Signal Recovery in Perturbed Fourier Compressed Sensing

In many applications in compressed sensing, the measurement matrix is a Fourier matrix, i.e., it measures the Fourier transform of the underlying signal at some specified `base' frequencies $\{u_i\}_{i=1}^M$, where $M$ is the number of measurements. However due to system calibration errors, the system may measure the Fourier transform at frequencies $\{u_i + \delta_i\}_{i=1}^M$ that are different from the base frequencies and where $\{\delta_i\}_{i=1}^M$ are unknown. Ignoring perturbations of this nature can lead to major errors in signal recovery. In this paper, we present a simple but effective alternating minimization algorithm to recover the perturbations in the frequencies \emph{in situ} with the signal, which we assume is sparse or compressible in some known basis. In many cases, the perturbations $\{\delta_i\}_{i=1}^M$ can be expressed in terms of a small number of unique parameters $P \ll M$. We demonstrate that in such cases, the method leads to excellent quality results that are several times better than baseline algorithms (which are based on existing off-grid methods in the recent literature on direction of arrival (DOA) estimation, modified to suit the computational problem in this paper). Our results are also robust to noise in the measurement values. We also provide theoretical results for (1) the convergence of our algorithm, and (2) the uniqueness of its solution under some restrictions.Comment: New theortical results about uniqueness and convergence now included. More challenging experiments now include

Analysis of Tomographic Reconstruction of 2D Images using the Distribution of Unknown Projection Angles

It is well known that a band-limited signal can be reconstructed from its uniformly spaced samples if the sampling rate is sufficiently high. More recently, it has been proved that one can reconstruct a 1D band-limited signal even if the exact sample locations are unknown, but given just the distribution of the sample locations and their ordering in 1D. In this work, we extend the analytical bounds on the reconstruction error in such scenarios for quasi-bandlimited signals. We also prove that the method for such a reconstruction is resilient to a certain proportion of errors in the specification of the sample location ordering. We then express the problem of tomographic reconstruction of 2D images from 1D Radon projections under unknown angles with known angle distribution, as a special case for reconstruction of quasi-bandlimited signals from samples at unknown locations with known distribution. Building upon our theoretical background, we present asymptotic bounds for 2D quasi-bandlimited image reconstruction from 1D Radon projections in the unknown angles setting, which commonly occurs in cryo-electron microscopy (cryo-EM). To the best of our knowledge, this is the first piece of work to perform such an analysis for 2D cryo-EM, even though the associated reconstruction algorithms have been known for a long time

A search for rotating radio transients and fast radio bursts in the Parkes high-latitude pulsar survey

Discoveries of rotating radio transients and fast radio bursts (FRBs) in pulsar surveys suggest that more of such transient sources await discovery in archival data sets. Here we report on a single-pulse search for dispersed radio bursts over a wide range of Galactic latitudes (|b| < $60^{\circ}$) in data previously searched for periodic sources by Burgay et al. We re-detected 20 of the 42 pulsars reported by Burgay et al. and one rotating radio transient reported by Burke-Spolaor. No FRBs were discovered in this survey. Taking into account this result, and other recent surveys at Parkes, we corrected for detection sensitivities based on the search software used in the analyses and the different backends used in these surveys and find that the all-sky FRB event rate for sources with a fluence above 4.0 Jy ms at 1.4 GHz to be ${\cal R} = 4.4^{+5.2}_{-3.1} \times 10^3$ FRBs day$^{-1}$ sky$^{-1}$, where the uncertainties represent a $99\%$ confidence interval. While this rate is lower than inferred from previous studies, as we demonstrate, this combined event rate is consistent with the results of all systematic FRB searches at Parkes to date and does not require the need to postulate a dearth of FRBs at intermediate latitudes.Comment: Accepted, 10 pages, 6 figure

Estimating Joint Probability Distribution With Low-Rank Tensor Decomposition, Radon Transforms and Dictionaries

In this paper, we describe a method for estimating the joint probability density from data samples by assuming that the underlying distribution can be decomposed as a mixture of product densities with few mixture components. Prior works have used such a decomposition to estimate the joint density from lower-dimensional marginals, which can be estimated more reliably with the same number of samples. We combine two key ideas: dictionaries to represent 1-D densities, and random projections to estimate the joint distribution from 1-D marginals, explored separately in prior work. Our algorithm benefits from improved sample complexity over the previous dictionary-based approach by using 1-D marginals for reconstruction. We evaluate the performance of our method on estimating synthetic probability densities and compare it with the previous dictionary-based approach and Gaussian Mixture Models (GMMs). Our algorithm outperforms these other approaches in all the experimental settings

The significance of promitochondrial structures in rat liver for mitochondrial biogenesis

1. The heavy, light and fluffy mitochondrial fractions obtained by differential centrifugation were further characterized with respect to their protein synthesizing ability in vitro, their nucleic acid content, buoyant density of their DNA and ultrastructure. 2. The light mitochondrial fraction synthesized proteins in vitro at a rate 4-5 times as high as heavy and fluffy mitochondria. The incorporation ability of this fraction was also maximally affected by the thyroid status of the animal. The radioactivity in leucyl-tRNA of the light mitochondrial fraction was about 3-4 times as high as that of the other two fractions. 3. The heavy, light and fluffy mitochondrial fractions contained small but consistent amounts of RNA and DNA. Although the DNA content was the same in all mitochondria fractions, the light mitochondria contained relatively more RNA. The buoyant density of DNA from all the fractions was 1.701g/cm3. 4. Electron microscopy revealed that the heavy mitochondria have a typical mitochondrial architecture, with densely packed cristae and a well developed double membrane. Light mitochondria were also surrounded by double membranes, but were smaller in size and contained less cristae. The fluffy fraction consisted of a mixture of well formed mitochondria and those in the process of degradation. 5. The significance of these findings in relation to mammalian mitochondrial genesis is discussed