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.

JF - ADVANCES IN DISCRETE TOMOGRAPHY AND ITS APPLICATIONS T3 - Applied and Numerical Harmonic Analysis PB - Birkhauser Boston CY - Cambridge SN - 978-0-8176-3614-2 N1 - UT: 000271523600010doi: 10.1007/978-0-8176-4543-4_8 ER - TY - CHAP T1 - Discrete Tomography Methods for Nondestructive Testing. T2 - Advances in Discrete Tomography and Its Applications Y1 - 2007 A1 - Joachim Baumann A1 - Zoltán Kiss A1 - Sven Krimmel A1 - Attila Kuba A1 - Antal Nagy A1 - Lajos Rodek A1 - Burkhard Schillinger A1 - Juergen Stephan ED - Gábor T Herman ED - Attila Kuba AB -The 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.

JF - Advances in Discrete Tomography and Its Applications T3 - Applied and Numerical Harmonic Analysis PB - Birkhauser SN - 978-0-8176-3614-2 N1 - doi: 10.1007/978-0-8176-4543-4_14 ER - TY - CHAP T1 - Emission discrete tomography. T2 - ADVANCES IN DISCRETE TOMOGRAPHY AND ITS APPLICATIONS Y1 - 2007 A1 - Elena Barcucci A1 - Andrea Frosini A1 - Attila Kuba A1 - Antal Nagy A1 - Simone Rinaldi A1 - Martin Samal A1 - Steffen Zopf ED - Gábor T Herman ED - Attila Kuba AB -

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.

JF - ADVANCES IN DISCRETE TOMOGRAPHY AND ITS APPLICATIONS T3 - Applied and Numerical Harmonic Analysis PB - Birkhauser Boston CY - Cambridge SN - 978-0-8176-3614-2 N1 - doi: 10.1007/978-0-8176-4543-4_15 ER -