Detecting discontinuity points from spectral data with the quotient-difference (qd) algorithm

AbstractThis paper introduces a new technique for the localization of discontinuity points from spectral data. Through this work, we will be able to detect discontinuity points of a 2π-periodic piecewise smooth function from its Fourier coefficients. This could be useful in detecting edges and reducing the effects of the Gibbs phenomenon which appears near discontinuities and affects signal restitution. Our approach consists in moving from a discontinuity point detection problem to a pole detection problem, then adapting the quotient-difference (qd) algorithm in order to detect those discontinuity points

Smooth and compactly supported viscous sub-cell shock capturing for Discontinuous Galerkin methods

In this work, a novel artificial viscosity method is proposed using smooth and compactly supported viscosities. These are derived by revisiting the widely used piecewise constant artificial viscosity method of Persson and Peraire as well as the piecewise linear refinement of Klöckner et al. with respect to the fundamental design criteria of conservation and entropy stability. Further investigating the method of modal filtering in the process, it is demonstrated that this strategy has inherent shortcomings, which are related to problems of Legendre viscosities to handle shocks near element boundaries. This problem is overcome by introducing certain functions from the fields of robust reprojection and mollififers as viscosity distributions. To the best of our knowledge, this is proposed for the first time in this work. The resulting $C_0^\infty$ artificial viscosity method is demonstrated to provide sharper profiles, steeper gradients and a higher resolution of small-scale features while still maintaining stability of the method

Sparse Gradient Image Reconstruction from Incomplete Fourier Measurements and Prior Edge Information

In many imaging applications, such as functional Magnetic Resonance Imaging (fMRI), full, uniformly- sampled Cartesian Fourier (frequency space) measurements are acquired to reconstruct an image. In order to reduce scan time and increase temporal resolution for fMRI studies, one would like to accurately reconstruct these images from the smallest possible set of Fourier measurements. The emergence of Compressed Sensing (CS) has given rise to techniques that can provide exact and stable recovery of sparse images from a relatively small set of Fourier measurements. In particular, if the images are sparse with respect to their gradient, e.g., piece-wise constant, total-variation minimization techniques can be used to recover those images from a highly incomplete set of Fourier measurements. In this paper, we propose a new algorithm to further reduce the number of Fourier measurements required for exact or stable recovery by utilizing prior edge information from a high resolution reference image. This reference image, or more precisely, the fully sampled Fourier measurements of this reference image, is obtained prior to an fMRI study in order to provide approximate edge information for the region of interest. By combining this edge information with CS techniques for sparse gradient images, numerical experiments show that we can further reduce the number of Fourier measurements required for exact or stable recovery by an additional factor of 1.6 − 3 , compared with CS techniques alone, without edge information