1. Introduction

The attenuated x-ray transform is a variant of the classical x-ray transform in which functions are integrated over straight lines with respect to an exponential weight. It arises as a model in single photon emission computed tomography (SPECT) and in the study of the stationary linear single speed transport equation. Let a, f be continuous functions of compact support in lit ℝn and let θ be a unit vector. We define the divergent beam x-ray transform of a at x in direction θ bywhere the integration is with respect to arc length. The attenuated x-ray transform of f is a function on the space of directed lines, whose value on the line l with direction θ is given by where Y( τ) is an arc length parametrization of l. When the attenuation, a, is identically zero, the attenuated x-ray transform reduces to the ordinary x-ray transform. In the model of single photon emission tomography, the function f represents the spatial density of emitters which are assumed to emit photons isotropically. The function a is the linear attenuation coefficient, and so the attenuated x-ray transform is supposed to represent the photon intensity at a detector, collimated to accept only photons which have travelled alung a tspecific line. A useful survey of the physics can be found in [91. (The density of emitters is called the activity distribution, and so some authors denote it by a, whereas we use a for attenuation.)