Title: Binary Tomography Reconstructions With Few Projections,
Abstract: We approach the tomographic problem in terms of linear system of equations
$A\mathbf{x}=\mathbf{p}$ in an $(M\times N)$-sized lattice grid $\mathcal{A}$.
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 $\mathcal{A}$ 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 $A\mathbf{x}=\mathbf{p}$ 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 $\mathbf{p}$ 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.