Reconstructing Singularities of a Function from Its Radon Transform

In many applications of tomography one is interested in finding the discontinuities of the unknown piecewise smooth function from the knowledge of the Radon transform of this function. For example, one could think about finding the boundaries of a crack in a solid, say aircraft wing or engine, or a rupture in a tissue in medical diagnostics. The aim of this paper is twofold. First, we want to study the relation between the singularities of a function f(x) and its Radon transform R(f). Secondly, we want to give a method for finding the singularities of f(x) given the singularities of R(f), and to study some numerical aspects of this problem, namely, finding the singularities of f(x), if the singularities of R(f) are given with some error. Although the literature on various numerical aspects of tomography is enormous (e.g., see [N]), the above problems were not studied sufficiently in the literature, as far as we know. In [Q1, p. 874] and [P, p. 132] it was noted that the singularities of R(f) in the two-dimensional case can be found at the values of parameters defining tangent lines to a curve across which the density f(x) is discontinuous. Our Theorems 1 and 3 give a complete and quantitative description of the set Qf of the singularities of R(f) and of the behavior of R(f) in a neighborhood of the set Qf . There is also a statement in [P, p. 132] concerning the singularities of R(f). This statement, given without proof in [P], does not include the result formulated in Theorem 1 of the present paper and proved in Section 2 below. In Theorem 1 a detailed description of the behavior of R(f) in a neighborhood of the