Super-Resolution of Positive Sources: the Discrete Setup

In single-molecule microscopy it is necessary to locate with high precision point sources from noisy observations of the spectrum of the signal at frequencies capped by $f_c$, which is just about the frequency of natural light. This paper rigorously establishes that this super-resolution problem can be solved via linear programming in a stable manner. We prove that the quality of the reconstruction crucially depends on the Rayleigh regularity of the support of the signal; that is, on the maximum number of sources that can occur within a square of side length about $1/f_c$. The theoretical performance guarantee is complemented with a converse result showing that our simple convex program convex is nearly optimal. Finally, numerical experiments illustrate our methods.Comment: 31 page, 7 figure

Quantification of High-Resolution Lattice Images and Electron Holograms

Progress towards the quantification of high-resolution electron microscopy and electron holograms has been achieved using digital acquisition with a slow-scan charge-coupled device (CCD) camera. Two applications are described: the precise measurement of lattice-fringe spacings and the determination of the mean inner potential. Lattice images can be characterized by a finite sum of two-dimensional sinusoids. A new method for measurement of the frequency, amplitude and phase of each sinusoid, based on an interpolation technique in reciprocal space, is presented. The method offers considerably higher precision for measurement of lattice fringes than the optical bench and is applicable to images recorded with an electron dose of less than 1 el / Å2 and specimen areas as small as 8 Å across. The attainable precision is dependent on specimen characteristics, electron dose and the size of the measured area, and ranges from 0.001 Å to 0.05 Å. An improved method has also been developed for measurement of mean inner potential using digital off-axis electron holograms from 90° crystal wedges. The value of (-14.21 ± 0.16) V obtained for GaAs represents the most accurate measurement yet reported for the mean inner potential

Application of the Non-Hermitian Singular Spectrum Analysis to the exponential retrieval problem

We present a new approach to solve the exponential retrieval problem. We derive a stable technique, based on the singular value decomposition (SVD) of lag-covariance and crosscovariance matrices consisting of covariance coefficients computed for index translated copies of an initial time series. For these matrices a generalized eigenvalue problem is solved. The initial signal is mapped into the basis of the generalized eigenvectors and phase portraits are consequently analyzed. Pattern recognition techniques could be applied to distinguish phase portraits related to the exponentials and noise. Each frequency is evaluated by unwrapping phases of the corresponding portrait, detecting potential wrapping events and estimation of the phase slope. Efficiency of the proposed and existing methods is compared on the set of examples, including the white Gaussian and auto-regressive model noise

Bayesian Interpolation and Parameter Estimation in a Dynamic Sinusoidal Model

In this paper, we propose a method for restoring the missing or corrupted observations of nonstationary sinusoidal signals which are often encountered in music and speech applications. To model nonstationary signals, we use a time-varying sinusoidal model which is obtained by extending the static sinusoidal model into a dynamic sinusoidal model. In this model, the in-phase and quadrature components of the sinusoids are modeled as first-order Gauss–Markov processes. The inference scheme for the model parameters and missing observations is formulated in a Bayesian framework and is based on a Markov chain Monte Carlo method known as Gibbs sampler. We focus on the parameter estimation in the dynamic sinusoidal model since this constitutes the core of model-based interpolation. In the simulations, we first investigate the applicability of the model and then demonstrate the inference scheme by applying it to the restoration of lost audio packets on a packet-based network. The results show that the proposed method is a reasonable inference scheme for estimating unknown signal parameters and interpolating gaps consisting of missing/corrupted signal segments