By Andrew Markoe
This e-book is a finished learn of the Radon rework, which operates on a functionality by means of integrating it over hyperplanes. The booklet starts with an easy and graphical advent to the Radon rework, tomography and CT scanners, by means of a rigorous improvement of the elemental houses of the Radon rework. subsequent the writer introduces Grassmann manifolds within the learn of the k-plane remodel (a model of the Radon rework) which integrates over k-dimensional planes instead of hyperplanes. the rest chapters are desirous about extra complicated issues.
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Additional info for Analytic tomography
Single x-ray projection. f in the given direction. The set of all such x-ray projections forms the Radon transform of the object. The problem in tomography is therefore to use the known x-ray data in the form of the Radon transform to reconstruct the unknown object from which it originates. The sinogram represents the known or given information for this problem. Soon we will give an explanation of how such a reconstruction can be accomplished. The explanation is based on the method used by CT scanners in medical applications.
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. Here we have θ = 0, π4 , π2 . orthogonal to angles 0, π4 , and π2 , namely the projections R0 f, R π4 f, and R π2 f, where f is the square object introduced before. Here are the results of backprojecting in 10 and then in 17 directions.
It is easy to pick up the difference between gray matter and the skull. But the difference in density between normal tissue and diseased tissue can be very small. 5% difference in density between a metastatic breast carcinoma and gray matter in the brain, as compared with a 50% difference between gray matter and the skull. In fact, the subtle difference in density between the gray matter and a carcinoma is almost impossible to detect by 42 1 Computerized Tomography, X-rays, and the Radon Transform the unaided eye in the images we have been presenting.
Analytic tomography by Andrew Markoe