Reconstruction of Preclinical PET Images via Chebyshev Polynomial Approximation of the Sinogram

Over the last decades, there has been an increasing interest in dedicated preclinical imaging modalities for research in biomedicine. Especially in the case of positron emission tomography (PET), reconstructed images provide useful information of the morphology and function of an internal organ. PET data, stored as sinograms, involve the Radon transform of the image under investigation. The analytical approach to PET image reconstruction incorporates the derivative of the Hilbert transform of the sinogram. In this direction, in the present work we present a novel numerical algorithm for the inversion of the Radon transform based on Chebyshev polynomials of the first kind. By employing these polynomials, the computation of the derivative of the Hilbert transform of the sinogram is significantly simplified. Extending the mathematical setting of previous research based on Chebyshev polynomials, we are able to efficiently apply our new Chebyshev inversion scheme for the case of analytic preclinical PET image reconstruction. We evaluated our reconstruction algorithm on projection data from a small-animal image quality (IQ) simulated phantom study, in accordance with the NEMA NU 4-2008 standards protocol. In particular, we quantified our reconstructions via the image quality metrics of percentage standard deviation, recovery coefficient, and spill-over ratio. The projection data employed were acquired for three different Poisson noise levels: 100% (NL1), 50% (NL2), and 20% (NL3) of the total counts, respectively. In the uniform region of the IQ phantom, Chebyshev reconstructions were consistently improved over filtered backprojection (FBP), in terms of percentage standard deviation (up to 29% lower, depending on the noise level). For all rods, we measured the contrast-to-noise-ratio, indicating an improvement of up to 68% depending on the noise level. In order to compare our reconstruction method with FBP, at equal noise levels, plots of recovery coefficient and spill-over ratio as functions of the percentage standard deviation were generated, after smoothing the NL3 reconstructions with three different Gaussian filters. When post-smoothing was applied, Chebyshev demonstrated recovery coefficient values up to 14% and 42% higher, for rods 1-3 mm and 4-5 mm, respectively, compared to FBP, depending on the smoothing sigma values. Our results indicate that our Chebyshev-based analytic reconstruction method may provide PET reconstructions that are comparable to FBP, thus yielding a good alternative to standard analytic preclinical PET reconstruction methods.