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Erschienen in:
2008 | OriginalPaper | Buchkapitel
Use of the Fourier Transform in the Distributions Sense for Creation Numerical Algorithms for Cone-Beam Tomography
verfasst von : O. E. Trofimov
Erschienen in: Progress in Industrial Mathematics at ECMI 2006
Verlag: Springer Berlin Heidelberg
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Let the homogeneous of
L
degree function
g
(
x
) be defined in
N
dimensional space, and let the function
G
(
y
) be its Fourier transform in the distribution sense. The theorem that allows to present the function
G
(
y
) using only the values of function
g
(
x
) on the unit sphere is proved in the chapter for the case
L
> −
N
. The case
N
=3 and
L
= −1 corresponds to the properties of beam transform in 3D space. In the chapter it is shown how the theorem may be used for creation of numerical algorithms for cone-beam tomography.
Let the homogeneous of
L
degree function
g
(
x
) be defined in
N
-dimensional space, and let the function
G
(
y
) be its Fourier transform. In view of homogeneity, the function
g
(
x
) and its Fourier transform in sense of distributions are defined by their values on the unit sphere [GS00]. We will prove the theorem that allows to present the function
G
(
y
) using only the value of function
g
(
x
) on the unit sphere for the case
L
> −
N
.