A fast compact algorithm for cubic spline smoothing
Computational Statistics & Data Analysis
Invariances, Laplacian-like wavelet bases, and the whitening of fractal processes
IEEE Transactions on Image Processing
Linear image reconstruction by Sobolev norms on the bounded domain
SSVM'07 Proceedings of the 1st international conference on Scale space and variational methods in computer vision
Low-noise dynamic reconstruction for X-ray tomographic perfusion studies using low sampling rates
Journal of Biomedical Imaging
Analytical footprints: compact representation of elementary singularities in wavelet bases
IEEE Transactions on Signal Processing
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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)}=Sigma kisinZa[k]delta(t-k), 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, parttau gamma, where gamma is the order of the derivative and tau is an additional phase factor. We specify the corresponding L-splines, which yield an extended class of fractional splines. The operator parttau gamma is used to define a scale-invariant energy measure-the squared L2-norm of the gammath 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 2gamma, 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 2gamma. We also establish a formal link between the regularization parameter lambda and the cutoff frequency of the smoothing spline filter: omega0aplambda-2gamma. 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