Reconstruction algorithms for a class of restricted ray transforms without added singularities

Abstract. Let X and X denote a restricted ray transform along curves and a corresponding backprojection operator, respectively. Theoretical analysis of reconstruction from the data Xf is usually based on a study of the composition X∗DX, where D is some local operator (usually a derivative). If X∗ is chosen appropriately, then X∗DX is a Fourier Integral Operator (FIO) with singular symbol. The singularity of the symbol leads to the appearance of artifacts (added singularities) that can be as strong as the original (or, useful) singularities. By choosing D in a special way one can reduce the strength of added singularities, but it is impossible to get rid of them completely. In the paper we follow a similar approach, but make two changes. First, we replace D with a nonlocal operator D̃ that integrates Xf along a curve in the data space. The result D̃Xf resembles the generalized Radon transform R of f . The function D̃Xf is defined on pairs (x0,Θ) ∈ U×S, where U ⊂ R is an open set containing the support of f , and S is the unit sphere in R. Second, we replace X∗ with a backprojection operator R∗ that integrates with respect to Θ over S. It turns out that if D̃ and R∗ are appropriately selected, then the composition R∗D̃X is an elliptic pseudodifferential operator of order zero with principal symbol 1. Thus, we obtain an approximate reconstruction formula that recovers all the singularities correctly and does not produce artifacts. The advantage of our approach is that by inserting D̃ we get access to the frequency variable Θ. In particular, we can incorporate suitable cut-offs in R∗ to eliminate bad directions Θ, which lead to added singularities.