Chebyshev Interpolation Using Almost Equally Spaced Points and Applications in Emission Tomography

Abstract

Since their introduction, Chebyshev polynomials of the first kind have been extensively investigated, especially in the context of approximation and interpolation. Although standard interpolation methods usually employ equally spaced points, this is not the case in Chebyshev interpolation. Instead of equally spaced points along a line, Chebyshev interpolation involves the roots of Chebyshev polynomials, known as Chebyshev nodes, corresponding to equally spaced points along the unit semicircle. By reviewing prior research on the applications of Chebyshev interpolation, it becomes apparent that this interpolation is rather impractical for medical imaging. Especially in clinical positron emission tomography (PET) and in single-photon emission computerized tomography (SPECT), the so-called sinogram is always calculated at equally spaced points, since the detectors are almost always uniformly distributed. We have been able to overcome this difficulty as follows. Suppose that the function to be interpolated has compact support and is known at q equally spaced points in −1,1. We extend the domain to −a,a, a>1, and select a sufficiently large value of a, such that exactlyq Chebyshev nodes are included in −1,1, which are almost equally spaced. This construction provides a generalization of the concept of standard Chebyshev interpolation to almost equally spaced points. Our preliminary results indicate that our modification of the Chebyshev method provides comparable, or, in several cases including Runge’s phenomenon, superior interpolation over the standard Chebyshev interpolation. In terms of the L∞ norm of the interpolation error, a decrease of up to 75% was observed. Furthermore, our approach opens the way for using Chebyshev polynomials in the solution of the inverse problems arising in PET and SPECT image reconstruction.