Significant Reduction of Gibbs’ Overshoot with Generalized Sampling Method

As is well-known, the use of Shannon sampling to interpolate functions with discontinuous jump points leads to the Gibbs’ overshoot. In image processing, it can lead to the problem of artifacts close to edges, known as Gibbs ringring. Its amplitude cannot be reduced by increasing the sample density. Here we consider a generalized Shannon sampling method which allows the use of timevarying sample densities so that samples can be taken at a varying rate adapted to the behavior of the function. Using this generalized sampling method to approximate a periodic step function, we observe a strong reduction of Gibbs’ overshoot. In a concrete example, the amplitude of the Gibbs’ overshoot is reduced by about 70%.