Opportunities to use synthetic aperture radar (SAR) in scientific studies and military operations are expanding with the development of small SAR systems that can be operated on small unmanned air vehicles (UAV)s. While the nimble nature of small UAVs make them an attractive platform for many reasons, small UAVs are also more prone to deviate from a linear course due autopilot errors and external forces such as turbulence and wind. Thus, motion compensation and improved processing algorithms are required to properly focus the SAR images. The work of this dissertation overcomes some of the challenges and addresses some of the opportunities of operating SAR on small UAVs. Several contributions to SAR backprojection processing for UAV SARs are developed including: 1. The derivation of a novel SAR backprojection algorithm that accounts for motion during the pulse that is appropriate for narrow or ultra-wide-band SAR. 2. A compensation method for SAR backprojection to enable radiometrically accurate image processing. 3. The design and implementation of a real-time backprojection processor on a commercially available GPU that takes advantage of the GPU texture cache. 4. A new autofocus method that improves the image focus by estimating motion measurement errors in three dimensions, correcting for both amplitude and phase errors caused by inaccurate motion parameters. 5. A generalization of factorized backprojection, which we call the Dually Factorized Backprojection method, that factorizes the correlation integral in both slow-time and fast-time in order to efficiently account for general motion during the transmit of an LFM-CW pulse. Much of this work was conducted in support of the Characterization of Arctic Sea Ice Experiment (CASIE), and the appendices provide substantial contributions for this project as well, including: 1. My work in designing and implementing the digital receiver and controller board for the microASAR which was used for CASIE. 2. A description of how the GPU backprojection was used to improved the CASIE imagery. 3. A description of a sample SAR data set from CASIE provided to the public to promote further SAR research.

Factorized backprojection is a processing algorithm for reconstructing images from data collected by synthetic aperture radar (SAR) systems. Factorized backprojection requires less computation than conventional time-domain backprojection with little loss in accuracy for straight-line motion. However, its implementation is not as straightforward as direct backprojection. Further, implementing an azimuth window has been difficult in previous versions of factorized backprojection. This thesis provides a new, easily parallelizable formulation of factorized backprojection designed for both pulsed and linearly frequency modulated continuous wave (LFM-CW) stripmap SAR data. A method of easily implementing an azimuth window as part of the factorized backprojection algorithm is introduced. The approximations made in factorized backprojection are investigated and a detailed analysis of the corresponding errors is provided. We compare the performance of windowed factorized backprojection to direct backprojection for simulated and actual SAR data.

Synthetic aperture radar (SAR) is a type of radar capable of high-resolution coherent imaging. In order to produce coherent imagery from raw SAR data, an image formation algorithm is employed. The various image formation algorithms have strengths and weaknesses. As this work shows, time-domain backprojection is one algorithm whose strengths are particularly well-suited to use at low-altitudes. This work presents novel research in three areas regarding time-domain backprojection. The first key contribution of this work is a detailed analysis of SAR time-domain backprojection. The work derives a general form of backprojection from first principles. It characterizes the sensitivities of backprojection to the various inputs as well as error sources and performance characteristics. This work then shows what situations are particularly well-suited to use of the backprojection algorithm, namely regimes with turbulent motion and wide variation in incidence angle across the range swath (e.g., low-altitude, airborne SAR).The second contribution of this work is an analysis of geometric signal correlation for multi-static, sometimes termed multiple-input and multiple-output (MIMO), imaging. Multi-static imaging involves forming multiple images using different combinations of transmitters and receivers. Geometric correlation is a measure of how alike observations of a target are from different aspect angles. This work provides a novel model for geometric correlation which may be used to determine the degree to which multi-static images are correlated. This in turn determines their applicable use: operating in the highly correlated regime is desirable for coherent processing whereas operating in a lower-correlation regime is desirable for obtaining independent looks. The final contribution of this work is a novel algorithm for interferometry based on backprojected data. Because of the way backprojected images are formed, they are less suited to traditional interferometric methods. This work derives backprojection interferometry and compares it to the traditional method of interferometry. The sensitivity and performance of backprojection interferometry are shown, as well as where backprojection interferometry offers superior results. This work finds that backprojection interferometry performs better with longer interferometric baseline lengths or systems with large measurement error in the baseline length or angle (e.g., low-altitude, airborne SAR).

This group project aims to investigate the possibility ofconstructing a prototype micro-CT-scanner, hopefullyfurthering the advancement in the field of small CT-scanners,with the ultimate goal to provide a structure useable in anambulance, to scan and send images to the hospital, so thattreatment can commence directly upon arrival.This report deals with the matter of reconstructing the takenimages to construct a three-dimensional model of thescanned object. An algorithm is developed in a mannerunderstandable to those not familiar with traditionalreconstruction techniques, simplifying the understanding ofCT backprojection while also providing a useful algorithm touse along with the scanner setup. The working principles ofthe reconstruction algorithm are explained along with itsdevelopment process, and the project ends with clear resultsin the form of an “Adaptive Backprojection” algorithm.

Synthetic Aperture Radar (SAR) is a type of radar system that can generate high-resolution images with which one can detect subtle changes on the scale of centimetres from space. It can operate in any weather condition and during both day and night, making it unique compared to optical sensors. SAR is used for applications such as environmental monitoring, surveillance, and earth observation. Its ability to penetrate clouds and, to some extent, vegetation, allows for insights into terrain, vegetation structure, and even subsurface features. The importance of modelling the generated data of a SAR system before initiating the construction and development of it cannot be overstated. This thesis presents the implementation of the Reverse BackProjection Algorithm (RBPA) designed to generate raw SAR data efficiently and accurately. The RBPA stands out with its flexibility, enabling researchers and designers to simulate and gauge the SAR system's effectiveness under diverse scenarios. This provides an easy way of fine-tuning configurations for distinct needs concerning scene geometries, orbits, and radar designs. Two versions of the RBPA were implemented, differing slightly in the theoretical approach of azimuth defocusing. On top of this, a bistatic mode and Terrain Observation by Progressive Scans (TOPS) acquisition mode was also implemented. The inclusion of these two modes were specifically due to their relevance for the upcoming European Space Agency (ESA) SAR mission, Harmony. The addition of the TOPS mode required a comprehensive design of the antenna framework. Moreover, this implementation also paves the way for simpler integration of modes in the future. The two versions of the RBPA were profiled, revealing the optimal system and parameter configurations.

State-of-the-art reconstruction algorithms for medical helical cone-beam Computed Tomography (CT) are of type non-exact Filtered Backprojection (FBP). They are attractive because of their simplicity and low computational cost, but they produce sub-optimal images with respect to artifacts, resolution, and noise. This thesis deals with possibilities to improve the image quality by means of iterative techniques. The first algorithm, Regularized Iterative Weighted Filtered Backprojection (RIWFBP), is an iterative algorithm employing the non-exact Weighted FilteredBackprojection (WFBP) algorithm [Stierstorfer et al., Phys. Med. Biol. 49, 2209-2218, 2004] in the update step. We have measured and compared artifact reduction as well as resolution and noise properties for RIWFBP and WFBP. The results show that artifacts originating in the non-exactness of the WFBP algorithm are suppressed within five iterations without notable degradation in terms of resolution versus noise. Our experiments also indicate that the number of required iterations can be reduced by employing a technique known as ordered subsets. A small modification of RIWFBP leads to a new algorithm, the Weighted Least Squares Iterative Filtered Backprojection (WLS-IFBP). This algorithm has a slightly lower rate of convergence than RIWFBP, but in return it has the attractive property of converging to a solution of a certain least squares minimization problem. Hereby, theory and algorithms from optimization theory become applicable. Besides linear regularization, we have examined edge-preserving non-linear regularization.In this case, resolution becomes contrast dependent, a fact that can be utilized for improving high contrast resolution without degrading the signal-to-noise ratio in low contrast regions. Resolution measurements at different contrast levels and anthropomorphic phantom studies confirm this property. Furthermore, an even morepronounced suppression of artifacts is observed. Iterative reconstruction opens for more realistic modeling of the input data acquisition process than what is possible with FBP. We have examined the possibility to improve the forward projection model by (i) multiple ray models, and (ii) calculating strip integrals instead of line integrals. In both cases, for linearregularization, the experiments indicate a trade off: the resolution is improved atthe price of increased noise levels. With non-linear regularization on the other hand, the degraded signal-to-noise ratio in low contrast regions can be avoided. Huge input data sizes make experiments on real medical CT data very demanding. To alleviate this problem, we have implemented the most time consuming parts of the algorithms on a Graphics Processing Unit (GPU). These implementations are described in some detail, and some specific problems regarding parallelism and memory access are discussed.

One of the most important challenges in dynamic magnetic resonance imaging (MRI) is to achieve high spatial and temporal resolution when it is limited by system performance. It is desirable to acquire data fast enough to capture the dynamics in the image time series without losing high spatial resolution and signal to noise ratio. Many techniques have been introduced in the recent decades to achieve this goal. Newly developed algorithms like Highly Constrained Backprojection (HYPR) and Compressed Sensing (CS) reconstruct images from highly undersampled data using constraints. Using these algorithms, it is possible to achieve high temporal resolution in the dynamic image time series with high spatial resolution and signal to noise ratio (SNR). In this thesis we have analyzed the performance of HYPR to CS algorithm. In assessing the reconstructed image quality, we considered computation time, spatial resolution, noise amplification factors, and artifact power (AP) using the same number of views in both algorithms, and that number is below the Nyquist requirement. In the simulations performed, CS always provides higher spatial resolution than HYPR, but it is limited by computation time in image reconstruction and SNR when compared to HYPR. HYPR performs better than CS in terms of SNR and computation time when the images are sparse enough. However, HYPR suffers from streaking artifacts when it comes to less sparse image data.

In this dissertation, two recent problems from tomographic imaging are studied, and results from numerical simulations with synthetic data are presented. The first part deals with ultrasound modulated optical tomography, a method for imaging interior optical properties of partially translucent media that combines optical contrast with ultrasound resolution. The primary application is the optical imaging of soft tissue, for which scattering and absorption rates contain important functional and structural information about the physiological state of tissue cells. We developed a mathematical model based on the diffusion approximation for photon propagation in highly scattering media. Simple reconstruction schemes for recovering optical absorption rates from boundary measurements with focused ultrasound are presented. We show numerical reconstructions from synthetic data generated for mathematical absorption phantoms. The results indicate that high resolution imaging with quantitatively correct values of absorption is possible. Synthetic focusing techniques are suggested that allow reconstruction from measurements with certain types of non-focused ultrasound signals. A preliminary stability analysis for a linearized model is given that provides an initial explanation for the observed stability of reconstruction. In the second part, backprojection schemes are proposed for the detection of small amounts of highly enriched nuclear material inside 3D volumes. These schemes rely on the geometrically singular structure that small radioactive sources represent, compared to natural background radiation. The details of the detection problem are explained, and two types of measurements, collimated and Compton-type measurements, are discussed. Computationally, we implemented backprojection by counting the number of particle trajectories intersecting each voxel of a regular rectangular grid covering the domain of detection. For collimated measurements, we derived confidence estimates indicating when voxel trajectory counts are deviating significantly from what is expected from background radiation. Monte Carlo simulations of random background radiation confirm the estimated confidence values. Numerical results for backprojection applied to synthetic measurements are shown that indicate that small sources can be detected for signal-to-noise ratios as low as 0.1%.

DIRA-3D is an iterative model-based reconstruction method for dual-energy helical CT whose goal is to determine the material composition of the patient from accurate linear attenuation coefficients (LACs). Possible applications are, for example, to aid in calculations of radiation transport and dose calculations in brachytherapy with low energy photons, and in proton therapy. There was a need to replace the current image reconstruction method, the PI-method, with a weighted filtered backprojection (wFBP) algorithm for image reconstruction, since wFBP is used for image reconstruction in Siemens's CT-scanners. The new DIRA-3D algorithm implemented the program take for cone-beam projection generation and the FreeCT wFBP algorithm for image reconstruction. Experiments showed that the accuracies of the resulting LACs for the DIRA-3D algorithm using wFBP for image reconstruction were comparable to the one using the PI-method for image reconstruction. The relative LAC errors reached a value below 0.2% after 10 iterations.

Synthetic aperture radar (SAR) was originally designed as an airborne ground-imaging radar technology. But it has long been desired to also be able to use SAR imaging systems to detect, locate, and track moving ground targets, a process called Ground Moving Target Indication (GMTI). Unfortunately, due to the nature of how SAR works, it is inherently poorly suited to the task of GMTI. SAR only focuses targets and image features that remain stationary during the data collection. A moving ground target therefore does not focus in a conventional SAR image, which complicates the process of performing GMTI with SAR systems. This thesis investigates the feasibility of performing GMTI with single-channel, unsquinted, broadside stripmap SAR despite this inherent limitation. This study focuses solely on the idealized case of direct energy returns from point targets on flat ground, where they and the airborne radar platform all move rectilinearly with constant speed. First, the various aspects of how SAR works, the signal processing used to collect the SAR data, and the backprojection image formation algorithm are explained. The effects of target motion are described and illustrated in actual and simulated SAR images. It is shown how the backprojection (BPJ) algorithm, typically used to image a stationary landscape scene, can also focus on moving targets when the target motion is known a priori. A SAR BPJ ambiguity function is also derived and presented. Next, the time-changing geometry between the airborne radar and a ground target is mathematically analyzed, and it is shown that the slant range between the radar and any ground target, moving or stationary, is a hyperbolic function of time. It is then shown that this hyperbolic range history causes the single-channel SAR GMTI problem to be underdetermined. Finally, a method is then presented for resolving the underdetermined nature of the problem. This is done by constraining a target's GMTI solution using contextual information in the SAR image. Using constraining information, a theoretical way is presented to perform limited GMTI with a single-channel SAR system by using a modified form of the BPJ imaging algorithm, and practical considerations are addressed that complicate the process. Instead of focusing on stationary pixels, this GMTI method uses the BPJ ambiguity function to search for moving targets on a straight path, such as a road, by performing matched filtering on a collection of moving pixels in a position-velocity image space. Nevertheless, it is concluded that for moving point targets, general GMTI with no path constraints is infeasible in practice with a single-channel SAR.