Reconstruction of an image using data collected from a tomography system requires inversion of a specific transform that models the data collection process.; In X-ray transmission tomography, the data is related to the attenuation function of the object of interest, and is modeled by the integral of the attenuation function along lines. In two dimensions, this is referred to as the Radon transform.; In emission tomography, the data is related both to the attenuation of the object and the emission distribution of a radiochemical substance within the object. If the attenuation of the object is uniform, the data is modeled by the integrals along exponentially weighted lines, which is referred to as the exponential Radon transform.; Given a function on the unit sphere, its Funk transform is defined by its integral over the great circles. Isometrically mapping the sphere onto the unit disk and half-plane models of the hyperbolic geometry enables representation of the x-ray transform on the hyperbolic disk and circular averages in the half plane in terms of the Funk transform. These transforms form a model for the linearized electrical impedance tomography, and synthetic aperture radar problems, respectively.; This thesis studies the inversion of Radon, exponential Radon, Funk and circular averages transforms with respect to their underlying invariance properties.; The first part of the thesis studies invariance of Radon and exponential Radon transforms with respect to the rigid body motions of the Euclidean space. The Radon and exponential Radon transforms are formulated as convolutions over the Euclidean motion group, M(N). As a result, they are block diagonalized in M(N)-Fourier domain, leading to new inversion algorithms, which can be implemented using fast M(N)-Fourier transform algorithms.; The second part of this thesis studies Funk transform and circular averages. The Funk transform is formulated as a convolution over the rotation group and an inversion algorithm is derived and implemented using a fast SO(3)-Fourier transform algorithm. This algorithm is used to invert the circular averages.
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