"Algorithms in Tomography 001.ps.gz" - читать интересную книгу автораAlgorithms in Tomography Frank Natterer Institut f"ur Numerische und instrumentelle Mathematik Universit"at M"unster Einsteinstrae 62, D-48149 M"unster (Germany) 1 Introduction The basic problem in computerized tomography is the reconstruction of a function from its line or plane integrals. Applications come from diagnostic radiology, astronomy, electron microscopy, seismology, radar, plasma physics, nuclear medicine and many other fields. More recent kinds of tomography replace the straight line model by an inverse problem for a partial differential equation. The outline of this paper is as follows. In section 2 we survey the mathematical models used in tomography. In section 3 we give a fairly detailed survey on 2D reconstruction algorithm which still are the work horse of tomography. In section 4 we describe recent developments in 3D reconstruction. In section 5 we make a few remarks on the beginning development of algorithms for nonstraight-line tomography. 2 Mathematical Models in Tomography In the description of the mathematical models we restrict ourselves to those features which are important for the mathematical scientist. For physical and medical aspects see [35], [6], [32]. (a) Transmission CT. This is the original and simplest case of CT. In transmission tomography one probes an object with non diffracting radiation, e.g. x-rays for the human body. If I0 is the intensity of the source, a(x) the linear attenuation coefficient of the object at point x, L the ray along which the radiation propagates, and I the intensity past the object, then I = I0 e \Gamma R a(x)dx : (2.1) In the simplest case the ray L may be thought of as a straight line. Modeling L as strip or cone, possibility with a weight factor to account for detector inhomogeneites may be more appropriate. In (2.1) we neglegt the dependence of a 1 2 Algorithms in Tomography from the energy (beam hardening effect) and other non linear phenomena (e.g. partial volume effect). The mathematical problem in transmission tomography is to determine a from measurements of I for a large set of rays L. If L is simply the straight line connecting the source x0 with the detector x1, (2.1) gives rise to the integrals `n II 0 = \Gamma x1Z |
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