# Reconstruction of Preclinical PET Images via Chebyshev Polynomial Approximation of the Sinogram
## 1 Introduction  
For the past two decades, there has been an increasing demand for preclinical imaging systems. In this direction, several dedicated commercial and research imaging modalities have emerged for the study of small animals [1], such as mice, rats, rabbits, etc., enabling the implementation of longitudinal studies on the same animal. These imaging technologies include computed tomography (CT) [2], magnetic resonance imaging (MRI) [3], ultrasound (US) [4], and the nuclear medicine imaging modalities of single-photon emission computed tomography (SPECT) [5] and positron emission tomography (PET) [6]. PET imaging of small animals is an important, noninvasive imaging tool that has been used systematically in modern biomedical research [7; 8]. Small-animal PET, based on radioisotopes, has been established as a powerful diagnostic tool in molecular biology, genetics, oncology, cardiology, and drug development [9; 10; 11]. In preclinical PET, for the imaging of an internal organ's morphology or function, a radiopharmaceutical, such as \({}^{18}\)F-2-deoxy-2-fluoro-D-glucose (FDG), is intravenously injected into the blood stream of the small animal and is distributed in its body [12]. A radiotracer targets the organ being imaged and localizes there: once the organ becomes radioactive, it emits \(\gamma\)-rays. The emitted radiation is detected by the detectors which, in turn, create an image of the radioisotope distribution within the organ under investigation. PET data are usually stored in the form of sinograms or projections. Analytic image reconstruction, considered as the solution of a corresponding _inverse problem_, is performed by inverting the projection data. Measured data consist of the celebrated two-dimensional Radon transform of the radioisotope distribution function. Therefore, analytical PET image reconstruction involves the analytical inversion of the Radon transform of the radioisotope distribution function. The analytic inversion of the Radon line integral involves the derivative of the Hilbert transform of the sinogram. To this end, there are several analytic reconstruction techniques, including the predominant method of the so-called _filtered backprojection_ (FBP) [13; 14]. Recently, in our lab, we have developed another approach to analytic image reconstruction, namely the _spline reconstruction technique_ (SRT) [15; 16; 17; 18], as well as its attenuated generalization, referred to as \"aSRT\" [19]. These analytic reconstruction methods focus on the use of third-degree polynomial splines. Other analytic approaches involve Chebyshev decomposition and backprojection, by utilizing Chebyshev polynomials of the second kind [20]. In the present work, we present a new numerical algorithm for the inversion of the Radon transform based on Chebyshev polynomials of the first kind. By employing first-kind Chebyshev polynomial approximation, the computation of the derivative of the Hilbert transform of the sinogram is significantly simplified. Extending the mathematical setting of our previous research based on Chebyshev polynomials [21], we are able to efficiently apply a new Chebyshev inversion scheme for analytic PET image reconstruction. The main novelty of this paper is the mathematical derivation of the analytic reconstruction formula involving Chebyshev polynomials of the first kind for the first time, and the investigation of its performance in comparison to FBP. The underlying hypotheses involved in our present work are that Chebyshev polynomials of the first kind: (i) operate solely in the physical space, without involving any Fourier transform, and (ii) are employed globally, instead of locally, on the domain of approximation which is beneficial in terms of noise in the reconstructed images. We tested our analytic reconstruction method on projection data from a small-animal, image quality (IQ) simulated phantom study [22], in accordance with the NEMA protocol NU 4-2008 [23].