Fractional and Complex Pseudo-Splines and the Construction of Parseval\n Frames

2016/02/27 by Ole Christensen, Christensen, Ole, Brigitte Forster +3
Engineering · Computer Science · #Advanced Numerical Analysis Techniques #Image and Signal Denoising Methods #Image Processing Techniques and Applications

paper · pdf · doi:10.48550/arxiv.1602.08580

Abstract

Pseudo-splines of integer order (m,\ℓ) were introduced by Daubechies,\nHan, Ron, and Shen as a family which allows interpolation between the classical\nB-splines and the Daubechies' scaling functions. The purpose of this paper is\nto generalize the pseudo-splines to fractional and complex orders (z, \ℓ)\nwith \α:= re z > 1. This allows increased flexibility in regard to\nsmoothness: instead of working with a discrete family of functions from C^m,\nm\∈ N₀, one uses a \continuous family of functions belonging to the\nH "older spaces C^\α-1. The presence of the imaginary part of z\nallows for direct utilization in complex transform techniques for signal and\nimage analyses. We also show that in analogue to the integer case, the\ngeneralized pseudo-splines lead to constructions of Parseval wavelet frames via\nthe unitary extension principle.\n

Citations

• Orthonormal bases of compactly supported wavelets

Related

• Exponential Splines of Complex Order
• Fractional Cone Splines and Hex Splines
• Bernuau spline wavelets and Sturmian sequences
• On B-spline framelets derived from the unitary extension principle
• Construction of scaling partitions of unity