Reconstruction from ray integrals with sources on a curve

The problem of reconstruction of the function f from data of ray integrals with sources on a curve ⊂ E is of practical importance; see, for example, [7]. Several methods are known [1–3, 6, 8]. Grangeat’s method [3] gives the first derivative of the 3D Radon transform of f by one-fold integration. Combining with the Radon inversion formula yields a reconstruction by means of three-fold integration. Katsevich [5] has adapted this method for numerical implementation by means of segmentation of and local reduction to two-fold integration. He introduced special weights to cope with multiple intersection of a hyperplane with . We state here a global two-fold integral reconstruction formula from data of ray integrals g(y, v), y ∈ . The function f can be evaluated on any chord of . No special weight is necessary. Theorem 1. Let = {y = y(s), 0 s 1} be a C1-curve in E. An arbitrary function f ∈ C2(E) such that supp f E\ can be recovered in any point x ∈ ]y(0), y(1)[ from its ray integrals g by