Some Geometric and Tomographic Results on Gray-Scale Images

Discrete tomography deals with the reconstruction of images from a (usually small) set of X-ray projections. This is achieved by modeling the tomographic problem as a linear system of equations and then applying a suitable discrete reconstruction algorithm based on iterations. In this paper we adopt the well-known grid model and prove some geometric properties of integer solutions consisting of $p\geq 2$ gray levels. In particular, we show that all gray-scale solutions having the same two-norm belong to a same hypersphere, centered at the uniform image related to the data and having radius ranging in an interval whose bounds are explicitly computed. Moving from a uniqueness theorem for gray-scale images, we compute special sets of directions that guarantee uniqueness of reconstruction and exploit them as the input of the Conjugate Gradient Least Squares algorithm. Then we apply an integer rounding to the resulting output and, basing on previously described geometric parameters, we test the quality of the obtained reconstructions for an increasing number of iterations, which leads to a progressive improvement of the percentage of correctly reconstructed pixels, until perfect reconstruction is achieved. Differently, using sets of directions which are classically employed, but far from being sets of uniqueness, only partial reconstructions are obtained.