Self-Similarity: Part I—Splines and Operators

M. Unser, T. Blu

IEEE Transactions on Signal Processing, in press.

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The central theme of this pair of papers 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)} = ∑_k∈Z a[k] δ(t−k) where L is a suitable pseudo-differential 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, ∂_τ^γ, 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 ∂_τ^γ is used to define a scale-invariant energy measure—the squared L₂-norm of the γth derivative of the signal—which 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.