The reconstruction of discrete sets from their projections is a frequently studied field in discrete tomography with applications in electron microscopy, image processing, radiology, and so on. Several efficient reconstruction algorithms have been developed for certain classes of discrete sets having some good geometrical properties. On the other hand, it has been shown that the reconstruction under certain circumstances can be very time-consuming, even NP-hard. In this chapter we show how prior information that the set to be reconstructed consists of several components can be exploited in order to facilitate the reconstruction. We present some general techniques to decompose a discrete set into components knowing only its projections and thus reduce the reconstruction of a general discrete set to the reconstruction of single components, which is usually a simpler task.

1 aBalázs, Péter1 aHerman, Gábor T1 aKuba, Attila uhttps://www.inf.u-szeged.hu/en/publication/decomposition-algorithms-for-reconstructing-discrete-sets-with-disjoint-components01835nas a2200241 4500008004100000020002200041245006000063210005900123260002100182300001400203520108200217100002101299700001801320700001801338700001701356700001601373700001701389700002601406700002101432700002101453700001701474856010201491 2007 eng d a978-0-8176-3614-200aDiscrete Tomography Methods for Nondestructive Testing.0 aDiscrete Tomography Methods for Nondestructive Testing bBirkhauserc2007 a303 - 3323 aThe industrial nondestructive testing (NDT) of objects seems to be an ideal application of discrete tomography. In many cases, the objects consist of known materials, and a lot of a priori information is available (e.g., the description of an ideal object, which is similar to the actual one under investigation). One of the frequently used methods in NDT is to take projection images of the objects by some transmitting ray (e.g., X- or neutron-ray) and reconstruct the cross sections. But it can happen that only a few number of projections can be collected, because of long and/or expensive data acquisition, or the projections can be collected only from a limited range of directions. The chapter describes two DT reconstruction methods used in NDT experiments, shows the results of a DT procedure applied in the reconstruction of oblong objects having projections only from a limited range of angles, and, finally, suggests a few further possible NDT applications of DT.

1 aBaumann, Joachim1 aKiss, Zoltán1 aKrimmel, Sven1 aKuba, Attila1 aNagy, Antal1 aRodek, Lajos1 aSchillinger, Burkhard1 aStephan, Juergen1 aHerman, Gábor T1 aKuba, Attila uhttps://www.inf.u-szeged.hu/en/publication/discrete-tomography-methods-for-nondestructive-testing01401nas a2200229 4500008004100000020002200041245003400063210003300097260003900130300001400169520074500183100002000928700002000948700001700968700001600985700002001001700001801021700001801039700002101057700001701078856007601095 2007 eng d a978-0-8176-3614-200aEmission discrete tomography.0 aEmission discrete tomography aCambridgebBirkhauser Bostonc2007 a333 - 3663 a

Three problems of emission discrete tomography (EDT) are presented. The first problem is the reconstruction of measurable plane sets from two absorbed projections. It is shown that Lorentz theorems can be generalized to this case. The second is the reconstruction of binary matrices from their absorbed row and columns sums if the absorption coefficient is μ0 = log((1+v^{/}5)/2). It is proved that the reconstruction in this case can be done in polynomial time. Finally, a possible application of EDT in single photon emission computed tomography (SPECT) is presented: Dynamic structures are reconstructed after factor analysis.

1 aBarcucci, Elena1 aFrosini, Andrea1 aKuba, Attila1 aNagy, Antal1 aRinaldi, Simone1 aSamal, Martin1 aZopf, Steffen1 aHerman, Gábor T1 aKuba, Attila uhttps://www.inf.u-szeged.hu/en/publication/emission-discrete-tomography