Radon transforms: Unitarization, Inversion and Wavefront sets

The first contribution of this thesis is a new approach based on the theory of group representations in order to solve in a general an unified way the unitarization and inversion problems for generalized Radon transform associated to dual pairs (G/K,G/H) of homogeneous spaces of a locally compact group G, where K and H are closed subgroups of G. Precisely, under some technical assumptions, if the quasi-regular representations of G acting on L^2(G/K) and L^2(G/H) are irreducible, then the Radon transform, up to a composition with a suitable pseudo-differential operator, can be extended to a unitary operator intertwining the two representations. If, in addition, the representations are square integrable, an inversion formula for the Radon transform based on the voice transform associated to these representations is given. Several examples are discussed. The second purpose of the thesis is to investigate the connection between the shearlet transform and the wavelet transform, which has to be found in the Radon transform in affine coordinates. This link yields a formula for the shearlet transform that involves only integral transforms applied to the affine Radon transform of the signal, thereby opening new perspectives both for finding a new algorithm to compute the shearlet transform of a signal and for the inversion of the Radon transform. Furthermore, we study the role of the Radon transform in microlocal analysis, especially in the resolution of the wavefront set in shearlet analysis. We propose a new approach based on the wavelet transform and on the Radon transform which clarifies how the ability of the shearlet transform to characterize the wavefront set of signals follows directly by the combination of the microlocal properties inhereted by the one-dimensional wavelet transform with a sensitivity for directions inhereted by the Radon transform. Finally, the last chapter of the thesis is devoted to the extension of the shearlet transform to distributions. Our main results are continuity theorems for the shearlet transform and its transpose, called the shearlet synthesis operator, on various test function spaces. Then, we use these continuity results to develop a distributional framework for the shearlet transform via a duality approach. This work arises from the lack in the theory of a complete distributional framework for the shearlet transform and from the link between the shearlet transform with the Radon and the wavelet transforms, whose distribution theory is a deeply investigated and well known subject in applied mathematics.