Tomographic reconstruction of functions from their singularities (Q1111116)

The reconstructive tomography problem is approached for density functions f(x) having jumps along smooth curves, of the form: \(f(x)=\phi (x)+\sum^{m}_{j=1}\alpha_ j\chi_{E_ i}\) (\(\phi\) (x) is compactly supported and infinitely differentiable, \(\alpha_ j\in {\mathbb{R}}^ 1\) and \(\chi_{E_ i}\) are characteristic functions of compact subsets \(E_ i\) of \({\mathbb{R}}^ n)\). It is shown that the location of boundaries may be determined from the smoothness of the Radon transform. It is also shown, without recourse to back projections, how to determine the singular support of f(x). At the end the problem of reconstructing radial density functions is considered.