Inversion of the attenuated radon transform using cubic splines. (September 2016) - Protonotarios, Nicholas E.; Fokas, Athanassios S.; Gaitanis, Anastasios; Kastis, George A.
Abstract : Introduction: The mathematics of single photon emission computed tomography (SPECT) involve the inversion of a certain integral, namely the attenuated Radon transform. This integral is stored in the form of the so-called attenuated sinogram . Purpose: By modifying the original formulation introduced by one of the authors in 2006, we have presented a slightly different formulation of the Inverse Attenuated Radon Transform (IART), which we refer to as attenuated Spline Reconstruction Technique (aSRT). The purpose of this study is to evaluate aSRT using simulated SPECT/CT phantoms and compare it with the industry standard, filtered backprojection (FBP). Materials and methods: We have rederived the analytic formula of the IART, which involves the Hilbert transform of the attenuation correction coefficient as well as of two functions of the attenuated sinogram. For the numerical implementation of these Hilbert transforms we have employed an approximation of custom-made third degree piecewise polynomials, namely cubic splines. Relevant sinograms were generated in STIR (Software for Tomographic Imaging Reconstruction) and were acquired for 30, 60, 120, 180, and 360 projections over 360 degrees. Comparisons with FBP were evaluated using contrast and bias. Results: Our numerical tests suggest that the new technique can efficiently produce accurate reconstructions for real phantoms. Conclusion: aSRT provides an alternative to FBP, which incorporates the attenuation correction within the algorithm itself. G.A. Kastis acknowledges the financial support of the "Alexander S. Onassis Public Benefit Foundation" under grant No. R ZL 001-0. Furthermore, this work was partially supported by the research program "Inverse Problems and Medical Imaging" (200/842) of the Research Committee of the Academy of Athens.