2016/02/27 by Ole Christensen, Christensen, Ole, Brigitte Forster +3
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Image Processing Techniques and Applications #Image and Signal Denoising Methods #math.FA #msc:42C15 #msc:65D07
paper · pdf · doi:10.48550/arxiv.1602.08580
arxiv created 2016/04/16 · arxiv updated 2016/04/19
Pseudo-splines of integer order (m,ℓ) were introduced by Daubechies, Han, Ron, and Shen as a family which allows interpolation between the classical B-splines and the Daubechies' scaling functions. The purpose of this paper is to generalize the pseudo-splines to fractional and complex orders (z, ℓ) with α:=\re z > 1. This allows increased flexibility in regard to smoothness: instead of working with a discrete family of functions from Cm, m∈ \N0, one uses a continuous family of functions belonging to the Hölder spaces Cα-1. The presence of the imaginary part of z allows for direct utilization in complex transform techniques for signal and image analyses. We also show that in analogue to the integer case, the generalized pseudo-splines lead to constructions of Parseval wavelet frames via the unitary extension principle.