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This work reports three reconstruction algorithms developed to address the practical issues encountered in 3D tomography diagnostics, such as the limited view angles available in many practical applications, the large scale and nonlinearity of the problems when they are in 3D, and the measurement uncertainty. These algorithms are: an algebraic reconstruction technique (ART) screening algorithm, a nonlinear iterative reconstruction technique (NIRT), and an iterative reconstruction technique integrating view registration optimization (IRT-VRO) algorithm. The ART screening algorithm was developed to enhance the performance of the traditional ART algorithm to solve linear tomography problems, the NIRT was to solve nonlinear tomography problems, and the IRT-VRO was to address the issue of view registration uncertainty in both linear and nonlinear problems. This dissertation describes the mathematical formulations, and the experimental and numerical validations for these algorithms. It is expected that the results obtained in this dissertation to lay the groundwork for their further development and expanded adaption in the deployment of tomography diagnostics in various practical applications. / Ph. D.

Nephrolithiasis can be a painful problem due to presence of kidney stones. Kidney stone is among the common painful disorders of the urinary system. Various imaging modalities are used to diagnose patients with symptoms of renal or urinary tract disease such as plain kidney, ureter, bladder x-ray (KUB), intravenous pyelography (IVP), and computed tomography (CT). As a traditional three-dimensional (3D) nephrolithiasis and kidney stones detection technique, computed tomography (CT) provides detailed cross-sectional images as well as 3D structure of kidney from moving the x-ray beam in a circle around the body. However, the risk of CT scans of the kidney is relatively higher exposure to radiation which is more than regular x-rays. C-arm technique is a new x-ray imaging modality that uses 2D array detector and cone shaped x-ray beam to create 3D information about the scanned object. Both x-ray source and 2D array detector cells mounted on C-shaped wheeled structure (C-arm). A series of projection images are acquired by rotating the C-arm around the patient in along circular path with a single rotation. The characteristic structure of C-arm allows to provide wide variety of movements around the patient that helps to remain the patient stationary during scanning time. In this work, we investigated a C-arm technique to generate a series of tomographic images for nephrolithiasis and detection of kidney stones. C-arm tomographic technique (C-arm tomosynthesis) as a new three dimensional (3D) kidney imaging method that provides a series of two dimensional (2D) images along partial circular orbit over limited view angle. Our experiments were done with kidney phantom which formed from a pig kidney with two embedded kidney stones inside it and low radiation dosage. Radiation dose and scanning time needed for kidney imaging are all dramatically reduced due to the cone beam geometry and also to limitation of angular rotation. To demonstrate the capability of our C-arm tomosynthesis to generate 3D kidney information for kidney stone detection, two groups of tomographic image reconstruction algorithms were developed for C-arm tomosynthesis: direct algorithms such as filtered back projection (FBP) and iterative algorithms such as simultaneous algebraic reconstruction technique (SART), maximum likelihood expectation maximization (MLEM), ordered- subset maximum likelihood expectation maximization (OS-MLEM) and Pre-computed penalized likelihood reconstruction (PPL). Three reconstruction methods were investigated including: pixel-driven method (PDM), ray-driven method (RDM) and distance driven method (DDM). Each method differs in their efficiency of calculation accuracy per computing time. Preliminary results demonstrated the capability of proposed technique to generate volumetric data about the kidney for nephrolithiasis and kidney stone detection by using all investigated reconstruction algorithms. In spite of each algorithms differs in their strategies, embedded kidney stone can be clearly visualized in all reconstruction results. Computer simulation studies were also done on simulated phantom to evaluate the results for each reconstruction algorithm. To mimic kidney phantom, simulated phantom was simulated with two different size kidney stones. Dataset of projection images was collated by using a virtual C-arm tomosynthesis with geometric configuration similar to real technique. All investigated algorithms were used to reconstruct 3D information. Different of image quality functions were applied to evaluate the imaging system and the reconstruction algorithms. The results show the capability of C-arm tomosynthesis to generate 3D information of kidney structures and to identify the size and location of kidney stones with limited amount of radiation dose.

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This thesis presents two quantitative holographic reconstruction techniques for the imaging of dielectric targets. The first method is a quasi-real-time holographic reconstruction technique, which is capable of imposing physically based constraints on the real and imaginary parts of the permittivity. The other method is a real-time holographic reconstruction technique that is faster than the constrained method but cannot accommodate constraints on the reconstructed permittivity in its current form. The goal of this thesis is to introduce both methods and recommend which is best. Microwave holography has been used by our research group to reconstruct images of a target’s shape and location from microwave scattering parameters. This thesis will demonstrate that holography can be extended to quantify the permittivity distribution in a region of interest. The problems presented in this thesis are generic and are meant to show that near-field quantitative holography is a valid approach for applications such as tissue imaging, baggage inspection, concealed weapon detection, etc. The holographic inversion is carried out in the spectral domain (Fourier space), which allows for the use of Fourier transform properties to expedite the algorithm. This differs from sensitivity-based imaging (another inversion method developed by Tu et al. (2015)) where the inversion is performed in real space and is unable to take advantage of the techniques proposed in this thesis to improve the speed of reconstruction. Mutual coupling is not taken into consideration in the forward model of scattering used here; however, this technique is meant to be viewed as a foundation for a more sophisticated reconstruction algorithm, like the iterative Born method, which can overcome such limitations. Iterative reconstruction methods require an accurate initial guess, which can be provided by the quantitative technique presented in this thesis. Moreover, this technique, implementing fast and efficient linearized inversion, can serve as a module, which is called repetitively by the iterative algorithm. Such a module will take the current estimate of the total field distribution inside the imaged volume as an input and will return an estimate of complex permittivity distribution. / Thesis / Master of Applied Science (MASc)

Filtered backprojection (FBP) reconstruction algorithms are very popular in the field of X-ray computed tomography (CT) because they give advantages in terms of the numerical accuracy and computational complexity. Ramp filter based fan-beam FBP reconstruction algorithms have the position dependent weight in the backprojection which is responsible for spatially non-uniform distribution of noise and resolution, and artifacts. Many algorithms based on shift variant filtering or spatially-invariant interpolation in the backprojection step have been developed to deal with this issue. However, these algorithms are computationally demanding. Recently, fan-beam algorithms based on Hilbert filtering with inverse distance weight and no weight in the backprojection have been derived using the Hamaker’s relation. These fan-beam reconstruction algorithms have been shown to improve noise uniformity and uniformity in resolution. In this thesis, fan-beam FBP reconstruction algorithms with inverse distance back-projection weight and no backprojection weight for 2D image reconstruction are presented and discussed for the two fan-beam scan geometries -equi-angular and equispace detector array. Based on the proposed and discussed fan-beam reconstruction algorithms with inverse distance backprojection and no backprojection weight, new 3D cone-beam FDK reconstruction algorithms with circular and helical scan trajectories for curved and planar detector geometries are proposed. To start with three rebinning formulae from literature are presented and it is shown that one can derive all fan-beam FBP reconstruction algorithms from these rebinning formulae. Specifically, two fan-beam algorithms with no backprojection weight based on Hilbert filtering for equi-space linear array detector and one new fan-beam algorithm with inverse distance backprojection weight based on hybrid filtering for both equi-angular and equi-space linear array detector are derived. Simulation results for these algorithms in terms of uniformity of noise and resolution in comparison to standard fan-beam FBP reconstruction algorithm (ramp filter based fan-beam reconstruction algorithm) are presented. It is shown through simulation that the fan-beam reconstruction algorithm with inverse distance in the backprojection gives better noise performance while retaining the resolution properities. A comparison between above mentioned reconstruction algorithms is given in terms of computational complexity. The state of the art 3D X-ray imaging systems in medicine with cone-beam (CB) circular and helical computed tomography scanners use non-exact (approximate) FBP based reconstruction algorithm. They are attractive because of their simplicity and low computational cost. However, they produce sub-optimal reconstructed images with respect to cone-beam artifacts, noise and axial intensity drop in case of circular trajectory scan imaging. Axial intensity drop in the reconstructed image is due to the insufficient data acquired by the circular-scan trajectory CB CT. This thesis deals with investigations to improve the image quality by means of the Hilbert and hybrid filtering based algorithms using redundancy data for Feldkamp, Davis and Kress (FDK) type reconstruction algorithms. In this thesis, new FDK type reconstruction algorithms for cylindrical detector and planar detector for CB circular CT are developed, which are obtained by extending to three dimensions (3D) an exact Hilbert filtering based FBP algorithm for 2D fan-beam beam algorithms with no position dependent backprojection weight and fan-beam algorithm with inverse distance backprojection weight. The proposed FDK reconstruction algorithm with inverse distance weight in the backprojection requires full-scan projection data while the FDK reconstruction algorithm with no backprojection weight can handle partial-scan data including very short-scan. The FDK reconstruction algorithms with no backprojection weight for circular CB CT are compared with Hu’s, FDK and T-FDK reconstruction algorithms in-terms of axial intensity drop and computational complexity. The simulation results of noise, CB artifacts performance and execution timing as well as the partial-scan reconstruction abilities are presented. We show that FDK reconstruction algorithms with no backprojection weight have better noise performance characteristics than the conventional FDK reconstruction algorithm where the backprojection weight is known to result in spatial non-uniformity in the noise characteristics. In this thesis, we present an efficient method to reduce the axial intensity drop in circular CB CT. The efficient method consists of two steps: the first one is reconstruction of the object using FDK reconstruction algorithm with no backprojection weight and the second is estimating the missing term. The efficient method is comparable to Zhu et al.’s method in terms of reduction in axial intensity drop, noise and computational complexity. The helical scanning trajectory satisfies the Tuy-smith condition, hence an exact and stable reconstruction is possible. However, the helical FDK reconstruction algorithm is responsible for the cone-beam artifacts since the helical FDK reconstruction algorithm is approximate in its derivation. In this thesis, helical FDK reconstruction algorithms based on Hilbert filtering with no backprojection weight and FDK reconstruction algorithm based on hybrid filtering with inverse distance backprojection weight are presented to reduce the CB artifacts. These algorithms are compared with standard helical FDK in-terms of noise, CB artifacts and computational complexity.

Magnetic Resonance Electrical Impedance Tomography (MREIT) reconstructs images of electrical conductivity distribution based on magnetic flux density (B) measurements. Magnetic flux density is generated by an externally applied current on the object and measured by a Magnetic Resonance Imaging (MRI) scanner. With the measured data and peripheral voltage measurements, the conductivity distribution of the object can be reconstructed. There are two types of reconstruction algorithms. First type uses current density distributions to reconstruct conductivity distribution. Object must be rotated in MRI scanner to measure three components of magnetic flux density. These types of algorithms are called J-based reconstruction algorithms. The second type of reconstruction algorithms uses only one component of magnetic flux density which is parallel to the main magnetic field of MRI scanner. This eliminates the need of subject rotation. These types of algorithms are called B-based reconstruction algorithms. In this study four of the B-based reconstruction algorithms, proposed by several research groups, are examined. The algorithms are tested by different computer models for noise-free and noisy data. For noise-free data, the algorithms work successfully. System SNR 30, 20 and 13 are used for noisy data. For noisy data the performance of algorithm is not as satisfactory as noise-free data. Twice differentiation of z component of B (Bz) is used for two of the algorithms. These algorithms are very sensitive to noise. One of the algorithms uses only one differentiation of Bz so it is immune to noise. The other algorithm uses sensitivity matrix to reconstruct conductivity distribution.

Fluorescence diffuse optical tomography provides non-invasive, in vivo imaging of molecular targets in small animals. While standard fluorescence microscopy is limited to shallow depths and small fields of view, tomographic methods allows recovery of the distribution of fluorescent probes throughout the small animal body. In this thesis, we present novel reconstruction algorithms for the tomographic separation of optical parameters using time-domain (TD) measurements. These technique are validated using simulations and with experimental phantom and mouse imaging studies. We outline the contributions of each chapter of the thesis below. First, we explore the TD fluorescence tomography reconstruction problem for single and multiple fluorophores with discrete lifetimes. We focus on late arriving photons and compare a direct inversion approach with a two-step, asymptotic approach operating on the same TD data. We show that for lifetime multiplexing, the two methods produce fundamentally different kinds of solutions. The direct inversion is computationally inefficient and results in poor separation but has overall higher resolution while the asymptotic approach provides better separation, relative quantitation of lifetime components and localization but has overall lower resolution. We verify these results with simulation and experimental phantoms. Second, we introduce novel high resolution lifetime multiplexing algorithms which combine asymptotic methods for separation of fluorophores with the high resolving power of early photon tomography. We show the effectiveness of such methods to achieve high resolution reconstructions of multiple fluorophores in simulations with complex-shaped phantoms, a digital mouse atlas and also experimentally in fluorescent tube phantoms. Third, we compare the performance of tomographic spectral and lifetime multiplexing. We show that both of these techniques involve a two-step procedure, consisting of a diffuse propagation step and a basis-function mixing step. However, in these two techniques, the order of the two steps is switched, which leads to a fundamental difference in imaging performance. As an illustration of this difference, we show that the relative concentrations of three colocalized fluorophores in a diffuse medium can accurately be retrieved with lifetime methods but cannot be retrieved with spectral methods. Fourth, we address the long standing challenge in diffuse optical tomography (DOT) of cross-talk between absorption and scattering. We extend the ideas developed from lifetime multiplexing algorithms by using a constrained optimization approach for separation of absorption and scattering in DOT. Using custom designed phantoms, we demonstrate a novel technique allows better separation of absorption and scattering inclusions compared to existing algorithms for CW and TD diffuse optical tomography. Finally, we show experimental validation of the lifetime multiplexing algorithms developed in this thesis using three experimental models. First, we show the reconstruction of overlapping complex shapes in a dish phantom. Second, we demonstrate the localization accuracy of lifetime based methods using fluorescent pellets embedded in a sacrificed mouse. Third, we show using planar imaging and tomography, the in vivo recovery of multiple anatomically targeted near-infrared fluorophores. In summary, we have presented novel reconstruction algorithms and experimental methods that extend the capability of time-domain fluorescence diffuse optical tomography systems. The methods developed in this thesis should also have applicability for general multi-parameter image reconstruction problems. / Engineering and Applied Sciences

A reconstruction algorithm in computerized tomography is a procedure for reconstructing the attenuation coefficientscient, a real-valued function associated with the object of interest, from the measured projection data. Generally speaking, reconstruction algorithms in CT fall into two categories: direct, e.g., filtered back-projection (FBP), or iterative. In this thesis, we discuss a new fast matrix-free iterative reconstruction method based on a polyenergetic model. While most modern x-ray CT scanners rely on the well-known filtered back-projection algorithm, the corresponding reconstructions can be corrupted by beam hardening artifacts. These artifacts arise from the unrealistic physical assumption of monoenergetic x-ray beams. In this thesis, to compensate, we use an alternative model that accounts for differential absorption of polyenergetic x-ray photons and discretize it directly. We do not assume any prior knowledge about the physical properties of the scanned object. We study and implement different solvers and nonlinear unconstrained optimization methods, such as a Newton-like method and an extension of the Levenberg-Marquardt-Fletcher algorithm. We explain how we can use the structure of the Radon matrix and the properties of FBP to make our method matrix-free and fast. Finally, we discuss how we regularize our problem by applying different regularization methods, such as Tikhonov and regularization in the 1-norm. We present numerical reconstructions based on the associated nonlinear discrete formulation incorporating various iterative optimization methods.