A convergent composite mapping Fourier domain iterative algorithm for 3-D discrete tomography Original Research Article

The discrete tomography problem is to reconstruct a binary function defined on a discrete lattice from its sums in directions parallel to the axes of the lattice. The problem has multiple solutions; we wish to determine a solution close, in the least-squares sense, to a specified starting function. We formulate the problem in the Fourier domain, and use the Agarwal–Cooley fast convolution or Good–Thomas fast Fourier transform to unwrap the multidimensional (2-D or 3-D) problem into a long 1-D problem, in which the given projection data specify some values of the DFT of the 1-D sequence of binary values. The z-transform of the DFT values, evaluated on the unit circle, yields these binary values. The problem is now to determine the minimum perturbation of the unconstrained DFT values of the starting function which cause a Toeplitz matrix to lose rank. This type of problem is well-known in spectral estimation. We obtain a new iterative algorithm which converges to a solution of the discrete tomography problem close to the starting function.