On the Stability of Reconstructing Lattice Sets from X-rays Along Two Directions

Abstract

Abstract. We consider the stability problem of reconstructing lattice sets from their noisy X-rays (i.e. line sums) taken along two directions. Stability is of major importance in discrete tomography because, in practice, these X-rays are affected by errors due to the nature of measurements. It has been shown that the reconstruction from noisy X-rays taken along more than two directions can lead to dramatically different reconstructions. In this paper we prove a stability result showing that the same instability result does not hold for the reconstruction from two directions. We also show that the derived stability result can be carried over by similar techniques to lattice sets with invariant points.

Stability results for the reconstruction of binary pictures from two projections

“…This shows that (1) is sharp. h A stability result of a similar type has been given in [1] (Theorem 17). However, Theorem 19 generalizes the previous result in several ways.…”

“…The main part of this section comprises the results of Sections 3-5 from [1]. However, Theorem 19 is a generalized version of its counterpart ( [1], Theorem 17), as will be discussed at the end of this section.…”

“…However, Theorem 19 is a generalized version of its counterpart ( [1], Theorem 17), as will be discussed at the end of this section.…”

“…The relevant results in these papers concerning projections from two directions either address reconstructions with smaller projection errors [20] or they require that even the reconstruction is uniquely determined by its projections [17,21]. Our Theorem 19 is a generalization of Theorem 5.1.18 in [16] and of Theorem 17 in [1]. More about the reconstruction of binary pictures from noise-free projections in the context of (0, 1)-matrices can be found in [22].…”

“…If one interprets F as the original binary picture, then Theorem 19 states that if one knows in advance that F is uniquely determined by its row and column sums (without actually knowing F ), and that the error in the measurements (projection error) is not larger than 2, then a certain upper bound is given -guaranteeing that any reconstruction F 0 cannot differ too much from the original picture. Theorem 17 of [1] was more restrictive in the sense that prior to reconstruction one had to know additionally the cardinality of F (cf. Theorem 17 (ii), [1]), and one had to know that a measurement error really occurred (with norm 2).…”