Research Article
On the Exponential Radon Transform and Its Extension to
Certain Functions Spaces
S. K. Q. Al-Omari
1
and A. KJlJçman
2
1
Department of Applied Sciences, Faculty of Engineering Technology, Al-Balqa Applied University, Amman 11134, Jordan
2
Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia (UPM),
43400 Serdang, Selangor, Malaysia
Correspondence should be addressed to A. Kılıc ¸man; akilic@upm.edu.my
Received 25 October 2013; Accepted 18 April 2014; Published 5 May 2014
Academic Editor: Abdullah Alotaibi
Copyright © 2014 S. K. Q. Al-Omari and A. Kılıc ¸man. Tis is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
We investigate the exponential Radon transform on a certain function space of generalized functions. We establish certain space
of generalized functions for the cited transform. Te transform that is obtained is well defned. More properties of consistency,
convolution, analyticity, continuity, and sufcient theorems have been established.
1. Introduction
Te Radon transform of a sufciently nice function defned
on R
is given by
(R
)()≡(R)(,)≡∫
⊥
(+) d, (1)
where (,) ∈
R
=∑
−1
, ∑
−1
is the unit sphere in
R
, and d is the Euclidean measure on the subspace
⊥
orthogonal to .
Applications of the Radon transform occur in a number
of areas, such as seismic signal processing, remote sensing,
and system identifcation from output data [1, 2]. Te Radon
transform is extended to various spaces of distributions,
rapidly decreasing and integrable Boehmians [3, 4]. More
about the Radon transform is given in [5–9].
Te discrete Radon transform is defned by [10, 11]. Te
attenuated Radon transform is defned in Mikusi´ nski et al.
[12, 13]. For a uniform attenuation coefcient ∈ C, the
exponential Radon transform of a compactly supported real
valued function , defned on R
2
, is given by Kurusa and
Hertle [7, 8]:
T
(,)=∫
R
2
(x)(x ⋅−)
x⋅
⊥
dx, ∈ R, (2)
where =(cos , sin )
is a unit vector on S
1
, ∈
[0,2),
⊥
=(− sin , cos ).
Te exponential Radon transform constitutes a mathe-
matical model for imaging modalities such as X-ray tomogra-
phy for =0, single photon emission tomography for ∈ R,
and optical polarization tomography of trass tensor feld [14].
However, if in addition is unknown, then one frst must fnd
and then fnd . Tis is the identifcation problem.
Te exponential Radon transform, as a generalization of
the Radon transform, is defned as a mapping of function
spaces and is also represented in terms of Fourier transforms
of its domain and range, and this leads to a characterization of
the range of the transform. For more information about the
exponential Radon transform, we refer to [15, 16].
2. General Construction of Boehmians
Te minimal structure necessary for the construction of
Boehmians consists of the following elements:
(i) a set a and a commutative semigroup (g,∗);
(ii) an operation ⊙: a × g → a such that for each ∈ a
and
1
,
2
,∈ g,
⊙(
1
∗
2
)=(⊙
1
)⊙
2
; (3)
Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2014, Article ID 612391, 6 pages
http://dx.doi.org/10.1155/2014/612391