Limited Data Problems for the Generalized Radon Transform in ${{\mathbb R}^n}$

Abstract

We consider the generalized Radon transform (defined in terms of smooth weight functions) on hyperplanes in ${{\mathbb R}^n}$. We analyze general filtered backprojection type reconstruction methods for limited data with filters given by general pseudodifferential operators. We provide microlocal characterizations of visible and added singularities in ${{\mathbb R}^n}$ and define modified versions of reconstruction operators that do not generate added artifacts. We calculate the symbol of our general reconstruction operators as pseudodifferential operators and provide conditions for the filters under which the reconstruction operators are elliptic for the visible singularities. If the filters are chosen according to those conditions, we show that almost all visible singularities can be recovered reliably. Our work generalizes the results for the classical line transforms in ${{{\mathbb R}^2}}$ and the classical reconstruction operators (that use specific filters). In our proofs, we employ a general paradig...