Binary Tomography Reconstructions With Few Projections

Discrete tomography deals with the reconstruction of images from projections collected along a few given directions. Different approaches can be considered, according to different models. In this paper we adopt the grid model, where pixels are lattice points with integer coordinates, X-rays are discrete lattice lines, and projections are obtained by counting the number of lattice points intercepted by X-rays taken in the assigned directions. We approach the tomographic problem in terms of linear system of equations Ax = p in an (M×N)-sized lattice grid A. Using a finite number of directions always yields the presence of ghosts, so preventing uniqueness. Ghosts can be managed by increasing the number of directions, which implies that also the number of collected projections (also called bins) increases. Therefore, for a best performing outcome, a kind of compromise should be sought among the number of employed directions, the number of collected projections, and the percentage of exactly reconstructed image. In this paper we wish to investigate such a problem in the case of binary images. We move from a theoretical result that allow uniqueness in A with just four suitably selected X-ray directions. This is exploited in studying the structure of the allowed ghosts in the given lattice grid. The knowledge of the ghost sizes, combined with geometrical information concerning the real valued solution of Ax = p having minimal Euclidean norm, leads to an explicit implementation of the previously obtained uniqueness theorem. This provides an easy binary algorithm (BRA) that, in the grid model, quickly returns perfect noise-free tomographic reconstructions. Then we focus on the tomography-side relevant problem of reducing the number of collected projections and, in the meantime, preserving a good quality of reconstruction. It turns out that, using sets of just four suitable directions, a high percentage of reconstructed pixel is preserved, even when the size of the projection vector p is considerably smaller than the size of the image to be reconstructed. Results are commented and discussed, also showing applications of BRA on phantoms with different features.