Multi-target Detection with an Arbitrary Spacing Distribution.

Motivated by the structure reconstruction problem in single-particle cryo-electron microscopy, we consider the multi-target detection model, where multiple copies of a target signal occur at unknown locations in a long measurement, further corrupted by additive Gaussian noise. At low noise levels, one can easily detect the signal occurrences and estimate the signal by averaging. However, in the presence of high noise, which is the focus of this paper, detection is impossible. Here, we propose two approaches-autocorrelation analysis and an approximate expectation maximization algorithm-to reconstruct the signal without the need to detect signal occurrences in the measurement. In particular, our methods apply to an arbitrary spacing distribution of signal occurrences. We demonstrate reconstructions with synthetic data and empirically show that the sample complexity of both methods scales as SNR-3 in the low SNR regime.

Bispectrum Inversion with Application to Multireference Alignment.

We consider the problem of estimating a signal from noisy circularly-translated versions of itself, called multireference alignment (MRA). One natural approach to MRA could be to estimate the shifts of the observations first, and infer the signal by aligning and averaging the data. In contrast, we consider a method based on estimating the signal directly, using features of the signal that are invariant under translations. Specifically, we estimate the power spectrum and the bispectrum of the signal from the observations. Under mild assumptions, these invariant features contain enough information to infer the signal. In particular, the bispectrum can be used to estimate the Fourier phases. To this end, we propose and analyze a few algorithms. Our main methods consist of non-convex optimization over the smooth manifold of phases. Empirically, in the absence of noise, these non-convex algorithms appear to converge to the target signal with random initialization. The algorithms are also robust to noise. We then suggest three additional methods. These methods are based on frequency marching, semidefinite relaxation and integer programming. The first two methods provably recover the phases exactly in the absence of noise. In the high noise level regime, the invariant features approach for MRA results in stable estimation if the number of measurements scales like the cube of the noise variance, which is the information-theoretic rate. Additionally, it requires only one pass over the data which is important at low signal-to-noise ratio when the number of observations must be large.

Toward Single Particle Reconstruction without Particle Picking: Breaking the Detection Limit.

Single-particle cryo-electron microscopy (cryo-EM) has recently joined X-ray crystallography and NMR spectroscopy as a high-resolution structural method to resolve biological macromolecules. In a cryo-EM experiment, the microscope produces images called micrographs. Projections of the molecule of interest are embedded in the micrographs at unknown locations, and under unknown viewing directions. Standard imaging techniques first locate these projections (detection) and then reconstruct the 3-D structure from them. Unfortunately, high noise levels hinder detection. When reliable detection is rendered impossible, the standard techniques fail. This is a problem, especially for small molecules. In this paper, we pursue a radically different approach: we contend that the structure could, in principle, be reconstructed directly from the micrographs, without intermediate detection. The aim is to bring small molecules within reach for cryo-EM. To this end, we design an autocorrelation analysis technique that allows one to go directly from the micrographs to the sought structures. This involves only one pass over the micrographs, allowing online, streaming processing for large experiments. We show numerical results and discuss challenges that lay ahead to turn this proof-of-concept into a complementary approach to state-of-the-art algorithms.

Random conical tilt reconstruction without particle picking in cryo-electron microscopy.

A method is proposed to reconstruct the 3D molecular structure from micrographs collected at just one sample tilt angle in the random conical tilt scheme in cryo-electron microscopy. The method uses autocorrelation analysis on the micrographs to estimate features of the molecule which are invariant under certain nuisance parameters such as the positions of molecular projections in the micrographs. This enables the molecular structure to be reconstructed directly from micrographs, completely circumventing the need for particle picking. Reconstructions are demonstrated with simulated data and the effect of the missing-cone region is investigated. These results show promise to reduce the size limit for single-particle reconstruction in cryo-electron microscopy.

Heterogeneous multireference alignment for images with application to 2-D classification in single particle reconstruction.

Motivated by the task of 2-D classification in single particle reconstruction by cryo-electron microscopy (cryo-EM), we consider the problem of heterogeneous multireference alignment of images. In this problem, the goal is to estimate a (typically small) set of target images from a (typically large) collection of observations. Each observation is a rotated, noisy version of one of the target images. For each individual observation, neither the rotation nor which target image has been rotated are known. As the noise level in cryo-EM data is high, clustering the observations and estimating individual rotations is challenging. We propose a framework to estimate the target images directly from the observations, completely bypassing the need to cluster or register the images. The framework consists of two steps. First, we estimate rotation-invariant features of the images, such as the bispectrum. These features can be estimated to any desired accuracy, at any noise level, provided sufficiently many observations are collected. Then, we estimate the images from the invariant features. Numerical experiments on synthetic cryo-EM datasets demonstrate the effectiveness of the method. Ultimately, we outline future developments required to apply this method to experimental data.

Improved fidelity of brain microstructure mapping from single-shell diffusion MRI.

Diffusion weighted imaging (DWI) is sensitive to alterations in the diffusion of water molecules caused by microstructural barriers. Different microstructural compartments are characterized by differences in DWI signal. Diffusion tensor imaging conflates the signal from these compartments into a single tensor, which poorly represents multiple white matter fascicles and extra-axonal space. Diffusion compartment imaging (DCI) models overcome this limitation by providing parametric representations for the signal contribution of each compartment, thereby improving the fidelity of brain microstructure mapping. However, current approaches fail to identify DCI model parameters from conventional single-shell DWI with the desired accuracy. It has been demonstrated that part of this inaccuracy is due to the ill-posedness of the estimation of DCI model parameters from conventional single-shell acquisitions. In this paper, we propose to regularize the estimation problem for single-shell DWI by learning a prior distribution of DCI model parameters from DWI acquired at multiple b-values in an external population of subjects. We demonstrate that this population-informed prior enables, for the first time, accurate estimation of DCI models from single-shell DWI typically acquired in clinical practice. We validated our approach on synthetic and in vivo data of healthy subjects and patients with autism spectrum disorder. We applied the approach to population studies of brain microstructure in autism and found that introducing a population-informed prior leads to reliable detection of group differences. Our algorithm enables novel investigation from large existing DWI datasets in normal development and in disease and injury.

Estimation of a multi-fascicle model from single B-value data with a population-informed prior.

Diffusion tensor imaging cannot represent heterogeneous fascicle orientations in one voxel. Various models propose to overcome this limitation. Among them, multi-fascicle models are of great interest to characterize and compare white matter properties. However, existing methods fail to estimate their parameters from conventional diffusion sequences with the desired accuracy. In this paper, we provide a geometric explanation to this problem. We demonstrate that there is a manifold of indistinguishable multi-fascicle models for single-shell data, and that the manifolds for different b-values intersect tangentially at the true underlying model making the estimation very sensitive to noise. To regularize it, we propose to learn a prior over the model parameters from data acquired at several b-values in an external population of subjects. We show that this population-informed prior enables for the first time accurate estimation of multi-fascicle models from single-shell data as commonly acquired in clinical context. The approach is validated on synthetic and in vivo data of healthy subjects and patients with autism. We apply it in population studies of the white matter microstructure in autism spectrum disorder. This approach enables novel investigations from large existing DWI datasets in normal development and in disease.