Generalized smoothing splines and the optimal discretization of the Wiener filter

We introduce an extended class of cardinal L/sup */L-splines, where L is a pseudo-differential operator satisfying some admissibility conditions. We show that the L/sup */L-spline signal interpolation problem is well posed and that its solution is the unique minimizer of the spline energy functional...

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Vydáno v:	IEEE transactions on signal processing Ročník 53; číslo 6; s. 2146 - 2159
Hlavní autoři:	Unser, M., Blu, T.
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York, NY IEEE 01.06.2005 Institute of Electrical and Electronics Engineers The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Témata:	Algorithms Applied sciences Detection, estimation, filtering, equalization, prediction Digital filters Discretization Energy conservation Estimation error Exact sciences and technology Filtering algorithms Gaussian noise Information, signal and communications theory Interpolation Least squares approximation Mean square error methods Mean square errors Nonparametric estimation Operators Optimization recursive filtering Signal and communications theory Signal processing Signal, noise Smoothing Smoothing methods smoothing splines Splines splines (polynomial and exponential) stationary processes Studies Telecommunications and information theory Transfer functions variational principle Wiener filter stationary processes Non parametric estimation Nonparametric estimation smoothing splines Exponential function Recursive filtering splines (polynomial and exponential) Polynomial function Interpolation Wiener filter Smoothing Variational principle Stationary process Bayes methods
ISSN:	1053-587X, 1941-0476
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Popis
Shrnutí:	We introduce an extended class of cardinal L/sup /L-splines, where L is a pseudo-differential operator satisfying some admissibility conditions. We show that the L/sup /L-spline signal interpolation problem is well posed and that its solution is the unique minimizer of the spline energy functional /spl par/Ls/spl par//sub L2//sup 2/, subject to the interpolation constraint. Next, we consider the corresponding regularized least squares estimation problem, which is more appropriate for dealing with noisy data. The criterion to be minimized is the sum of a quadratic data term, which forces the solution to be close to the input samples, and a "smoothness" term that privileges solutions with small spline energies. Here, too, we find that the optimal solution, among all possible functions, is a cardinal L/sup /L-spline. We show that this smoothing spline estimator has a stable representation in a B-spline-like basis and that its coefficients can be computed by digital filtering of the input signal. We describe an efficient recursive filtering algorithm that is applicable whenever the transfer function of L is rational (which corresponds to the case of exponential splines). We justify these algorithms statistically by establishing an equivalence between L/sup /L smoothing splines and the minimum mean square error (MMSE) estimation of a stationary signal corrupted by white Gaussian noise. In this model-based formulation, the optimum operator L is the whitening filter of the process, and the regularization parameter is proportional to the noise variance. Thus, the proposed formalism yields the optimal discretization of the classical Wiener filter, together with a fast recursive algorithm. It extends the standard Wiener solution by providing the optimal interpolation space. We also present a Bayesian interpretation of the algorithm.
Bibliografie:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 ObjectType-Article-2 ObjectType-Feature-1 content type line 23
ISSN:	1053-587X 1941-0476
DOI:	10.1109/TSP.2005.847821