Uniqueness and reconstruction of finite lattice sets from their line sums (Q6585268)

Let \(\mathcal A=\{(\xi,\eta)\in{\mathbb Z}^2: 0\le\xi<M,\ 0\le\eta<N\}\) be a (finite) grid in the digital plane \({\mathbb Z}^2\). The classical problem of unique reconstruction of subsets \(\mathcal C\) of \(\mathcal A\) from the knowledge of the number of points of \(\mathcal C\) on lines parallel to a lattice direction from a given finite set, has a huge literature. Extending earlier results of Brunetti, Dulio, Peri and others, the authors prove several theorems in this direction. First, they describe special sets of lattice directions (which they call simple cycles), and present various properties of them. Then they prove that uniqueness of reconstruction is guaranteed if and only if the line sums (i.e., the number of points of \(\mathcal C\) on the lines in the given directions) are computed along suitable simple cycles having even cardinality. Beside these, the authors apply their uniqueness theorem to get a reconstruction algorithm for binary images, which significantly performs better than similar earlier algorithms. To prove their theorems, the authors combine various arguments of geometric, algebraic and combinatorial nature. The results are nicely illustrated by many figures and examples, as well.