The Attenuated X-Ray Transform

Matias Courdurier
University of Washington

Date: April 18, 2005

Abstract: In two dimensions, given a continuous function $f$, its Weighted X-ray Transform is $(R_p)f (L) := \int_L f(x)p(L,x)ds(x)$, where $L$ is any line and $p(L,x)$ is a (known) weight function. The inmediate inverse problem that arises in this situation is trying to recover $f$ (if possible) from its Weighted X-ray Transform.

This is not possible in general [2], but for a particular important family of weights, for the so called Attenuated X-Ray Transform, Arbuzov, Bukgheim and Kazantev in 1998 [1] and R.G. Novikov in 2000 [5] provided in two dimensions an explicit inversion formula. The Attenuated X-ray Transform is motivated by the medical image technique of Single Positron Emission Tomography (SPECT), where the interest is in recovering the density map $f$ of some radioactive material inside the body and where the weight on the attenuated X-ray transform comes mainly from scattering and absortion of the photons as they travel along lines inside the body.

We will review the '04 paper of Boman and Stromberg [3], where it is proven Novikov's formula for a somewhat larger class of weight functions using a completely different and more elementary method than Novikov's one.

A good reference on the development on the Attenuated X-ray Transform is the survey article by D. Finch [4].

References:

[1] Arbuzov, E.V., Burkhgeim, A.L., and Kazantev, S.G. Two dimensional tomography problems and the theory of A-analytic functions, Siberian Adv. Math., 8, 1-20 (1998).

[2] Boman, J. An Example of non-uniqueness for a generalized Radon Transform, J. Anal. Math., 61, 395-401, (1993).

[3] Boman, J. and Stromberg, J-O., Novikov's inversion formula for the Attenuated Radon Transform - A new approach, J. Geom. Anal., 14, 185-198, (2004).

[4] Finch, D. The attenuated X-ray transform: recent developments, in "Inside Out: Inverse problems and Applications", Uhlmann, G., Ed., Cambridge University Press, (2003).

[5] Novikov, R.G. An inversion formula for the attenuated X-ray transformation, Ark. Math., 40, 145-167, (2002)