6 research outputs found
Heterogeneous multireference alignment: a single pass approach
Multireference alignment (MRA) is the problem of estimating a signal from
many noisy and cyclically shifted copies of itself. In this paper, we consider
an extension called heterogeneous MRA, where signals must be estimated, and
each observation comes from one of those signals, unknown to us. This is a
simplified model for the heterogeneity problem notably arising in cryo-electron
microscopy. We propose an algorithm which estimates the signals without
estimating either the shifts or the classes of the observations. It requires
only one pass over the data and is based on low-order moments that are
invariant under cyclic shifts. Given sufficiently many measurements, one can
estimate these invariant features averaged over the signals. We then design
a smooth, non-convex optimization problem to compute a set of signals which are
consistent with the estimated averaged features. We find that, in many cases,
the proposed approach estimates the set of signals accurately despite
non-convexity, and conjecture the number of signals that can be resolved as
a function of the signal length is on the order of .Comment: 6 pages, 3 figure
Rotationally Invariant Image Representation for Viewing Direction Classification in Cryo-EM
We introduce a new rotationally invariant viewing angle classification method
for identifying, among a large number of Cryo-EM projection images, similar
views without prior knowledge of the molecule. Our rotationally invariant
features are based on the bispectrum. Each image is denoised and compressed
using steerable principal component analysis (PCA) such that rotating an image
is equivalent to phase shifting the expansion coefficients. Thus we are able to
extend the theory of bispectrum of 1D periodic signals to 2D images. The
randomized PCA algorithm is then used to efficiently reduce the dimensionality
of the bispectrum coefficients, enabling fast computation of the similarity
between any pair of images. The nearest neighbors provide an initial
classification of similar viewing angles. In this way, rotational alignment is
only performed for images with their nearest neighbors. The initial nearest
neighbor classification and alignment are further improved by a new
classification method called vector diffusion maps. Our pipeline for viewing
angle classification and alignment is experimentally shown to be faster and
more accurate than reference-free alignment with rotationally invariant K-means
clustering, MSA/MRA 2D classification, and their modern approximations
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