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TTR algorithm for the inversion of the exponential X-ray transform

Jean-Marc WAGNER and Frédéric NOO


This work presents new mathematical results concerning three-dimensional (3-D) image reconstruction from exponential X-ray (parallel-beam) projections.

The exponential X-ray transform is a mathematical tool useful in SPECT imaging and also in Intensity Modulated Radiation Therapy [1]. In 2-D SPECT, it allows fast analytical reconstructions to be achieved with accurate correction for attenuation and depth-dependent collimator-response [2,3]. In fully 3-D 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 Rotating-Slant-Hole scanner [5].

2-D image reconstruction from exponential X-ray projections has been widely studied over the last twenty years and is now well-understood, especially thanks to the significant work of Pan and Metz [6,7]. In fully 3-D geometry, the situation is quite different. To our knowledge, only two works concerning exact 3-D reconstruction from exponential X-ray 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 un-attenuated case [10]. Such a question is mathematically difficult to answer because 3-D reconstruction theory for un-attenuated X-ray projections [11] is not readily modified to handle exponential X-ray projections.

In this paper, we present a filtered-backprojection (FBP) algorithm suitable for image reconstruction from exponential X-ray 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 un-attenuated projections. From the theory for the 2D Radon transform, Cho et al. derived the first True Three-dimensional Reconstruction (TTR) algorithm of FBP-type for un-attenuated X-ray projections. From the theory for the 2D exponential Radon transform, we have derived an algorithm of FBP-type for 3-D image reconstruction from exponential X-Ray projections. We call this new algorithm the EXP-TTR 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 3-D exponential X-ray transform.


Let \( \Omega \) be the equatorial band of aperture angle \( \theta _{0} \) on the unit sphere (see figure [*]a). The data used for reconstruction are the exponential X-ray projections

p(\underline{\theta },\underline{s})=\int ^{+\infty }_{-\inf...
..., \, \, \, \, \, \textrm{ }\underline{s}.\underline{\theta }=0
\end{displaymath} (1)

corresponding to the directions \( \underline{\theta }\in \Omega \). The 3-D image to be reconstructed is \( f \) and \( \mu \) is the constant attenuation factor.

Figure: (a) Description of the equatorial band \( \Omega \) (b) Description of the set \( A\protect \protect \).

In the EXP-TTR algorithm, \( f \) is reconstructed at point \( \underline{x} \) of \( R^{3} \) using the filtered backprojection formula

f(\underline{x})=\int _{\Omega }d\underline{\theta }\, e^{\m...
...ine{x}-(\underline{x}.\underline{\theta })\underline{\theta })
\end{displaymath} (2)

where the filtered projections \( p^{F}(\underline{\theta },\underline{s}) \) are calculated as
p^{F}(\underline{\theta },\underline{s})=p(\underline{\theta...
...\, \otimes ^{2}\, h_{\mu }(\underline{\theta },\underline{s}).
\end{displaymath} (3)

The symbol \( \otimes ^{2} \) in this formula represents a 2D convolution and \( h_{\mu }(\underline{\theta },\underline{s}) \) is the spatial response of the filter to be applied to each projection \( p(\underline{\theta },\underline{s}) \).

Following the same ideas as in [11] and [12], we have obtained the frequency expression \( H_{\mu }(\underline{\theta },\underline{S}) \) of the filter \( h_{\mu }(\underline{\theta },\underline{s}) \). This expression is

H_{\mu }(\underline{\theta },\underline{S})=\frac{1}{8\pi (1...
\end{displaymath} (4)

where \( \underline{S}.\underline{\theta }=0 \) and
A^{*}=A\setminus \left\{ \underline{n}\in A\, :\, \vert\unde...
...rline{S}.\underline{n})\underline{n}\vert<\mu /2\pi \right\} .
\end{displaymath} (5)

The set \( A\protect \protect \) is the set of all unit vectors \( \underline{n} \) orthogonal to great circles included in \( \Omega \). It is represented at figure [*]b.

We have expanded the calculation of \( H_{\mu }(\underline{\theta },\underline{S}) \) in ([*]) for the particular case of the equatorial band and thereby obtained an analytical expression of \( H_{\mu }(\underline{\theta },\underline{S}) \) 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 \( \theta _{0}=90 \) degrees, for which the equatorial band is identical to the unit sphere; the filter \( H_{\mu }(\underline{\theta },\underline{S}) \) is then equivalent to the one derived by Hazou [8].

Figure: Graphical representation of the filter \( H_{\mu }(\underline{\theta },\underline{S}) \) ( \( \mu =0.04\, mm^{-1}\protect \protect \)). Polar angle of \( \underline{\theta }\protect \protect \) is equal to 90 degrees. Left : \( \theta _{0}=90 \) degrees. Right : \( \theta _{0}=30\protect \protect \) degrees.
\resizebox*{4cm}{!}{\includegraphics{} } \resizebox*{4cm}{!}{\includegraphics{} }

In figure [*], we show reconstructions of a simulated phantom modelling the heart. This phantom consists of three ellipsoids, two of which model the ventricules with 20\( \% \) of activity. The reconstruction was achieved on a grid of \( 100^{3} \) cubic voxels of side \( 1.5mm \), using \( \theta _{0}=45 \) degrees and \( \mu =0.0152\, mm^{-1} \). The set \( \Omega \) was uniformly sampled in spherical coordinates with a pitch of 3 degrees and the projections were sampled on grids of \( 128^{2} \) pixels of side \( 1.5mm \). The quality of the reconstruction in figure [*] demonstrates the exactness of the algorithm.

Figure: Reconstruction of a simulated phantom of the heart for \( \theta _{0}=45 \) degrees and \( \mu =0.0152\, mm^{-1} \).


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Jean-Marc Wagner and Frédéric Noo. All rights reserved.