algorithms to the 68 "natural" images of the CBSD-68 dataset and to a cryo-EM image of the E. coli 70S ribosome, available at the Electron Microscopy Data Bank [1].…”

An adaptive variational model for multireference alignment with mixed noise

“…This mathematical model arises in many scientific and engineering fields, for instance, structural biology [5,8,26,30,37,38], single cell genomic sequencing [27], radar [17,28], robotics [9], crystalline simulations [6],image registration and super-resolution [10,14,22] and some algorithms and theoretical analysis [1, 2, 7, 11-13, 15, 23, 32, 39]. There are some variants of MRA problem, such as heterogeneous MRA [25], super-resolution MRA [33].…”