Self-similarity: Part I - Splines and operators
Journal
IEEE Transactions on Signal Processing
Journal Volume
55
Journal Issue
4
Date Issued
2007-04-01
Author(s)
Unser, Michael
Abstract
The central theme of this pair of papers (Parts I and II in this issue) is self-similarity, which is used as a bridge for connecting splines and fractals. The first part of the investigation is deterministic, and the context is that of L-splines; these are defined in the following terms: s(t) is a cardinal L-spline iff L{s(t) = σκεζ α[κ]δ(t - κ), where L is a suitable pseudodifferential operator. Our starting point for the construction of self-similar splines is the identification of the class of differential operators L that are both translation and scale invariant. This results into a two-parameter family of generalized fractional derivatives, ∂τγ, where γ is the order of the derivative and τ is an additional phase factor. We specify the corresponding L-splines, which yield an extended class of fractional splines. The operator ∂τγ is used to define a scale-invariant energy measurethe squared L2-norm of the γth derivative of the signalwhich provides a regularization functional for interpolating or fitting the noisy samples of a signal. We prove that the corresponding variational (or smoothing) spline estimator is a cardinal fractional spline of order 2γ, which admits a stable representation in a B-spline basis. We characterize the equivalent frequency response of the estimator and show that it closely matches that of a classical Butterworth filter of order 2γ. We also establish a formal link between the regularization parameter λ and the cutoff frequency of the smoothing spline filter: ω0 ≈ λ-2γ. Finally, we present an efficient computational solution to the fractional smoothing spline problem: It uses the fast Fourier transform and takes advantage of the multiresolution properties of the underlying basis functions. © 2007 IEEE.
Subjects
Fractals | Fractional derivatives | Fractional splines | Interpolation | Self-similarity | Smoothing splines | Tikhonov regularization
Type
journal article