Reconstruction of Integers from Pairwise Distances

Given a set of integers, one can easily construct the set of their pairwise distances. We consider the inverse problem: given a set of pairwise distances, find the integer set which realizes the pairwise distance set. This problem arises in a lot of fields in engineering and applied physics, and has confounded researchers for over 60 years. It is one of the few fundamental problems that are neither known to be NP-hard nor solvable by polynomial-time algorithms. Whether unique recovery is possible also remains an open question. In many practical applications where this problem occurs, the integer set is naturally sparse (i.e., the integers are sufficiently spaced), a property which has not been explored. In this work, we exploit the sparse nature of the integer set and develop a polynomial-time algorithm which provably recovers the set of integers (up to linear shift and reversal) from the set of their pairwise distances with arbitrarily high probability if the sparsity is O(n^{1/2-\eps}). Numerical simulations verify the effectiveness of the proposed algorithm.Comment: 14 pages, 4 figures, submitted to ICASSP 201

Multiple Illumination Phaseless Super-Resolution (MIPS) with Applications To Phaseless DOA Estimation and Diffraction Imaging

Phaseless super-resolution is the problem of recovering an unknown signal from measurements of the magnitudes of the low frequency Fourier transform of the signal. This problem arises in applications where measuring the phase, and making high-frequency measurements, are either too costly or altogether infeasible. The problem is especially challenging because it combines the difficult problems of phase retrieval and classical super-resolutionComment: To appear in ICASSP 201

Recovery of sparse 1-D signals from the magnitudes of their Fourier transform

The problem of signal recovery from the autocorrelation, or equivalently, the magnitudes of the Fourier transform, is of paramount importance in various fields of engineering. In this work, for one-dimensional signals, we give conditions, which when satisfied, allow unique recovery from the autocorrelation with very high probability. In particular, for sparse signals, we develop two non-iterative recovery algorithms. One of them is based on combinatorial analysis, which we prove can recover signals up to sparsity o(n^(1/3)) with very high probability, and the other is developed using a convex optimization based framework, which numerical simulations suggest can recover signals upto sparsity o(n^(1/2)) with very high probability

Sparse Phase Retrieval: Uniqueness Guarantees and Recovery Algorithms

The problem of signal recovery from its Fourier transform magnitude is of paramount importance in various fields of engineering and has been around for over 100 years. Due to the absence of phase information, some form of additional information is required in order to be able to uniquely identify the signal of interest. In this work, we focus our attention on discrete-time sparse signals (of length n). We first show that, if the DFT dimension is greater than or equal to 2n, then almost all signals with aperiodic support can be uniquely identified by their Fourier transform magnitude (up to time-shift, conjugate-flip and global phase). Then, we develop an efficient Two-stage Sparse Phase Retrieval algorithm (TSPR), which involves: (i) identifying the support, i.e., the locations of the non-zero components, of the signal using a combinatorial algorithm (ii) identifying the signal values in the support using a convex algorithm. We show that TSPR can provably recover most O(n^(1/2-ϵ)-sparse signals (up to a timeshift, conjugate-flip and global phase). We also show that, for most O(n^(1/4-ϵ)-sparse signals, the recovery is robust in the presence of measurement noise. These recovery guarantees are asymptotic in nature. Numerical experiments complement our theoretical analysis and verify the effectiveness of TSPR

Sparse Phase Retrieval: Convex Algorithms and Limitations

We consider the problem of recovering signals from their power spectral densities. This is a classical problem referred to in literature as the phase retrieval problem, and is of paramount importance in many fields of applied sciences. In general, additional prior information about the signal is required to guarantee unique recovery as the mapping from signals to power spectral densities is not one-to-one. In this work, we assume that the underlying signals are sparse. Recently, semidefinite programming (SDP) based approaches were explored by various researchers. Simulations of these algorithms strongly suggest that signals upto O(n^(1/2−ϵ) sparsity can be recovered by this technique. In this work, we develop a tractable algorithm based on reweighted ℓ_1-minimization that recovers a sparse signal from its power spectral density for significantly higher sparsities, which is unprecedented. We also discuss the limitations of the existing SDP algorithms and provide a combinatorial algorithm which requires significantly fewer ”phaseless” measurements to guarantee recovery

STFT Phase Retrieval: Uniqueness Guarantees and Recovery Algorithms

The problem of recovering a signal from its Fourier magnitude is of paramount importance in various fields of engineering and applied physics. Due to the absence of Fourier phase information, some form of additional information is required in order to be able to uniquely, efficiently, and robustly identify the underlying signal. Inspired by practical methods in optical imaging, we consider the problem of signal reconstruction from the short-time Fourier transform (STFT) magnitude. We first develop conditions under, which the STFT magnitude is an almost surely unique signal representation. We then consider a semidefinite relaxation-based algorithm (STliFT) and provide recovery guarantees. Numerical simulations complement our theoretical analysis and provide directions for future work

Non-prescription sale of antibiotics in pharmacies across Puducherry, India

Background: Despite being illegal, non-prescription sales of antibiotics in India continue to be a major contributor to antibiotic abuse, which fosters antibiotic resistance.Methods: Two trained actors simulated symptoms of acute gastroenteritis (AGE) and upper respiratory tract infections (URTI) in 60 pharmacies each randomly selected in the Union Territory of Pondicherry, India. The ease of obtaining antibiotics, any additional enquiries made by the pharmacies, and any additional advise given were noted.Results: Only 33/120 pharmacies (27.5%) declined to dispense antibiotics without prescription; all 33 were attached to a hospital or nursing home. The most frequently dispensed antibiotics for AGE and URTI were ciprofloxacin (41.4%) and coamoxiclav (41.3%) respectively. Out of the 87 pharmacies which dispensed antibiotics without prescription, the presence of additional symptoms and previous drug allergy were enquired by 20 (22.9%) and 9 (10.3%) pharmacies respectively. While over half of the pharmacies gave instructions regarding dose, duration and frequency of antibiotic consumption, none of the pharmacies provided information regarding adverse reaction profile of antibiotics. Non-pharmacological measures for symptomatic improvement were advised by 24/120 pharmacies (20%).Conclusions: Non-prescription sales of antibiotics are unacceptably high in Pondicherry. Stricter implementation of the law and public awareness of the perils of inappropriate antibiotic usage are the need of the hour

STFT Phase Retrieval: Uniqueness Guarantees and Recovery Algorithms

The problem of recovering a signal from its Fourier magnitude is of paramount importance in various fields of engineering and applied physics. Due to the absence of Fourier phase information, some form of additional information is required in order to be able to uniquely, efficiently, and robustly identify the underlying signal. Inspired by practical methods in optical imaging, we consider the problem of signal reconstruction from the short-time Fourier transform (STFT) magnitude. We first develop conditions under, which the STFT magnitude is an almost surely unique signal representation. We then consider a semidefinite relaxation-based algorithm (STliFT) and provide recovery guarantees. Numerical simulations complement our theoretical analysis and provide directions for future work