TY - JOUR
T1 - Ghosts in Discrete Tomography
AU - Brunetti, Sara
AU - Peri, Carla
AU - Hajdu, Lajos
AU - Dulio, Paolo
PY - 2015
Y1 - 2015
N2 - Switching components, also named as bad configurations, interchanges, and ghosts (according to different scenarios), play a key role in the study of ambiguous configurations, which often appear in Discrete Tomography and in several other areas of research. In this paper we give an upper bound for the minimal size bad configurations associated to a given set $$S$$S of lattice directions. In the special but interesting case of four directions, we show that the general argument can be considerably improved, and we present an algebraic method which provides such an improvement. Moreover, it turns out that finding bad configurations is in fact equivalent to finding multiples of a suitable polynomial in two variables, having only coefficients from the set $$\{-1,0,1\}$${-1,0,1}. The general problem of describing all polynomials having such multiples seems to be very hard (Borwein and Erdélyi, in Ill J Math 41(4):667–675, 1997). However, in our particular case, it is hopeful to give some kind of solution. In the context of Digital Image Analysis, it represents an explicit method for the construction of ghosts, and consequently might be of interest in image processing, also in view of efficient algorithms to encode data.
AB - Switching components, also named as bad configurations, interchanges, and ghosts (according to different scenarios), play a key role in the study of ambiguous configurations, which often appear in Discrete Tomography and in several other areas of research. In this paper we give an upper bound for the minimal size bad configurations associated to a given set $$S$$S of lattice directions. In the special but interesting case of four directions, we show that the general argument can be considerably improved, and we present an algebraic method which provides such an improvement. Moreover, it turns out that finding bad configurations is in fact equivalent to finding multiples of a suitable polynomial in two variables, having only coefficients from the set $$\{-1,0,1\}$${-1,0,1}. The general problem of describing all polynomials having such multiples seems to be very hard (Borwein and Erdélyi, in Ill J Math 41(4):667–675, 1997). However, in our particular case, it is hopeful to give some kind of solution. In the context of Digital Image Analysis, it represents an explicit method for the construction of ghosts, and consequently might be of interest in image processing, also in view of efficient algorithms to encode data.
KW - Bad configuration
KW - Discrete tomography
KW - Ghost
KW - X-ray
KW - Bad configuration
KW - Discrete tomography
KW - Ghost
KW - X-ray
UR - http://hdl.handle.net/10807/68675
U2 - 10.1007/s10851-015-0571-2
DO - 10.1007/s10851-015-0571-2
M3 - Article
SN - 0924-9907
VL - 2015/53
SP - 210
EP - 224
JO - Journal of Mathematical Imaging and Vision
JF - Journal of Mathematical Imaging and Vision
ER -