Reconstruction of Images with Discontinuities via Variably Scaled Discontinuous Kernels: Applications to MPI

We consider the problem of reconstructing a 2D image, i.e. the computational issue consisting in approximating an underlying function given a set of data points and a set of data values. More precisely, we consider the interpolation problem; we look for a function such that it exactly matches the data values at the nodes. Since in applications we usually know the real data values at just few points (possibly scattered), in this work we focus on Radial Basis Function (RBF) interpolation. Indeed, it is meshfree and easy to implement in any dimension, differently from polynomial-based methods. Therefore, we consider the function space determined by a strictly positive definite and symmetric kernel which depends on a shape parameter . When dealing with images whose underlying functions have discontinuities or steep gradients, a fundamental task is to reduce the well-known Gibbs phenomenon, which affects the quality of the reconstruction causing oscillations and distortions in the whole image. A possible approach is to consider Variably Scaled Kernels (VSKs) [2], where the shape parameter is substituted by a scale function . VSKs turn out to be particularly effective when considering steep gradients, but the choice of the scale function is crucial. In our setting, in order to obtain a mask matrix of the image as scale function, we use an edge detector on a preliminary RBF reconstruction. We present some results involving different functions and sets of nodes. Finally, we apply this procedure in the context of Magnetic Particle Imaging (MPI) [1], using Lissajous nodes as data sites and showing that the proposed method outperforms the classical polynomial reconstruction in [3]. References 1. T.M. Buzug and T. Knopp, Magnetic Particle Imaging, Springer (2012) . 2. M. Bozzini et al., Interpolation with variably scaled kernels, IMA J. Numer. Anal. 35 (2015) , 199-219. 3. S. De Marchi et al. , Spectral filtering for the reduction of the Gibbs phenomenon for polynomial approximation methods on Lissajous curves with applications in MPI, Dolomites Res. Notes Approx. 10 (2017) , 128-137.