Electromagnetic Fields and Waves
Electromagnetic Fields and Waves ECEN 3400
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This 3 page Class Notes was uploaded by Mrs. Lacy Schneider on Thursday October 29, 2015. The Class Notes belongs to ECEN 3400 at University of Colorado at Boulder taught by Edward Kuester in Fall. Since its upload, it has received 19 views. For similar materials see /class/231773/ecen-3400-university-of-colorado-at-boulder in ELECTRICAL AND COMPUTER ENGINEERING at University of Colorado at Boulder.
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Date Created: 10/29/15
Supplementary Notes for ECEN 3400 Edward E Kuester Department of Electrical and Computer Engineering 425 UCB University of Colorado Boulder CO 80309 0425 19th April 2004 1 Computed Tomography CAT Scans The word tomography comes from the Greek words rouoq tomos a slice or cross section and ypwg sw graphein to write It means a technique for producing cross sectional images usually of the human body for medical diagnostics but it is also used in many other applications The data acquired by X ray scans of biological tissues and other materials is essentially a set of projections the shadows cast by obstructions to the propagation of the X rays Such projec tions provide some information but important objects can be hidden behind others and remain undetected in the traditional X ray diagnostics The data from projections can be enhanced in its usefulness by doing supplemental computations with it We describe below a vastly oversimpli ed method for doing this Computed Axial Tomography or Computer Aided Tomography The acronym CAT is used for both more familiarly they are called CAT scans The method outlined here for computer processing of the data is called the Algebraic Reconstruction Technique ART We divide the region to be scanned into N pixels as shown in the gure below Each pixel is assumed to be small enough that it is approximately uniform in its material properties At X ray frequencies a plane wave is re ected very little from most materials the atoms and molecules cannot respond fast enough to have an e or a signi cantly different from their free space values However because typical samples are huge in size compared to a wavelength at X ray frequencies the wavelength ranges from 025 to 100 A 1 A 10 10 m the attenuation of a plane wave in a material is important and we denote by 04 the attenuation constant of the material in the ith pixel An X ray emitter sends waves through this region along various paths called rays as indicated Even though the waves are in the form of narrow beams they still behave locally like plane waves The intensity i e power carried by the wave is measured by a detector after it has passed through the region to the opposite side from the emitter The ratio of power measured to power emitted will be due to the cumulative effect of attenuation through each of the pixels the ray passes through Pmeasured E T sizaldlleizagdm 67204Nduv P emitted rayl where d11 d12 diN 121 122 d2N D 7 dN1 dN2 dNN 041 042 lal 04N and A1 A2 Al w This matrix equation is solved for the unknown attenuation constants From the computed attenuation data we use known information regarding the attenuation properties of different kinds of biological tissue bone fat muscle skin etc to determine what is at each pixel location The attenuation in medical applications is characterized by the mdz39oopacz39ty The radioopacities of various biological materials are measured on the Houns eld scale named after one of the inventors of the CAT scan Radioopacity varies on a scale from 1000 very low attenuation to 0 water to 1000 very high attenuation Common values for other types of tissue are Bone or calci cation 80 to 1000 Congealed blood 56 to 76 Grey matter 36 to 46 White matter 22 to 32 Fat 100 Air 1000 Obviously this method is not limited to two dimensional images although the number of pixels can become quite large in the threedimensional case making numerical computation of the solution to the matrix equation dif cult and requiring more sophisticated mathematical techniques Also it can be applied to materials other than biological tissue such as mineral and oil deposits and other situations needing nondestructive sensing techniques Finally the method is not limited to waves of X ray frequency although in its unmodi ed form it does need the re ected waves to be small Many variations of the idea have been proposed in the more than 30 years since it was rst proposed References Computed Tomography 1 G T Herman A Lent and S W Rowland ART Mathematics and applications A re port on the mathematical foundations and on the applicability to real data of the algebraic reconstruction techniques77 J Theor Biology vol 42 pp 1 32 1973 2 R Gordon A tutorial on ART Algebraic Reconstruction Techniques77 IEEE Trans Nuclear Science vol 21 no 3 pp 78 93 1974 3 K A Dines and R J Lytle Computerized geophysical tomography77 Proc IEEE vol 67 pp 10651073 1979 4 G T Herman Image Reconstruction from Projections The Fundamentals of Computerized Tomography New York Academic Press 1980 5 W R Hendee The Physical Principles of Computed Tomography Boston Little Brown 1983 6 A C Kak and M Slaney Principles of Computerized Tomographic Imaging New York IEEE Press 1988 7 J Hsieh Computed Tomography Bellingham WA SPlE Press 2003