The Radon Transform and the Mathematics of Medical Imaging PDF Download
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Author: Jen Beatty
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 52
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Book Description
Tomography is the mathematical process of imaging an object via a set of finite slices. In medical imaging, these slices are defined by multiple parallel X-ray beams shot through the object at varying angles. The initial and final intensity of each beam is recorded, and the original image is recreated using this data for multiple slices. I will discuss the central role of the Radon transform and its inversion formula in this recovery process.
Author: Jen Beatty
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 52
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Book Description
Tomography is the mathematical process of imaging an object via a set of finite slices. In medical imaging, these slices are defined by multiple parallel X-ray beams shot through the object at varying angles. The initial and final intensity of each beam is recorded, and the original image is recreated using this data for multiple slices. I will discuss the central role of the Radon transform and its inversion formula in this recovery process.
Author: Peter Kuchment
Publisher: SIAM
ISBN: 1611973295
Category : Computers
Languages : en
Pages : 256
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Book Description
This book surveys the main mathematical ideas and techniques behind some well-established imaging modalities such as X-ray CT and emission tomography, as well as a variety of newly developing coupled-physics or hybrid techniques, including thermoacoustic tomography. The Radon Transform and Medical Imaging emphasizes mathematical techniques and ideas arising across the spectrum of medical imaging modalities and explains important concepts concerning inversion, stability, incomplete data effects, the role of interior information, and other issues critical to all medical imaging methods. For nonexperts, the author provides appendices that cover background information on notation, Fourier analysis, geometric rays, and linear operators. The vast bibliography, with over 825 entries, directs readers to a wide array of additional information sources on medical imaging for further study.
Author: Ronny Ramlau
Publisher: Walter de Gruyter GmbH & Co KG
ISBN: 311055951X
Category : Mathematics
Languages : en
Pages : 469
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Book Description
The series is devoted to the publication of high-level monographs, surveys and proceedings which cover the whole spectrum of computational and applied mathematics. The books of this series are addressed to both specialists and advanced students. Interested authors may submit book proposals to the Managing Editor or to any member of the Editorial Board. Managing EditorUlrich Langer, Johannes Kepler University Linz, Austria Editorial BoardHansj rg Albrecher, University of Lausanne, SwitzerlandRonald H. W. Hoppe, University of Houston, USAKarl Kunisch, RICAM, Linz, Austria; University of Graz, AustriaHarald Niederreiter, RICAM, Linz, AustriaChristian Schmeiser, University of Vienna, Austria
Author: Stanley R. Deans
Publisher: Courier Corporation
ISBN: 0486462412
Category : Mathematics
Languages : en
Pages : 306
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Book Description
Of value to mathematicians, physicists, and engineers, this excellent introduction to Radon transform covers both theory and applications, with a rich array of examples and literature that forms a valuable reference. This 1993 edition is a revised and updated version by the author of his pioneering work.
Author: Roger L. Easton Jr.
Publisher: John Wiley & Sons
ISBN: 1119991862
Category : Technology & Engineering
Languages : en
Pages : 1005
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Book Description
Fourier Methods in Imaging introduces the mathematical tools for modeling linear imaging systems to predict the action of the system or for solving for the input. The chapters are grouped into five sections, the first introduces the imaging “tasks” (direct, inverse, and system analysis), the basic concepts of linear algebra for vectors and functions, including complex-valued vectors, and inner products of vectors and functions. The second section defines "special" functions, mathematical operations, and transformations that are useful for describing imaging systems. Among these are the Fourier transforms of 1-D and 2-D function, and the Hankel and Radon transforms. This section also considers approximations of the Fourier transform. The third and fourth sections examine the discrete Fourier transform and the description of imaging systems as linear "filters", including the inverse, matched, Wiener and Wiener-Helstrom filters. The final section examines applications of linear system models to optical imaging systems, including holography. Provides a unified mathematical description of imaging systems. Develops a consistent mathematical formalism for characterizing imaging systems. Helps the reader develop an intuitive grasp of the most common mathematical methods, useful for describing the action of general linear systems on signals of one or more spatial dimensions. Offers parallel descriptions of continuous and discrete cases. Includes many graphical and pictorial examples to illustrate the concepts. This book helps students develop an understanding of mathematical tools for describing general one- and two-dimensional linear imaging systems, and will also serve as a reference for engineers and scientists
Author: Charles L. Epstein
Publisher: SIAM
ISBN: 9780898717792
Category : Mathematics
Languages : en
Pages : 794
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Book Description
At the heart of every medical imaging technology is a sophisticated mathematical model of the measurement process and an algorithm to reconstruct an image from the measured data. This book provides a firm foundation in the mathematical tools used to model the measurements and derive the reconstruction algorithms used in most of these modalities. The text uses X-ray computed tomography (X-ray CT) as a 'pedagogical machine' to illustrate important ideas and its extensive discussion of background material makes the more advanced mathematical topics accessible to people with a less formal mathematical education. This new edition contains a chapter on magnetic resonance imaging (MRI), a revised section on the relationship between the continuum and discrete Fourier transforms, an improved description of the gridding method, and new sections on both Grangreat's formula and noise analysis in MR-imaging. Mathematical concepts are illuminated with over 200 illustrations and numerous exercises.
Author: Alexander G. Ramm
Publisher: CRC Press
ISBN: 1000151778
Category : Mathematics
Languages : en
Pages : 516
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Book Description
Over the past decade, the field of image processing has made tremendous advances. One type of image processing that is currently of particular interest is "tomographic imaging," a technique for computing the density function of a body, or discontinuity surfaces of this function. Today, tomography is widely used, and has applications in such fields as medicine, engineering, physics, geophysics, and security. The Radon Transform and Local Tomography clearly explains the theoretical, computational, and practical aspects of applied tomography. It includes sufficient background information to make it essentially self-contained for most readers.
Author: Timothy G. Feeman
Publisher: Springer Science & Business Media
ISBN: 0387927115
Category : Computers
Languages : en
Pages : 150
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Book Description
Medical imaging is a major part of twenty-first century health care. This introduction explores the mathematical aspects of imaging in medicine to explain approximation methods in addition to computer implementation of inversion algorithms.
Author: Gaik Ambartsoumian
Publisher: World Scientific
ISBN: 9811242453
Category : Mathematics
Languages : en
Pages : 248
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Book Description
A generalized Radon transform (GRT) maps a function to its weighted integrals along a family of curves or surfaces. Such operators appear in mathematical models of various imaging modalities. The GRTs integrating along smooth curves and surfaces (lines, planes, circles, spheres, amongst others) have been studied at great lengths for decades, but relatively little attention has been paid to transforms integrating along non-smooth trajectories. Recently, an interesting new class of GRTs emerged at the forefront of research in integral geometry. The two common features of these transforms are the presence of a 'vertex' in their paths of integration (broken rays, cones, and stars) and their relation to imaging techniques based on physics of scattered particles (Compton camera imaging, single scattering tomography, etc).This book covers the relevant imaging modalities, their mathematical models, and the related GRTs. The discussion of the latter comprises a thorough exploration of their known mathematical properties, including injectivity, inversion, range description and microlocal analysis. The mathematical background required for reading most of the book is at the level of an advanced undergraduate student, which should make its content attractive for a large audience of specialists interested in imaging. Mathematicians may appreciate certain parts of the theory that are particularly elegant with connections to functional analysis, PDEs and algebraic geometry.
Author: Andrew E. Todd-Pokropek
Publisher: Springer Science & Business Media
ISBN: 3642778887
Category : Medical
Languages : en
Pages : 700
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Book Description
Medical imaging is a very important area in diagnostic (and increasingly therapeutic) medicine. Many new techniques are being developed or extended which depend on digital methods. Although conventional x-radiographs still comprise the bulk of the medical images acquired in a hospital, digital methods such as computerized tomography and magnetic resonance imaging are now often claimed to have a more significant clinical impact. This book is concerned with three aspects of such digital images: their formation, or how they can be acquired; their handling, or how they may be manipulated to increase their clinical value; and their evaluation, or how their impact and value may be assessed. The book is divided into three parts. Part 1 comprises a series of reviews in the general subject area written by authorities in the field. Part 2 includes papers on theoretical aspects: 3D images, reconstruction, perception, and image processing. Part 3includes papers on applications in nuclear medicine, magnetic resonance, andradiology.
