Sampling and Reconstruction of Spatial Signals

Digital processing of signals f may start from sampling on a discrete set Γ, f →(f(ϒη))ϒηεΓ. The sampling theory is one of the most basic and fascinating topics in applied mathematics and in engineering sciences. The most well known form is the uniform sampling theorem for band-limited/wavelet signals, that gives a framework for converting analog signals into sequences of numbers. Over the past decade, the sampling theory has undergone a strong revival and the standard sampling paradigm is extended to non-bandlimited signals including signals in reproducing kernel spaces (RKSs), signals with finite rate of innovation (FRI) and sparse signals, and to nontraditional sampling methods, such as phaseless sampling. In this dissertation, we first consider the sampling and Galerkin reconstruction in a reproducing kernel space. The fidelity measure of perceptual signals, such as acoustic and visual signals, might not be well measured by least squares. In the first part of this dissertation, we introduce a fidelity measure depending on a given sampling scheme and propose a Galerkin method in Banach space setting for signal reconstruction. We show that the proposed Galerkin method provides a quasi-optimal approximation, and the corresponding Galerkin equations could be solved by an iterative approximation-projection algorithm in a reproducing kernel subspace of Lp. A spatially distributed network contains a large amount of agents with limited sensing, data processing, and communication capabilities. Recent technological advances have opened up possibilities to deploy spatially distributed networks for signal sampling and reconstruction. We introduce a graph structure for a distributed sampling and reconstruction system by coupling agents in a spatially distributed network with innovative positions of signals. We split a distributed sampling and reconstruction system into a family of overlapping smaller subsystems, and we show that the stability of the sensing matrix holds if and only if its quasi-restrictions to those subsystems have l_2 uniform stability. This new stability criterion could be pivotal for the design of a robust distributed sampling and reconstruction system against supplement, replacement and impairment of agents, as we only need to check the uniform stability of affected subsystems. We also propose an exponentially convergent distributed algorithm for signal reconstruction, that provides a suboptimal approximation to the original signal in the presence of bounded sampling noises. Phase retrieval (Phaseless Sampling and Reconstruction) arises in various fields of science and engineering. It consists of reconstructing a signal of interest from its magnitude measurements. Sampling in shift-invariant spaces is a realistic model for signals with smooth spectrum. We consider phaseless sampling and reconstruction of real-valued signals in a shift-invariant space from their magnitude measurements on the whole Euclidean space and from their phaseless samples taken on a discrete set with finite sampling density. We find an equivalence between nonseparability of signals in a shift-invariant space and their phase retrievability with phaseless samples taken on the whole Euclidean space. We also introduce an undirected graph to a signal and use connectivity of the graph to characterize the nonseparability of high-dimensional signals. Under the local complement property assumption on a shift-invariant space, we find a discrete set with finite sampling density such that signals in shift-invariant spaces, that are determined by their magnitude measurements on the whole Euclidean space, can be reconstructed in a stable way from their phaseless samples taken on that discrete set. We also propose a reconstruction algorithm which provides a suboptimal approximation to the original signal when its noisy phaseless samples are available only

Applied Harmonic Analysis and Sparse Approximation

Efficiently analyzing functions, in particular multivariate functions, is a key problem in applied mathematics. The area of applied harmonic analysis has a significant impact on this problem by providing methodologies both for theoretical questions and for a wide range of applications in technology and science, such as image processing. Approximation theory, in particular the branch of the theory of sparse approximations, is closely intertwined with this area with a lot of recent exciting developments in the intersection of both. Research topics typically also involve related areas such as convex optimization, probability theory, and Banach space geometry. The workshop was the continuation of a first event in 2012 and intended to bring together world leading experts in these areas, to report on recent developments, and to foster new developments and collaborations

Convex reconstruction from structured measurements

Convex signal reconstruction is the art of solving ill-posed inverse problems via convex optimization. It is applicable to a great number of problems from engineering, signal analysis, quantum mechanics and many more. The most prominent example is compressed sensing, where one aims at reconstructing sparse vectors from an under-determined set of linear measurements. In many cases, one can prove rigorous performance guarantees for these convex algorithms. The combination of practical importance and theoretical tractability has directed a significant amount of attention to this young field of applied mathematics. However, rigorous proofs are usually only available for certain "generic cases"---for instance situations, where all measurements are represented by random Gaussian vectors. The focus of this thesis is to overcome this drawback by devising mathematical proof techniques can be applied to more "structured" measurements. Here, structure can have various meanings. E.g. it could refer to the type of measurements that occur in a given concrete application. Or, more abstractly, structure in the sense that a measurement ensemble is small and exhibits rich geometric features. The main focus of this thesis is phase retrieval: The problem of inferring phase information from amplitude measurements. This task is ubiquitous in, for instance, in crystallography, astronomy and diffraction imaging. Throughout this project, a series of increasingly better convex reconstruction guarantees have been established. On the one hand, we improved results for certain measurement models that mimic typical experimental setups in diffraction imaging. On the other hand, we identified spherical t-designs as a general purpose tool for the derandomization of data recovery schemes. Loosely speaking, a t-design is a finite configuration of vectors that is "evenly distributed" in the sense that it reproduces the first 2t moments of the uniform measure. Such configurations have been studied, for instance, in algebraic combinatorics, coding theory, and quantum information. We have shown that already spherical 4-designs allow for proving close-to-optimal convex reconstruction guarantees for phase retrieval. The success of this program depends on explicit constructions of spherical t-designs. In this regard, we have studied the design properties of stabilizer states. These are configurations of vectors that feature prominently in quantum information theory. Mathematically, they can be related to objects in discrete symplectic vector spaces---a structure we use heavily. We have shown that these vectors form a spherical 3-design and are, in some sense, close to a spherical 4-design. Putting these efforts together, we establish tight bounds on phase retrieval from stabilizer measurements. While working on the derandomization of phase retrieval, I obtained a number of results on other convex signal reconstruction problems. These include compressed sensing from anisotropic measurements, non-negative compressed sensing in the presence of noise and identifying improved convex regularizers for low rank matrix reconstruction. Going even further, the mathematical methods I used to tackle ill-posed inverse problems can be applied to a plethora of problems from quantum information theory. In particular, the causal structure behind Bell inequalities, new ways to compare experiments to fault-tolerance thresholds in quantum error correction, a novel benchmark for quantum state tomography via Bayesian estimation, and the task of distinguishing quantum states

1-D broadside-radiating leaky-wave antenna based on a numerically synthesized impedance surface

A newly-developed deterministic numerical technique for the automated design of metasurface antennas is applied here for the first time to the design of a 1-D printed Leaky-Wave Antenna (LWA) for broadside radiation. The surface impedance synthesis process does not require any a priori knowledge on the impedance pattern, and starts from a mask constraint on the desired far-field and practical bounds on the unit cell impedance values. The designed reactance surface for broadside radiation exhibits a non conventional patterning; this highlights the merit of using an automated design process for a design well known to be challenging for analytical methods. The antenna is physically implemented with an array of metal strips with varying gap widths and simulation results show very good agreement with the predicted performance

Beam scanning by liquid-crystal biasing in a modified SIW structure

A fixed-frequency beam-scanning 1D antenna based on Liquid Crystals (LCs) is designed for application in 2D scanning with lateral alignment. The 2D array environment imposes full decoupling of adjacent 1D antennas, which often conflicts with the LC requirement of DC biasing: the proposed design accommodates both. The LC medium is placed inside a Substrate Integrated Waveguide (SIW) modified to work as a Groove Gap Waveguide, with radiating slots etched on the upper broad wall, that radiates as a Leaky-Wave Antenna (LWA). This allows effective application of the DC bias voltage needed for tuning the LCs. At the same time, the RF field remains laterally confined, enabling the possibility to lay several antennas in parallel and achieve 2D beam scanning. The design is validated by simulation employing the actual properties of a commercial LC medium