Reconstruction of densities from their projections, with applications in radiological physics

The precise determination of body attenuation for X-rays or its stopping power for heavy charged particles, positron annihilation scanning, and, to a lesser extent, single gamma -ray scanning all contain the same mathematical problem, namely, to determine a density distribution in space from its known projections on to one or more planes. Present methods of solving this problem involve taking slices through the distribution and considering the projected densities to be line integrals along lines through the slices and then using Fourier transforms, or orthogonal expansions of the line integrals, or a matrix inversion to determine the density distribution. An alternative method enables the density to be inferred by integration. In addition, a generalization of the method to surface integrals is given and possible applications are suggested including an application to positron annihilation scanning.