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We study the limited data problem of the spherical Radon transform in two and three-dimensional spaces with general acquisition surfaces. In such situations, it is known that the application of filtered-backprojection reconstruction formulas might generate added artifacts and degrade the quality of reconstructions. In this article, we explicitly analyze a family of such inversion formulas, depending on a smoothing function that vanishes to order k on the boundary of the acquisition surfaces. We show that the artifacts are k orders smoother than their generating singularity. Moreover, in two-dimensional space, if the generating singularity is conormal satisfying a generic condition then the artifacts are even k+1/2 orders smoother than the generating singularity. Our analysis for three-dimensional space contains an important idea of lifting up space. We also explore the theoretical findings in a series of numerical experiments. Our experiments show that a good choice of the smoothing function leads to a significant improvement of reconstruction quality.
We consider the generalized Radon transform (defined in terms of smooth weight functions) on hyperplanes in 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 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 R-2 and the classical reconstruction operators (that use specific filters). In our proofs, we employ a general paradigm that is based on the calculus of Fourier integral operators. Since this technique does not rely on explicit expressions of the reconstruction operators, it enables us to analyze more general imaging situations.