Compressed Sensing (CS) is an emerging paradigm in which signals are recovered from undersampled nonadaptive linear measurements taken at a rate proportional to the signal's true information content as opposed to its ambient dimension. The resulting problem consists in finding a sparse solution to an underdetermined system of linear equations. It has now been established, both theoretically and empirically, that certain optimization algorithms are able to solve such problems. Iterative Hard Thresholding (IHT) (Blumensath and Davies, 2007), which is the focus of this thesis, is an established CS recovery algorithm which is known to be effective in practice, both in terms of recovery performance and computational efficiency. However, theoretical analysis of IHT to date suffers from two drawbacks: state-of-the-art worst-case recovery conditions have not yet been quantified in terms of the sparsity/undersampling trade-off, and also there is a need for average-case analysis in order to understand the behaviour of the algorithm in practice. In this thesis, we present a new recovery analysis of IHT, which considers the fixed points of the algorithm. In the context of arbitrary matrices, we derive a condition guaranteeing convergence of IHT to a fixed point, and a condition guaranteeing that all fixed points are 'close' to the underlying signal. If both conditions are satisfied, signal recovery is therefore guaranteed. Next, we analyse these conditions in the case of Gaussian measurement matrices, exploiting the realistic average-case assumption that the underlying signal and measurement matrix are independent. We obtain asymptotic phase transitions in a proportional-dimensional framework, quantifying the sparsity/undersampling trade-off for which recovery is guaranteed. By generalizing the notion of xed points, we extend our analysis to the variable stepsize Normalised IHT (NIHT) (Blumensath and Davies, 2010). For both stepsize schemes, comparison with previous results within this framework shows a substantial quantitative improvement. We also extend our analysis to a related algorithm which exploits the assumption that the underlying signal exhibits tree-structured sparsity in a wavelet basis (Baraniuk et al., 2010). We obtain recovery conditions for Gaussian matrices in a simplified proportional-dimensional asymptotic, deriving bounds on the oversampling rate relative to the sparsity for which recovery is guaranteed. Our results, which are the first in the phase transition framework for tree-based CS, show a further significant improvement over results for the standard sparsity model. We also propose a dynamic programming algorithm which is guaranteed to compute an exact tree projection in low-order polynomial time.

X-ray crystallography is a technique for determining the structure (positions of atoms in space) of molecules. It is a well developed technique, and is applied routinely to both small inorganic and large organic molecules. However, the determination of the structures of large biological molecules by x-ray crystallography can still be an experimentally and computationally expensive task. The data in an x-ray experiment are the amplitudes of the Fourier transform of the electron density in the crystalline specimen. The structure determination problem in x-ray crystallography is therefore identical to a phase retrieval problem in image reconstruction, for which iterative transform algorithms are a common solution method. This thesis is concerned with iterative projection algorithms, a generalized and more powerful version of iterative transform algorithms, and their application to macromolecular x-ray crystallography. A detailed study is made of iterative projection algorithms, including their properties, convergence, and implementations. Two applications to macromolecular crystallography are then investigated. The first concerns reconstruction of binary image and the application of iterative projection algorithms to determining molecular envelopes from x-ray solvent contrast variation data. An effective method for determining molecular envelopes is developed. The second concerns the use of symmetry constraints and the application of iterative projection algorithms to ab initio determination of macromolecular structures from crystal diffraction data. The algorithm is tested on an icosahedral virus and a protein tetramer. The results indicate that ab initio phasing is feasible for structures containing 4-fold or 5-fold non-crystallographic symmetry using these algorithms if an estimate of the molecular envelope is available.

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This work focuses on using the linear prediction based projection completion for the fan-beam and cone-beam reconstruction algorithm with no backprojection weight. The truncated data problems are addressed in the computed tomography research. However, the image reconstruction from truncated data perfectly has not been achieved yet and only approximately accurate solutions have been obtained. Thus research in this area continues to strive to obtain close result to the perfect. Linear prediction techniques are adopted for truncation completion in this work, because previous research on the truncated data problems also have shown that this technique works well compared to some other techniques like polynomial fitting and iterative based methods. The Linear prediction technique is a model based technique. The autoregressive (AR) and moving average (MA) are the two important models along with autoregressive moving average (ARMA) model. The AR model is used in this work because of the simplicity it provides in calculating the prediction coefficients. The order of the model is chosen based on the partial autocorrelation function of the projection data proved in the previous researches that have been carried out in this area of interest. The truncated projection completion using linear prediction and windowed linear prediction show that reasonably accurate reconstruction is achieved. The windowed linear prediction provide better estimate of the missing data, the reason for this is mentioned in the literature and is restated for the reader’s convenience in this work. The advantages associated with the fan-beam reconstruction algorithms with no backprojection weights compared to the fan-beam reconstruction algorithm with backprojection weights motivated us to use the fan-beam reconstruction algorithm with no backprojection weight for reconstructing the truncation completed projection data. The results obtained are compared with the previous work which used conventional fan-beam reconstruction algorithms with backprojection weight. The intensity plots and the noise performance results show improvements resulting from using the fan-beam reconstruction algorithm with no backprojection weight. The work is also extended to the Feldkamp, Davis, and Kress (FDK) reconstruction algorithm with no backprojection weight for the helical scanning geometry and the results obtained are compared with the FDK reconstruction algorithm with backprojection weight for the helical scanning geometry.