Ph.D Student Shechtman Yoav Aspects of Sparsity-Based Reconstruction of Optical Information Department of Physics ? 18? Mordechai Segev Abstract

Many problems in optics are inverse problems - i.e. the recovery of an unknown optical signal (object, communication signal, pulse, etc.) from measurements related to it.  Many of these problems are ill posed - i.e. there are many signals that would yield the measurements.  For example, consider the problem of sub-wavelength imaging:  when light of wavelength propagates in media with refractive index n, only spatial frequencies below can propagate, whereas all frequencies above are rendered evanescent and decay exponentially with propagation distance. Hence, for propagation distances larger than a few wavelengths, diffraction in a homogeneous medium acts as a low-pass filter - an effect known as diffraction limit. Consequently, optical features of sub-wavelength resolution are blurred in a microscope.  As a result, given a blurred image, there is are many possible objects that would yield this image when being imaged using the optical system.  In this case, inverting the problem means finding the true object out of the many possible objects that would yield the measurements.  The only way to select the true object is by using some prior information on the object.

Here, I present my research work describing the use of signal sparsity as such prior knowledge.  A signal is said to be sparse, when it can be represented compactly in a known mathematical basis.  I show in my work that the prior knowledge of sparsity can be used to solve various ill-posed optical inverse problems: For example, a sparse sub-wavelength object can be recovered from measurements of its blurred image, under spatially coherent, incoherent or partially incoherent illumination.  In addition, the prior information of sparsity can allow the recovery of an object from its far-field diffraction pattern intensity, overcoming both the diffraction limit and the lack of optical phase measurement (the "phase-retrieval" problem). Other applications include coherent diffractive imaging of sparsely varying dynamic objects, and communication through a waveguide array.

There are many algorithmic methods that solve sparse linear equations.  However, most of the inverse problems I have dealt with require solving a sparse quadratic problem.  Namely, the relation between the measurements and the unknown signal is quadratic.  This is because optical fields oscillate much faster than the typical response time of an optical detector, leading to the measurement of time averaged intensity, which contains no phase information.  Consequently, part of my work also required developing a method (called GESPAR) for finding sparse solutions to quadratic equations.