By Peter W. Hawkes

ISBN-10: 0123742188

ISBN-13: 9780123742186

Advances in Imaging and Electron Physics merges long-running serials-Advances in Electronics and Electron Physics and Advances in Optical and Electron Microscopy. This sequence gains prolonged articles at the physics of electron units (especially semiconductor devices), particle optics at low and high energies, microlithography, photo technology and electronic photo processing, electromagnetic wave propagation, electron microscopy, and the computing tools utilized in some of these domain names.
An vital characteristic of those Advances is that the themes are written in this sort of approach that they are often understood via readers from different specialities.

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Traversing these boundaries from right to left the corresponding s-value increases. In particular, two points on the lower and upper Pi-window boundaries at the same uP -coordinate are separated by an angular distance of 2π [compare with Eq. (28)]. Since the points of sunrise and sunset now are separated by less than 2π , the Pi-line projected onto the planar detector must have positive slope. Point y H (s1 ) is the first IP of the exemplary Radon plane in Figure 33, which is reached when traversing the helix.

With these conventions the Radon transform of object 1 Johann Radon, Austrian mathematician, 1887–1956. 16 BONTUS AND KÖHLER F IGURE 12. An exemplary Radon plane with normal vector ω. The dashed line has the length ρ, which is the shortest distance from the origin to the plane. Points x on the plane fulfill the relation ω · x = ρ. function μ(x) is given by ∞ Rμ(ρ, ω) = d3 x μ(x)δ(ρ − ω · x). (34) −∞ 1. Inverse Radon Transform The inverse of the Radon transform can be obtained via the following formula (Natterer, 1986) π −1 μ(x) = 8π 2 2π dα sin α R μ(ω · x, ω), dϑ 0 (35) 0 where ω = (cos α sin ϑ, sin α sin ϑ, cos ϑ) (36) and R μ(ρ, ω) = ∂ 2 Rμ(ρ, ω) ∂ρ 2 (37) is the second derivative of the Radon transform with respect to ρ.

Intersections of the Radon plane. Projections of the Pi-line result in the solid lines indicated by π . The other solid line L1 corresponds to the asymptote on the helix. The asymptote on the projected helix plays a crucial role. The asymptote is indicated by the solid line L1 in Figure 34. It is parallel to y˙ H (s). Line L1 contains the origin of the planar detector and its gradient is equal to h/R. The ¯ latter follows from Eq. (32) using arctan x → (π/2 − 1/x) for x → ∞. The solid lines indicated by π in Figure 34 are the projections of the Piline.

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Advances in Imaging and Electron Physics, Vol. 151 by Peter W. Hawkes


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