"computed tomography 001.ps.gz" - читать интересную книгу автораMathematical problems in computed tomography Adel Faridani \Lambda y 1 Introduction Computed tomography (CT) entails the reconstruction of a function f from line integrals of f . This mathematical problem is encountered in a growing number of diverse settings in medicine, science, and technology, ranging from the famous application in diagnostic radiology to research in quantum optics. As a consequence, many aspects of CT have been extensively studied and are now well understood, thus providing an interesting model case for the study of other inverse problems. Other aspects, notably three-dimensional reconstructions, still provide numerous open problems. This article is based on a series of talks given by the author at the 1999 Mathematical Geophysics Summer School (MGSS) held at Stanford University. Its purpose is to give an introduction to the topic, treat some aspects in more detail, and to point out references for further study. The reader interested in a broader overview of the field, its relation to various branches of pure and applied mathematics, and its development over the years may wish to consult the monographs [6, 30, 31, 35, 61, 75], the volumes [20, 21, 27, 32, 33, 74], and review articles [41, 48, 55, 57, 64, 65, 81, 86]. In practice only integrals over finitely many lines can be measured, and the distribution of these lines is sometimes restricted. The following presentation is centered around the question: What features of f can be stably recovered from a given collection of line integrals of f ? For example, we may ask what resolution can be achieved with the available data. If a full reconstruction of f is not possible, we may try to detect the location of boundaries (jump discontinuities of f ), or also the sizes of the jumps. The exposition is divided into three parts. The first part is concerned with general theory. Its main themes are questions of uniqueness, stability and inversion for the x-ray transform, as well as detection of singularities. The second part is devoted to local tomography and is centered around a discussion of recently developed methods for computing jumps of a function from local tomographic data. The third part treats optimal sampling and has at its core a detailed error analysis of the parallel-beam filtered \Lambda Dept. of Mathematics, Oregon State University, Corvallis, OR 97331. Email: [email protected]; Homepage: http://ucs.orst.edu/~faridana yThis work was supported by NSF grant DMS-9803352 and NIH grant R01 RR 11800-4. 1 Part I: Some General Aspects 2 The x-ray and Radon transforms We begin by introducing some notation and background material. IRn consists of n-tuples of real numbers, usually designated by single letters, x = (x1; : : : ; xn), y = (y1; : : : ; yn), etc. The inner product and absolute value are defined by ! x; y ?= Pn1 xiyi and jxj =p ! x; x ?. The unit sphere Sn\Gamma 1 consists of the points with absolute value 1. C10 (IRn) denotes the set of infinitely differentiable functions on IRn with compact support. A continuous linear functional on C10 is called a distribution. If X is a set, Xffi denotes its interior, X its closure, and X c its complement. O/X and O/n denote the characteristic functions (indicator functions) of X, and of the unit ball in IRn, respectively. I.e., O/X (x) = 1 if x 2 X, and O/X(x) = 0 if x 62 X. jXj denotes the n-dimensional Lebesgue measure of X ae IRn. However, when it is clear that X should be treated as a set of dimension m ! n, jXj is the m-dimensional area measure. Thus jSk\Gamma 1j = 2ssk=2=\Gamma (k=2) is the (k \Gamma 1)-dimensional area of the (k \Gamma 1)-dimensional sphere. The convolution of two functions is given by f \Lambda g(x) = Z IRn f (x \Gamma y)g(y)dy: The Fourier transform is defined by ^f (,) = (2ss)\Gamma n=2 Z |
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