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

By Peter W. Hawkes

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 beneficial properties prolonged articles at the physics of electron units (especially semiconductor devices), particle optics at low and high energies, microlithography, photograph technology and electronic snapshot processing, electromagnetic wave propagation, electron microscopy, and the computing tools utilized in most of these domain names.
An vital characteristic of those Advances is that the topics are written in one of these approach that they are often understood by way of readers from different specialities.

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This ensures that a certain trajectory is complete if every Radon plane that contains x intersects with CBP (x). 4. Reconstruction Algorithm In the following we assume that vectors eν and weights μν have been defined such that Eq. (58) results in a constant value of 1. With this assumption, Eqs. (47) and (46) define a reconstruction algorithm, which recovers the μvalues in a mathematically exact way. The reconstruction steps are as follows: 1. Compute the derivative in the sense of Eq. (43). 2.

The weights must be set such that μν = 1 for the case shown in the left image of Figure 29, while μν = −1 for the case shown in the right image of Figure 29. a. Circular Full Scan. For the two possibilities of CBP (x) shown in Figure 25, the two segments on the circle together cover the complete circle. This provides an easy “recipe” for defining a full-scan algorithm, which uses data from the complete circle. We could apply just the algorithm defined above for both cases and average the results.

18 BONTUS AND KÖHLER F IGURE 14. Radon values located on the surface of a sphere are required for the evaluation of the inverse Radon transform at x according to Eq. (35). The center of the Radon sphere is located at x/2 and the radius of the sphere is equal to |x/2|. 2. Fourier Slice Theorem The Fourier slice theorem gives an important relation between the Radon transform and the object function in the Fourier domain. For its derivation we consider Eq. (34) and perform the Fourier transform along ρ: ∞ FRμ(ρ, ˜ ω) = dρ exp(−2π iρρ) ˜ Rμ(ρ, ω) −∞ ∞ = ∞ dρ exp(−2π iρρ) ˜ −∞ ∞ = d3 x μ(x)δ(ρ − ω · x) −∞ d3 x μ(x) exp(−2πiρω ˜ · x), (40) −∞ where in the last step the δ-function was evaluated for the integration over ρ.

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