## Development & Implementation of Algorithms for Fast Image Reconstruction

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##### Author

##### Date

2011##### Permanent Link

http://hdl.handle.net/10092/5998##### Thesis Discipline

Mathematics##### Degree Grantor

University of Canterbury##### Degree Level

Doctoral##### Degree Name

Doctor of PhilosophySignal and image processing is important in a wide range of areas, including medical and astronomical imaging, and speech and acoustic signal processing. There is often a need for the reconstruction of these objects to be very fast, as they have some cost (perhaps a monetary cost, although often it is a time cost) attached to them. This work considers the development of algorithms that allow these signals and images to be reconstructed quickly and without perceptual quality loss.

The main problem considered here is that of reducing the amount of time needed for images to be reconstructed, by decreasing the amount of data necessary for a high quality image to be produced. In addressing this problem two basic ideas are considered. The first is a subset selection problem where the aim is to extract a subset of data, of a predetermined size, from a much larger data set. To do this we first need some metric with which to measure how `good' (or how close to `best') a data subset is. Then, using this metric, we seek an algorithm that selects an appropriate data subset from which an accurate image can be reconstructed. Current algorithms use a criterion based upon the trace of a matrix. In this work we derive a simpler criterion based upon the determinant of a matrix. We construct two new algorithms based upon this new criterion and provide numerical results to demonstrate their accuracy and efficiency. A row exchange strategy is also described, which takes a given subset and performs interchanges to improve the quality of the selected subset.

The second idea is, given a reduced set of data, how can we quickly reconstruct an accurate signal or image? Compressed sensing provides a mathematical framework that explains that if a signal or image is known to be sparse relative to some basis, then it may be accurately reconstructed from a reduced set of data measurements. The reconstruction process can be posed as a convex optimization problem. We introduce an algorithm that aims to solve the corresponding problem and accurately reconstruct the desired signal or image. The algorithm is based upon the Barzilai-Borwein algorithm and tailored specifically to the compressed sensing framework. Numerical experiments show that the algorithm is competitive with currently used algorithms.

Following the success of compressed sensing for sparse signal reconstruction, we consider whether it is possible to reconstruct other signals with certain structures from reduced data sets. Specifically, signals that are a combination of a piecewise constant part and a sparse component are considered. A reconstruction process for signals of this type is detailed and numerical results are presented.