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

Investigation of registration algorithms for the automatic tile processing system

The Robotic Tile Inspection System (RTPS), under development in NASA-KSC, is expected to automate the processes of post-flight re-water-proofing and the process of inspection of the Shuttle heat absorbing tiles. An important task of the robot vision sub-system is to register the 'real-world' coordinates with the coordinates of the robot model of the Shuttle tiles. The model coordinates relate to a tile data-base and pre-flight tile-images. In the registration process, current (post-flight) images are aligned with pre-flight images to detect the rotation and translation displacement required for the coordinate systems rectification. The research activities performed this summer included study and evaluation of the registration algorithm that is currently implemented by the RTPS, as well as, investigation of the utility of other registration algorithms. It has been found that the current algorithm is not robust enough. This algorithm has a success rate of less than 80% and is, therefore, not suitable for complying with the requirements of the RTPS. Modifications to the current algorithm has been developed and tested. These modifications can improve the performance of the registration algorithm in a significant way. However, this improvement is not sufficient to satisfy system requirements. A new algorithm for registration has been developed and tested. This algorithm presented very high degree of robustness with success rate of 96%

Structure of catalase determined by MicroED.

MicroED is a recently developed method that uses electron diffraction for structure determination from very small three-dimensional crystals of biological material. Previously we used a series of still diffraction patterns to determine the structure of lysozyme at 2.9 Å resolution with MicroED (Shi et al., 2013). Here we present the structure of bovine liver catalase determined from a single crystal at 3.2 Å resolution by MicroED. The data were collected by continuous rotation of the sample under constant exposure and were processed and refined using standard programs for X-ray crystallography. The ability of MicroED to determine the structure of bovine liver catalase, a protein that has long resisted atomic analysis by traditional electron crystallography, demonstrates the potential of this method for structure determination