Identifying rotational radon transforms
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IDENTIFYING ROTATIONAL RADON TRANSFORMS ´ a ´d Kurusa1 Arp Bolyai Institute, Aradi v´ertan´ uk tere 1 H-6720 Szeged, Hungary E-mail: [email protected] (Received October 25, 2011; Accepted July 13, 2012) [Communicated by M´ aria B. Szendrei]
Abstract We show classes of test functions so that dilational and rotational invariances of the image RS,µ f of such a test function f determines dilational and rotational invariances of rotational Radon transform RS,µ . Then we determine the defining flower S and weight µ of a conformal Radon transform RS,µ in terms of the image RS,µ f of an unknown function that is a sum of an L2 -function and finitely many Dirac distributions if the flower S is not selftangent.
1. Introduction Let Sω be a set of hypersurfaces Sω,t in Rn so that ω ∈ Sn−1 and t ∈ [0, ∞). The Radon transform RS,µ of functions f : Rn → R integrable on each Sω,t is defined by Z RS,µ f (ω, t) =
f (x)µω,t (x) dx,
(1.1)
Sω,t
where dx is the natural surface measure on Sω,t and µω,t is a strictly positive continuous function on Sω,t that depends continuously on ω and t. In this definition, S the hypersurfaces Sω,t are called the petals, the set S = ω∈Sn−1 Sω of them is called flower and µω,t is called the weight on the petal Sω,t . In terms of this definition the “classic” Radon transform RH,µ is defined by the flower H = {Hω,t : ω ∈ Sn−1 , t ∈ [0, ∞)} of the petals Hω,t = {x : hx, ωi = t} with weight µω,t ≡ 1. Mathematics subject classification number : 44A12. Key words and phrases: invariant Radon transform, identification of Radon transform, rotational Radon transform, conformal Radon transform. 1 Supported by the European Union and co-funded by the European Social Fund under the project “Telemedicine-focused research activities on the field of Matematics, Informatics ´ and Medical sciences” of project number “TAMOP-4.2.2.A-11/1/KONV-2012-0073”. 0031-5303/2013/$20.00 c Akad´emiai Kiad´o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
188
´ KURUSA A.
The problem considered in this article is to identify an a priori unknown Radon transform RS,µ or the properties of its flower and/or weight by means of the known images RS,µ f of (partially) (un)known test functions f . In the special case where the petals are total geodesics of a certain Riemannian manifold Mukhometov proves in [9] and [10] that RS,1 1 determines S. Considering the attenuated Radon transform RH,µ which was arisen in single photon emission computed tomography Natterer shows in [11] and [12] that the weight µ can be computed up to a multiplicative function in the support of the test function f if f is known to be a finite sum of Dirac measures. Hertle proves in [5] that for constantly attenuated Radon transforms RH,µ the map (f, µ) 7→ RH,µ f is injective on the set of the compactly supported, not radial distributions f . Solmon shows in [16] that the weight µ of an exponential Radon transform (which is a synonym for constantly attenuated Radon transforms) RH,µ can be computed from RH,µ f if and only if the compactly supported
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