Spectral filtering for the resolution of the Gibbs phenomenon in MPI applications by Lissajous sampling

Polynomial interpolation and approximation methods on sampling points along Lissajous curves using Chebyshev series is an e-ffective way for a fast image reconstruc-tion in Magnetic Particle Imaging. Due to the nature of spectral methods, a Gibbs phenomenon occurs in the reconstructed image if the underlying function has discon-tinuities. A possible solution for this problem are spectral filtering methods acting on the coefficients of the approximating polynomial. In this work, after a description of the Gibbs phenomenon and classical filtering techniques in one and several dimensions, we present an adaptive spectral filtering process for the resolution of this phenomenon and for an improved approximation of the underlying function or image. In this adaptive filtering technique, the spectral filter depends on the distance of a spatial point to the nearest discontinuity. We show the e-ffectiveness of this filtering approach in theory, in numerical simulations as well as in the application in Magnetic Particle Imaging.