JeanMarc WAGNER and Frédéric NOO
This work presents new mathematical results concerning threedimensional (3D) image reconstruction from exponential Xray (parallelbeam) projections.
The exponential Xray transform is a mathematical tool useful in SPECT imaging and also in Intensity Modulated Radiation Therapy [1]. In 2D SPECT, it allows fast analytical reconstructions to be achieved with accurate correction for attenuation and depthdependent collimatorresponse [2,3]. In fully 3D SPECT, it provides a way to perform accurate attenuation correction without transmission measurements [4], which is highly attractive for sophisticated imaging systems such as the RotatingSlantHole scanner [5].
2D image reconstruction from exponential Xray projections has been widely studied over the last twenty years and is now wellunderstood, especially thanks to the significant work of Pan and Metz [6,7]. In fully 3D geometry, the situation is quite different. To our knowledge, only two works concerning exact 3D reconstruction from exponential Xray projections have been published so far. These two works (see [8] and [9]) assume both that the projections are finely sampled on the unit sphere. It is currently unknown if exact reconstruction can be achieved from more general data sets, such as those satisfying Orlov's condition for reconstruction in the unattenuated case [10]. Such a question is mathematically difficult to answer because 3D reconstruction theory for unattenuated Xray projections [11] is not readily modified to handle exponential Xray projections.
In this paper, we present a filteredbackprojection (FBP) algorithm suitable for image reconstruction from exponential Xray projections sampled on any subset of the unit sphere that includes great circles. A basic example of such a subset is the equatorial band illustrated in figure 1a  the discussion in section 2 will be focussed on that data set.
The derivation of our algorithm follows the same lines as the work of Cho et al. [12] for image reconstruction from unattenuated projections. From the theory for the 2D Radon transform, Cho et al. derived the first True Threedimensional Reconstruction (TTR) algorithm of FBPtype for unattenuated Xray projections. From the theory for the 2D exponential Radon transform, we have derived an algorithm of FBPtype for 3D image reconstruction from exponential XRay projections. We call this new algorithm the EXPTTR algorithm.
Our results generalize those published by Hazou [8] and Weng et al. [9]. However, they remain modest as they only apply to specific sets of measurements on the unit sphere  those including great circles. As was the work of Cho et al., we believe that our work is one step further towards full understanding of the 3D exponential Xray transform.
Let be the equatorial band of aperture angle
on the unit sphere (see figure a). The data used for
reconstruction are the exponential Xray projections
In the EXPTTR algorithm, is reconstructed at point
of using the filtered backprojection formula
(2) 
(3) 
Following the same ideas as in [11] and [12], we have obtained the
frequency expression
of the
filter
. This expression is
(5) 
We have expanded the calculation of in () for the particular case of the equatorial band and thereby obtained an analytical expression of more tractable for numerical implementations. For conciseness reasons, this expression is not given in the abstract. Only plots of the filter are given in figure . The left graph of figure illustrates the particular case degrees, for which the equatorial band is identical to the unit sphere; the filter is then equivalent to the one derived by Hazou [8].
