The main focus of this thesis is on the phase retrieval problem. This problem has a broad range of applications in advanced imaging systems, such as X-ray crystallography, coherent diffraction imaging, and astrophotography. Thanks to its broad applications and its mathematical elegance and sophistication, phase retrieval has attracted researchers with diverse backgrounds. Formally, phase retrieval is the problem of recovering a signal 𝔁 ∈ ℂⁿ from its phaseless linear measurements of the form |𝛼ᵢ∗𝔁| + 𝜖ᵢ where sensing vectors 𝛼ᵢ, 𝑖 = 1, 2, ..., 𝓶, are in the same vector space as 𝔁 and 𝜖ᵢ denotes the measurement noise. Finding an effective recovery method in a practical setup, analyzing the required sample complexity and convergence rate of a solution, and discussing the optimality of a proposed solution are some of the major mathematical challenges that researchers have tried to address in the last few years. In this thesis, our aim is to shed some light on some of these challenges and propose new ways to improve the imaging systems that have this problem at their core. Toward this goal, we focus on the high-dimensional setting where the ratio of the number of measurements to the ambient dimension of the signal remains bounded. This regime differs from the classical asymptotic regime in which the signal's dimension is fixed and the number of measurements is increasing. We obtain sharp results regarding the performance of the existing algorithms and the algorithms that are introduced in this thesis. To achieve this goal, we first develop a few sharp concentration inequalities. These inequalities enable us to obtain sharp bounds on the performance of our algorithms. We believe such results can be useful for researchers who work in other research areas as well. Second, we study the spectrum of some of the random matrices that play important roles in the phase retrieval problem, and use our tools to study the performance of some of the popular phase retrieval recovery schemes. Finally, we revisit the problem of structured signal recovery from phaseless measurements. We propose an iterative recovery method that can take advantage of any prior knowledge about the signal that is given as a compression code to efficiently solve the problem. We rigorously analyze the performance of our proposed method and provide extensive simulations to demonstrate its state-of-the-art performance.

Non-destructive imaging modalities for evaluating the internal properties of materials can be formulated as physics-driven inverse problems. Model-based Iterative reconstruction (MBIR) methods that integrate a forward model of the imaging system and a prior model of the object being imaged can provide superior reconstruction quality relative to conventional methods. However, making MBIR feasible for practical applications faces two key challenges. First, we require efficient computational methods for MBIR that allow large-scale reconstructions in real-time. Second, we must develop forward models that accurately capture the physics and geometry of the imaging system, and, support the use of advanced denoisers that enhance image quality as prior models.<br><br>This thesis attempts to address the aforementioned challenges and is divided into three main chapters, each corresponding to a different inverse imaging application. <br><br>In the first chapter of this thesis, we propose a novel 4D model-based iterative reconstruction (MBIR) algorithm for low-angle coherent-scatter X-ray Diffraction (XRD) tomography that can substantially increase the SNR. Our forward model is based on a Poisson photon counting model that incorporates a spatial point-spread function, detector energy response and energy-dependent attenuation correction. Our prior model uses a Markov random field (MRF) together with a reduced spectral bases set determined using non-negative matrix factorization. Our algorithm efficiently computes the Bayesian estimate by exploiting the sparsity of the measurement data. We demonstrate the ability of our method to achieve sufficient spatial resolution from sparse photon-starved measurements and also discriminate between materials of similar densities with real datasets.<br><br>In the second chapter of this thesis, we propose a multi-agent consensus equilibrium (MACE) algorithm for distributing both the computation and memory of <br>MBIR for Computed Tomographic (CT) reconstruction across a large number of parallel nodes. In MACE, each node stores only a sparse subset of views and a small portion of the system matrix, and each parallel node performs a local sparse-view reconstruction, which based on repeated feedback from other nodes, converges to the global optimum. Our distributed approach can also incorporate advanced denoisers as priors to enhance reconstruction quality. In this case, we obtain a parallel solution to the serial framework of Plug-n-play (PnP) priors, which we call MACE-PnP. In order to make MACE practical, we introduce a partial update method that eliminates nested iterations and prove that it converges to the same global solution. Finally, we validate our approach on a distributed memory system with real CT data. We also demonstrate an implementation of our approach on a massive supercomputer that can perform large-scale reconstruction in real-time. <br><br>In the third chapter of this thesis, we propose a method that makes MBIR feasible for real-time single-shot holographic imaging through deep turbulence. Our method uses surrogate optimization techniques to simplify and speedup the reflectance and phase-error updates in MBIR. Further, our method accelerates computation of the surrogate-updates by leveraging cache-prefetching and SIMD vector processing units on a single CPU core. We analyze the convergence and real CPU time of our method using simulated datasets, and demonstrate its dramatic speedup over the original MBIR approach. <br>

NiTi-based shape memory alloys (SMAs) offer a good combination of high-strength, ductility, corrosion resistance, and biocompatibility that has served them well and attracted the attention of many researchers and industries. The alloys unique thermo-mechanical ability to recover their initial shape after relatively large deformations by heating or upon unloading due to a characteristic reversible phase transformation makes them useful as damping devices, solid state actuators, couplings, etc. However, there is a need to increase the temperature of the characteristic phase transformation above 150 °C, especially in the aerospace industry where high temperatures are often seen. Prior researchers have shown that adding ternary elements (Pt, Pd, Au, Hf and Zr) to NiTi can increase transformation temperatures but most of these additions are extremely expensive, creating a need to produce cost-effective high temperature shape memory alloys (HTSMAs). Thus, the main objective of this research is to examine the relatively unstudied NiTiZr system for the ability to produce a cost effective and formable HTSMA. Transformation temperatures, precipitation paths, processability, and high-temperature oxidation are examined, specifically using high energy X-ray Diffraction (XRD) measurements, in NiTi-20 at.% Zr. This is followed by an in situ XRD study of the phase growth kinetics of the favorable H-phase nano precipitates, formed in NiTiHf and NiTiZr HTSMAs, based on prior thermo-mechanical processing in a commercial NiTi-15 at.% Hf HTSMA to examine the final processing methods and aging characteristics. Through this research, knowledge of the precipitation paths in NiTiZr and NiTiHf HTSMAs is extended and methods for characterization of phases and strains using high energy XRD are elucidated for future work in the field.

L'objectif de cette thèse était de synthétiser des particules de carbonate de calcium (CaCO3) poreuses pour applications industrielles comme charge dans du papier à cigarette, ainsi que pour l'encapsulation d’'arômes. Nous avons cherché à maîtriser les paramètres de synthèse pour obtenir de la vatérite de taille contrôlée. Nous avons étudié sa transformation à haute température et dans l’'eau, car ce polymorphe est métastable. La transition de phase vatérite/calcite a été étudié par DRX et imagerie par diffraction des rayons X cohérents qui permet d’accéder à l’'image en 3D des particules. Nous avons montré que la vatérite de taille 1 à 2µm présentant 20% de porosité peut être synthétisée de façon reproductible. Les particules préparées ont été introduites comme charge dans du papier à cigarette pour évaluer l’'impact de nouvelles formes de CaCO3 sur les propriétés physiques du papier ainsi que sur la réduction des certains composées nocifs contenus dans la fumée. Nous avons développé l’analyse conjointe de l’'absorption et de la diffraction des rayons X pour estimer la charge réelle introduite ainsi que la porosité des papiers. Nous avons démontré que l’'utilisation de CaCO3 sous forme des sphères poreuses permet d’'augmenter la diffusivité du papier et de réduire l’'émission de CO dans la fumée principale.L’encapsulation d'arômes par la co-cristallisation et l'inclusion moléculaire dans le carbonate de calcium a été aussi étudiée. Nous avons montré que CaCO3 peut être utilisé comme matrice d’'imprégnation d'arômes avec une efficacité d’'encapsulation de plus que 55%. Les particules aromatiques ont été après ajoutées dans le papier pour évaluation sensorielle. / The aim of this thesis was to synthesize porous calcium carbonate (CaCO3) particles for industrial applications as fillers for cigarette paper as well as a matrix for flavour encapsulation. We show that we can control the fabrication of porous particles of vaterite with a given size by tuning the parameters of synthesis. After the synthesis, the stability of vaterite in aqueous solution and at high temperature was studied. The phase transition was analyzed by XRD and coherent X-ray diffraction imaging that allows us to have a 3D-image of the particles. Finally, particles of 1-2 μm size with 20% porosity were reproducibly synthesized. Prepared vaterite particles were introduced as a filler in cigarette paper, with the goal to evaluate their impact on the physical properties of papers as well as on the reduction of some harmful compounds during the smoking. It was demonstrated that the use of vaterite can increase the diffusivity of paper and reduce the CO emission in the mainstream smoke. We also show that the use of X-ray absorption and diffraction can provide an estimation of the filler fraction and porosity of the papers in a non-destructive way. The encapsulation of flavours in CaCO3 particles was performed by co-crystallization and molecular inclusion. It was demonstrated that CaCO3 can be used as a matrix for flavour impregnation with more than 55% of encapsulation efficiency. Flavoured particles was added in paper for sensory evaluation. We shown that it is possible, to flavour the final product with flavoured calcium carbonate particles.

The combination of X-ray imaging and diffraction techniques provides a unique tool for structural and mechanical analysis of engineering components. A variety of modes can be employed in terms of the spatial resolution (length-scale), time resolution (frequency), and the nature of the physical quantity being interrogated. This thesis describes my contributions towards the development of novel X-ray “rich” imaging experimental techniques and data interpretation. The experimental findings have been validated via comparison with other experimental methods and numerical modelling. The combination of fast acquisition rate and high penetration properties of X-ray beams allows the collection of high-resolution 3-D tomographic data sets at submicron resolution during in situ deformation experiments. Digital Volume Correlation analysis tools developed in this study help understand crack propagation mechanisms in quasi-brittle materials and elasto-plastic deformation in co-sprayed composites. For the cases of crystalline specimens where the knowledge of “live” or residual elastic strain distributions is required, diffraction techniques have been advanced. Diffraction Strain Tomography (DST) allows non-destructive reconstruction of the 2-D (in-plane) variation of the out-of-plane strain component. Another diffraction modality dubbed Laue Orientation Tomography (LOT), a grain mapping approach has been proposed and developed based on the translate-rotate tomographic acquisition strategy. It allows the reconstruction of grain shape and orientation within polycrystalline samples, and provides information about intragranular lattice strain and distortion. The implications of this method have been thoroughly investigated. State-of-the-art engineering characterisation techniques evolve towards scrutinising submicron scale structural features and strain variation using the complementarity of X-ray imaging and diffraction. The first successful feasibility study is reported of in operando stress analysis in an internal combustion engine. Finally, further advancement of ‘rich’ imaging techniques is illustrated via the first successful application of Time-of-Flight Neutron Diffraction Strain (TOF-NDST) tomography for non-destructive reconstruction of the complete strain tensor using an inverse eigenstrain formulation.