Stability in Discrete Tomography: Some Positive Results

Sara Brunetti (1) , Alain Daurat (2)

1 DSMI - Department of Mathematics and Computer Science / Dipartimento di Scienze Matematiche e Informatiche "Roberto Magari"
2 LSIIT - Laboratoire des Sciences de l'Image, de l'Informatique et de la Télédétection

The problem of reconstructing finite subsets of the integer lattice from X-rays has been studied in discrete mathematics and applied in several fields like image processing, data security, electron microscopy. In this paper we focus on the stability of the reconstruction problem for some special lattice sets. First we prove that if the sets are additive, then a stability result holds for very small errors. Then, we study the stability of reconstructing convex sets from both an experimental and a theoretical point of view. Numerical experiments are conducted by using linear programming that support the conjecture that convex sets are additive with respect to a set of suitable directions, and consequently the reconstruction problem is stable. The theoretical investigation provides a stability result for lattice sets. It is used to prove the following property: if a sequence of lattice convex sets have X-rays in suitable directions which converge to X-rays of a convex body, then it converges to this convex body.

Mots clés: Discrete Tomography; Stability; Linear Programming; Additivity; Convexity

Domaines: Mathématique discrète [cs.DM]

Sara Brunetti, Alain Daurat. Stability in Discrete Tomography: Some Positive Results. Discrete Applied Mathematics, 2005, 147, pp.207-226. ⟨10.1016/j.dam.2004.09.012⟩. ⟨hal-00023085⟩