Download e-book for iPad: Introduction to the mathematics of medical imaging by Charles L. Epstein

By Charles L. Epstein

Preface to the second one version; Preface; how you can use this corner; Notational conventions; 1. Measurements and modeling; 2. Linear versions and linear equations; three. A uncomplicated version for tomography; four. creation to the Fourier remodel; five. Convolution; 6. The radon rework; 7. creation to Fourier sequence; eight. Sampling; nine. Filters; 10. imposing shift invariant filters; eleven. Reconstruction in X-ray tomography; 12. Imaging artifacts in X-ray tomography; thirteen. Algebraic reconstruction strategies; 14. Magnetic resonance imaging; 15. likelihood and random variables; sixteen. purposes of likelihood; 17. Random techniques; A. history fabric; B. simple research; Bibliography; Index

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Find an example of a planar region such that at least one particle trajectory is trapped forever. ” In this section we consider the analogous but simpler problem of determining the outline of a convex object from its shadows. As is also the case in 20 Chapter 1. Measurements and Modeling medical applications, we consider a two-dimensional problem. Let D be a convex region in the plane. Imagine that a light source is placed very far from D. Since the light source is very far away, the rays of light are all traveling in essentially the same direction.

2 Reconstructing an Object from Its Shadows Now we can quantitatively describe the shadow. Because there are two lines in each family of parallel lines that are tangent to the boundary of D, we need a way to select one of them. To do this we choose an orientation for the boundary of D; this operation is familiar from Green’s theorem in the plane. 13). Fix a source position ω(θ). 13). Examining the diagram, it is clear that the orientation of the boundary at the point of tangency and that of the oriented line agree for lt1 ,ω , and are opposite for lt0 ,ω .

The object is lying inside a disk and we can fire particles at the object. Assume that the particles bounce off according to a simple scattering process. 7). The outline of the object can be determined by knowing the correspondence between incoming lines, lin , and outgoing lines, lout . Each intersection point lin ∩ lout lies j j on the boundary of the object. Measuring {lout } for finitely many incoming directions {lin } j j determines finitely many points {lin ∩ lout } on the boundary of D. In order to use this finite collection of points to make any assertions about the rest of the boundary of D, more information is required.

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