TY - JOUR
T1 - A rounding theorem for unique binary tomographic reconstruction
AU - Dulio, Paolo
AU - Pagani, Silvia Maria Carla
PY - 2019
Y1 - 2019
N2 - 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.
AB - 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.
KW - Binary tomography
KW - Discrete tomography
KW - Lattice direction
KW - Lattice grid
KW - Minimum norm solution
KW - Uniqueness of reconstruction
KW - Binary tomography
KW - Discrete tomography
KW - Lattice direction
KW - Lattice grid
KW - Minimum norm solution
KW - Uniqueness of reconstruction
UR - http://hdl.handle.net/10807/142463
U2 - 10.1016/j.dam.2019.05.005
DO - 10.1016/j.dam.2019.05.005
M3 - Article
SN - 0166-218X
VL - 268
SP - 54
EP - 69
JO - Discrete Applied Mathematics
JF - Discrete Applied Mathematics
ER -