By Andrew Markoe
This e-book is a accomplished learn of the Radon remodel, which operates on a functionality by means of integrating it over hyperplanes. The publication starts off with an straight forward and graphical creation to the Radon rework, tomography and CT scanners, through a rigorous improvement of the fundamental homes of the Radon rework. subsequent the writer introduces Grassmann manifolds within the learn of the k-plane rework (a model of the Radon remodel) which integrates over k-dimensional planes instead of hyperplanes. the rest chapters are all for extra complex subject matters.
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Extra info for Analytic tomography
The first example is a simple object. We do not know what it looks like at the start, but we do know it’s Radon transform. At the end of the reconstruction process we will see the original object to which we will compare the reconstruction. The second example is a reconstruction of a human brain section from its x-ray projections. 1 Filtered Backprojection Reconstruction of a Simple Object We are presented with an unknown object but are given its Radon transform. Let us call the unknown object f .
Here are the results of backprojecting in 10 and then in 17 directions. 17. Successive backprojections of the Radon transform of a square. It seems that the more directions used to back project, the more the result resembles the original square. 17 shows a backprojection formed by averaging 20 individual backprojections. The first picture in the upper left of the figure is precisely the single backprojection discussed previously (fig. 15). The second picture illustrates the average of this backprojection with one from a neighboring 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 . Recall that Rθ f is the notation for the x-ray projection of f along the line perpendicular to the direction θ.