Download e-book for kindle: Analytic tomography by Andrew Markoe

By Andrew Markoe

ISBN-10: 0511530013

ISBN-13: 9780511530012

ISBN-10: 0521793475

ISBN-13: 9780521793476

This booklet is a accomplished research of the Radon remodel, which operates on a functionality by means of integrating it over hyperplanes. The publication starts with an straight forward and graphical creation to the Radon rework, tomography and CT scanners, by way of a rigorous improvement of the elemental houses of the Radon rework. subsequent the writer introduces Grassmann manifolds within the research of the k-plane remodel (a model of the Radon remodel) which integrates over k-dimensional planes instead of hyperplanes. the remainder chapters are inquisitive about extra complex issues.

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107–119) and has very succinct guided exercises in which the reader can develop Radon inversion; volume two of Terras [598] has a brief mention of applications to partial differential equation; and Dym and McKean [138] provide a short introduction to the Fourier inversion method for the Radon transform. 9. 3 Notes Tomography and the theory of the Radon transform lie at the intersection of the mathematical fields of integral geometry and inverse problems. An area of integral geometry is concerned with geometric properties of objects that are determined by some set of integrals over portions of the objects.

If we fix a particular θ, then g(θ , t) is a function of one variable that plays the role of g in backprojection in a single direction. This device allows us to backproject in any direction. Let us now backproject the Radon transform of a simple object: the square f of side 2 centered at the origin. 15. Let us now start backprojecting in more directions. The following diagram shows the effect of averaging the backprojections from three directions. 16. Three averaged backprojections of the Radon transform of the square object f .

This graph is plotted by computing the x4 ray projection R 3π f (s) for each real number s and then placing a point on the graph 4 at a height of R 3π f (s) units above the point s on the horizontal axis. You can see √ 4 as s goes from − 2 to 0 as predicted. Also the increase in the values of R 3π f (s) √ 4 note the decrease as s goes from 0 to − 2. The next step in the study of the Radon transform is to devise a way of visualizing the entire Radon transform, instead of a single projection. 11 Therefore we can create a sinogram by creating a density plot of the function of two variables R f (θ, s) .

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Analytic tomography by Andrew Markoe

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