By Guillaume Bal, David Finch, Visit Amazon's Peter Kuchment Page, search results, Learn about Author Central, Peter Kuchment, , John Schotland, Plamen Stefanov
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Title: Tomography and Inverse shipping Theory
Author: Bal, Guillaume (EDT)/ Finch, David (EDT)/ Kuchment, Peter (EDT)/ Schotland, John (EDT)/ Stefanov, P
Publisher: Amer Mathematical Society
Publication Date: 2012/01/01
Number of Pages: 180
Binding variety: PAPERBACK
Library of Congress: 2011033070
Read or Download Tomography and Inverse Transport Theory: International Workshop on Mathematical Methods in Emerging Modalities of Medical Imaging October 25-30, 2009, ... Workshop o PDF
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Extra info for Tomography and Inverse Transport Theory: International Workshop on Mathematical Methods in Emerging Modalities of Medical Imaging October 25-30, 2009, ... Workshop o
The above reconstruction works for arbitrary Gaussian sources of the x 2 form e−| η | for all η > 0. When η → 0 and after proper rescaling, this corresponds to a source term that approximates the delta distribution with support at x = 0. We have seen that reconstructions of a were not possible in this limit and thus cannot expect reconstructions from peaked Gaussian source terms to be stable. 3 using a diﬀerent reconstruction procedure that also involves inverting a Laplace transform. 4. We prove by induction that In,m > 0 when n + m is odd, (n, m) ∈ N2 , m > 0.
For such a source and absorption (f, a), a can be completely reconstructed from Rf a provided that Pθ0 a(s) is also known for some θ0 ∈ S1 and for any s ∈ R. In X-ray tomography, these integrals Pθ0 a(s) are known for all s ∈ R when a full transversal scan in the ﬁxed direction θ0 is performed on the object of interest. Such results show that combined with very limited tomographic projections of a, unique reconstructions of both a and f may be feasible. 3. Formulas for radial sources f . 18) (resp.
18) (resp. 19)), we ﬁrst give a general formula that relates the Fourier decomposition of Rf a (resp. Rf a) to the Fourier decomposition of a (resp. a ˆ) when f is a radial function. Then we provide an example of a smooth (and Gaussian) radial source f such that Rf a uniquely determines a up to its radial part. However, the stability of the reconstruction is very poor as we shall see. x + t˜ x⊥ Let f (x) = f1 (|x|) ∈ L2 (R2 ). 16) g(r) = 0 f (s)s √1 ds, for r > 0. r 2 − s2 22 10 GUILLAUME BAL AND ALEXANDRE JOLLIVET Let us introduce the Fourier decomposition of Rf a and a given by Rf a(rω) = imθ and a(rω) = m∈Z am (r)eimθ , where ω = (cos(θ), sin(θ)).