A Discrete X-Ray Transform for Chromotomographic HyperspectraI Imaging
AIR FORCE INST OF TECH WRIGHT-PATTERSON AFB OH GRADUATE SCHOOL OF ENGINEERING AND MANAGEMENT
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The United States Air Force has a pressing need for new methods of hyperspectral imaging. All current hyperspectral imaging technologies require long exposure times, since each involves filtering the available light, either spatially or according to color. We consider a recently proposed method for hypserspectral imaging that promises shorter exposure times. This new method applies the mathematical principles of tomography to the hyperspectral data cube. Known as chromotomography, this method uses a spinning prism to essentially capture the integrals of this cube over many rotations of a single line. This thesis addresses some of the mathematical issues that arise when trying to reconstruct a hyperspectral image from chromotomographic measurements. After reviewing some of the mathematical shortcomings of the current state of the artwhich arise from the technical difficulties of working with the continuous-variable X-ray transformwe make three contributions. First, we introduce a mathematically rigorous, discrete, X-ray transform that is somewhat faithful to its continuous cousin. Second, we show how under a few simplifying assumptions, our discrete transform can be generalized so as to provide a good approximation of the continuous one. This discretization allows us to apply modern finite-dimensional optimization methods to the chromotomographic reconstruction problem. Our third contribution is to apply a popular new example of such a method, known as Split Bregman iteration.
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