Dynamic angle selection in binary tomography

Abstract

a b s t r a c tIn this paper, we present an algorithm for the dynamic selection of projection angles in binary tomography. Based on the information present in projections that have already been measured, a new projection angle is computed, which aims to maximize the information gained by adding this projection to the set of measurements. The optimization model used for angle selection is based on a characterization of solutions of the binary reconstruction problem, and a related definition of information gain. From this formal model, an algorithm is obtained by several approximation steps. Results from a series of simulation experiments demonstrate that the proposed angle selection scheme is indeed capable of finding angles for which the reconstructed image is much more accurate than for the standard angle selection scheme.

Ghosts in Discrete Tomography

“…In [28] it was shown that, for |S| = 4, the unique situations where F S has some coefficient outside {−1, 0, 1} correspond, precisely, to the choices 4 } in the first case, and S 1 = {u 1 , u 2 }, S 2 = {u 3 , u 4 } in the second case. Also, we define the set S 1 − S 2 as follows…”

“…The related digital image is shown in Figure 3 A complete reduction of P (x, y) is obtained just by three choices, corresponding to the selected pairs (1, 2), (2, 1), and (1, 0). This provides the following reduction Therefore, the points (2, 2), (3,3), and (4, 4), having multiplicity 4, are covered 3 times, the points (4, 3), (3,4), (3,2), and (2, 3), having multiplicity 3, are covered 3 or 2 times, and all the double points are covered 1 or 2 times. Consequently (H(x, y) + xy 2 + x 2 y + x)F S (x, y) represents the corresponding reduced polynomial.…”

“…In particular, we assume that S = {u 1 , u 2 , u 3 , u 4 }, with u 1 + u 2 + u 3 = u 4 or u 1 +u 2 −u 3 = u 4 . Motivation for this choice relies on the fact that by [30, Theorem 2.4], less than four directions are never sufficient to distinguish all the subsets of a given grid A, and, for |S| = 4, the unique situations where F S represents a weakly bad configuration correspond, precisely, to the choices u 1 + u 2 + u 3 = u 4 or u 1 + u 2 − u 3 = u 4 (see [28]). Therefore, for a set of four directions, these are the unique cases where the problem is not trivial.…”

“…geometric and computerized tomography, combinatorics, computer vision, image analysis (see for an overview [2,3] and for some recent works [4][5][6]). Moreover, as a sub-area of Discrete Geometry, DT should also provide both a theoretical and a computational frame-work for digital images ( [7]).…”

“…Of course, with fewer projections, such a phenomenon becomes more significant. Thus, the study of ghost artifacts is of main importance in DT, where the usual approach requires a small number of projections (4 or fewer) so that the choice of the particular set of projection angles can have a large influence on the quality of the reconstructions (see for instance [4,[14][15][16][17][18]). The missing information can be only in part overcome by exploiting the knowledge that digital images consist of a low number (2-5) of different materials.…”