"numerical methods in tomography 001.ps.gz" - читать интересную книгу автораActa Numerica (1999), pp. 1-000 Numerical Methods in Tomography Frank NattererInstitut f"ur Numerische und instrumentelle Mathematik Universit"at M"unster Einsteinstrasse 62 D-48149 M"unster, Germany E-mail: [email protected] CONTENTS 1 Introduction 1 2 The filtered backprojection algorithm 3 3 3D reconstruction formulas 15 4 Iterative methods 21 5 Circular harmonic algorithms 25 6 Fourier reconstruction 28 7 Outlook 32 References 33 In this article we review the image reconstruction algorithms used in tomography. We restrict ourselves to the standard problems in the reconstruction of function from line or plane integrals as they occur in X-ray tomography, nuclear medicine, magnetic resonance imaging, and electron microscopy. Non-standard situations, such as incomplete data, unknown orientations, local tomography, discrete tomography are not dealt with. Nor do we treat nonlinear tomographic techniques such as impedance, ultrasound, and near infrared imaging. 1. Introduction By "Tomography" we mean a technique for imaging 2D cross sections of 3D objects. It is derived from the greek word o/ o_os = slice. Tomographic techniques are used in radiology and in many branches of science and technology. 1.1. The basic example In the simplest case, let us consider an object whose attenuation coefficient with respect to X-rays at the point x is f (x). We scan the cross section by a thin X-ray beam L of unit intensity. The intensity past the object is \Gamma RL f(x)dx : 2 Frank Natterer This intensity is measured, providing us with the line integral g(L) = Z L f (x)dx : (1.1) The problem is to compute f from g. In principle this problem has been solved by Radon (1917). Let L be the straight line x \Delta ` = s where ` = (cos '; sin ')T and s 2 IR1. Then, (1.1) can be written as g(`; s) = Z x\Delta `=s |
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