Spectral Dynamics and Regularization of Incompletely and Irregularly Measured Data

Abstract

Constrained by practical and economical aspects, in many applications, one often deals with data sampled irregularly and incompletely. The use of irregularly sampled data may result in some artifacts and poor spatial resolution. Therefore, the preprocessing of the measurements onto a regular grid plays an important step. One of the methods achieving this objective is based on the Fourier reconstruction, which involves an underdetermined system of equations. The recent Uniform Uncertainty Principle (UUP) uses convex optimization through l ₁ minimization for solving underdetermined systems. The l ₁ minimization admits certain theoretical guarantees and simpler implementation. The present work applies UUP to the Fourier-based data regularization problem. For the signals having sparse Fourier spectra, our method replaces the incomplete and irregular coordinate grid with the grid that is a subset of equispaced complete grid. It then generates error resulting from the stated replacement. Finally, it applies UUP to realize its objective. To justify the applicability of our method, we present the empirical performance of it on different sets of measurement coordinates as a function of number of nonzero Fourier coefficients.