Binary Tomography Reconstructions With Few Projections

TL;DR

This paper investigates binary image reconstruction from very few projections in discrete tomography, establishing theoretical uniqueness conditions and proposing an exact retrieval method from the minimum Euclidean norm solution.

Contribution

It introduces a new theoretical result ensuring unique binary solutions with four projections and develops an iterative algorithm for practical reconstruction.

Findings

Unique binary solutions can be retrieved from the minimum Euclidean norm solution.

The proposed rounding method is effective on various phantom images.

Theoretical conditions for uniqueness are established with four projections.

Abstract

Discrete tomography deals with the reconstruction of images from projections collected along a few given directions. Different approaches can be considered, according to different models. In this paper we adopt the grid model, where pixels are lattice points with integer coordinates, X-rays are discrete lattice lines, and projections are obtained by counting the number of lattice points intercepted by X-rays taken in the assigned directions. We move from a theoretical result that allows uniqueness of reconstruction in the grid with just four suitably selected X-ray directions. In this framework, the structure of the allowed ghosts is studied and described. This leads to a new result, stating that the unique binary solution can be explicitly and exactly retrieved from the minimum Euclidean norm solution by means of a rounding method based on some special entries, which are precisely determined. A corresponding iterative algorithm has been implemented, and tested on a few phantoms having different characteristics and structure.