Iterative Tomography

Below is a trivial two-dimensional density distribution and six of its projections. The aim of tomography is to estimate this distribution - which would normally be unknown - using only the information in the profiles.

Back projection is a sort of inverse process to projection. However, it admits no two-dimensional knowledge so the contents of one bin is shared along a "track" of all cells which could have contributed to that bin. The back projection of all bins of all profiles yields a first approximation to the original distribution.

Algebraic Reconstruction Techniques (ART algorithms) exploit the fact that the redistribution coefficients for back projecting the contents of bins along a "track" of cells can also be used to project the contents of cells into bins. Hence a set of projections (below in red) can be obtained from the first approximation.

Back projection onto the approximate distribution of the bin-by-bin difference between the original profiles (black) and the new set (red) yields an improved approximation. Further iterations converge more rapidly if any cell whose contents has become negative is reset to zero. Below is the result after 50 such iterations.