Boundary Ghosts for Discrete Tomography

A linear time approach to three-dimensional reconstruction by discrete tomography

“…The extension in [1] means that not only the f -values of the integer points outside C can be computed, but also the f -values of the points inside C which are not in T . Ceko, Petersen, Svalbe and Tijdeman [2] studied socalled boundary ghosts which are switching functions defined on c n points with 1 < c < 2 enclosing a set of almost 2 n points, for any n. The new algorithm enables one to compute the f -values of those interior points in linear time. In the literature alternative names for switching function are ghost and phantom.…”

“…The switching function is elementary. By (4) we have T = {(0, 2, 2), (1, 0, 3), (1, 2, 2), (1, 3, 0), (1,3,4), (2, 0, 3), (2, 1, 1), (2, 1, 5), (2, 3, 0), (2,3,4), (2, 4, 2), (3, 1, 1), (3, 1, 5), (3, 2, 3), (3, 4, 2), (4, 2, 3)} .…”

“…This fact was exploited by Dulio and Pagani in [6], wherein a rounding theorem was proven which allowed exact and unique binary tomographic reconstructions from the minimum Euclidean norm solution. It also motivated the construction of boundary ghosts by Ceko, Petersen, Svalbe and Tijdeman [2]. Boundary ghosts are switching components that consist of a thin boundary of switching elements, and a largely empty interior of uniquely determinable values.…”

Algorithms for linear time reconstruction by discrete tomography II

“…If we order them according to increasing weights, then we get (0, 5), (0, 6), (4, 2), (7, 0), (3, 3), (6, 1), (2,4).…”

“…The crux of the result of Gardner, Gritzmann and Prangenberg is therefore the requirement that the solution g should have nonnegative values. Recently Ceko, Petersen, Svalbe and Tijdeman [4] constructed switching components, called boundary ghosts, with a relatively large interior of points having values uniquely determined by their line sums, see e.g. Figure 1.…”