Project Details
Projekt
Splines of complex order, fractional operators and applications to signal and image processing
Subject Area
Mathematics
Term
from 2013 to 2018
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 246954628
Schoenberg's splines have found their way into a large number of applied mathematical areas, for example, signal processing, computer graphics, computer assisted geometric design, the numerics of partial differential equations and many more. With their flexible parameters they are adaptable to the particular task at hand and because of their simple representation they are easily implementable. These are properties that are imperative for a successful application. In addition, as splines are piecewise polynomials, they facilitate the interpretation of numerical results. In recent years, splines of fractional and complex order were defined, which, due to the existence of additional parameters, are even more adaptable. The continuous order parameter allows a precise adjustment to the regularity of the problem and the complex degree the extraction of phase and amplitude. These splines generate new mathematical interconnections ranging from fractional differential operators, Dirichlet averages, Riesz transformations and phase in higher dimensions, to curvature detection operators for image analysis.For concrete applications, the complete theoretical and numerical analyses of the approximation-theoretic properties of the spaces generated by splines of complex order are still missing. We want to close this gap by providing a comprehensive analysis of splines of complex order. For this purpose, we will emphasize the theoretical results and numerical analysis, as well as the algorithmic implementation.
DFG Programme
Research Grants
