Eliyahu Osherovich

Approximate Fourier phase information in the phase retrieval problem: what it gives and how to use it.

This work evaluates the importance of approximate Fourier phase information in the phase retrieval problem. The main discovery is that a rough phase estimate (up to π/2 rad) allows development of very efficient algorithms whose reconstruction time is an order of magnitude faster than that of the current method of choice--the hybrid input-output (HIO) algorithm. Moreover, a heuristic explanation is provided of why continuous optimization methods like gradient descent or Newton-type algorithms fail when applied to the phase retrieval problem and how the approximate phase information can remedy this situation. Numerical simulations are presented to demonstrate the validity of our analysis and success of our reconstruction method even in cases where the HIO algorithm fails, namely, complex-valued signals without tight support information.

Covering a continuous domain by distributed, limited robots

We present an algorithm for covering continuous domains by primitive robots whose only ability is to mark visited places with pheromone and to sense the level of the pheromone in their neighborhood. These pheromone marks can be sensed by all robots and thus provide a way for indirect communication between the robots. Apart from this, the robots have no means to communicate. Additionally they are memoryless, have no global information such as the domain map, own position, coverage percentage, etc. Despite the robots’ simplicity, we show that they are able to cover efficiently any connected domains, including non-planar ones.

All triangulations are reachable via sequences of edge-flips: an elementary proof

A simple proof is provided for the fact that the set of all possible triangulations of a planar point set in a polygonal domain is closed under the basic diagonal flip operation.

Fast Reconstruction Method for Diffraction Imaging

We present a fast image reconstruction method for two- and three-dimensional diffraction imaging. Provided that very little information about the phase is available, the method demonstrates convergence rates that are several orders of magnitude faster than current reconstruction techniques. Unlike current methods, our approach is based on convex optimization. Besides fast convergence, our method allows great deal of flexibility in choosing most appropriate objective function as well as introducing additional information about the sought signal, e.g., smoothness. Benefits of good choice of the objective function are demonstrated by reconstructing an image from noisy data.

Sparsity based super-resolution in optical measurements

We propose and experimentally demonstrate a method of exploiting prior knowledge of a signals sparsity to perform super-resolution in various optical measurements, including: single-shot sub-wavelength Coherent Diffractive Imaging (CDI), i.e. algorithmic object reconstruction from Fourier amplitude measurements, and ultra-fast pulse measurement, i.e. exceeding the temporal resolution imposed by the rise time of the photodiode. The prior knowledge of the signals sparsity compensates for the loss of phase information and the loss of high spatial frequencies in the case of CDI, and for the loss of temporal frequencies accompanying the photodiode measurement process.