DISCRETE TOMOGRAPHY AND FUZZY INTEGER PROGRAMMING

Abstract

We study the problem of reconstructing binary images from four projections data in a fuzzy environment. Given the uncertainly projections,we want to find a binary image that respects as best as possible these projections. We provide an iterative algorithm based on fuzzy integer programming and linear membership functions. Discrete Tomography (DT) deals with the reconstruction of digital images from their projections. The projections of an image are defined by the number of pixels of the colors on each line. Digital images are most commonly represented by inte- ger matrices. The problem of reconstructing monocolored images (blackand white) from two projections is well known (14). However, image processing in general and image reconstruction in particular is often characterized by uncertain and possi- bly inconsistent information. Since fuzzy programming is considered appropriate for solving real-world problems it seems reasonable to apply fuzzy programming methods to the reconstruction problem. In this paper, we propose a fuzzy approach for reconstructing binary images, which, instead of reconstructing projections exactly as in the deterministic recon- struction problem, recovers them as much as possible. The remainder of this paper is organized as follows. First, in Section 2, we briefly review fuzzy set theory and fuzzy programming and then, in Section 3, we present the problem of discrete tomography and image reconstruction. In Section 4, we discuss the application of fuzzy programming in image reconstruction and finally conclude the last section with a summary of our results.