Application of fast backprojection techniques for some inverse problems of integral geometry

Abstract: In this thesis we study some mathematical inverse problems concerning the determination of a function from circular averages, which appear in certain radar applications, and the determination of a function from its line integrals, which appear in the well known problem of computerized tomographic imaging (CT). The thesis is divided into three parts.In part I we investigate the problem of inverting circular averages, when the centers of the circles are situated on a straight line. First we present an inversion formula, based on Fourier and Hankel transforms, which we reformulate as aso-called backprojection followed by a ramp filter. A major disadvantage here is the computational complexity of a direct implementation of the backprojection,?(N3 ) for an image of N x N pixels. A new fast geometrical backprojection is developed, which reduces the number of operations to ?(N2 log N). The method can be applied to a large variety of similar problems in integral geometry.In part II we study the filtered backprojection method for determining a function from its line integrals (computerized tomographic imaging, CT). Fast algorithms, with ?(N2 log N) operations, are presented for parallel scanning as well as for fan-beam scanning.In part III we give results from numerical experiments. For circular averages artificial data have been used and for the CT case the Shepp-Logan and The G. Herman phantoms. The experiments show that the fast backprojection algorithm gives an image quality which is quite comparable to that obtained by conventional backprojections.

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