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A Unifying Spline Formulation for Stochastic Signal Processing [Or How Schoenberg Meets Wiener, with the Help of Tikhonov]

M. Unser, T. Blu

Plenary talk, Second International Conference on Computational Harmonic Analysis, Nineteenth Annual Shanks Lecture (CHA'04), Nashville TN, USA, May 24-30, 2004.



We introduce an extended class of cardinal L-splines where L is a pseudo-differential—but not necessarily local—operator satisfying some admissibility conditions. This family is quite general and includes a variety of standard constructions including the polynomial, elliptic, exponential, and fractional splines. In order to fit such splines to the noisy samples of a signal, we specify a corresponding smoothing spline problem which involves an L-semi-norm regularization term. We prove that the optimal solution, among all possible functions, is a cardinal L*L-spline which has a stable representation in a B-spline-like basis. We show that the coefficients of this spline estimator can be computed by digital filtering of the input samples; we also describe an efficient recursive filtering algorithm that is applicable whenever the transfer function of L is rational.

We justify this procedure statistically by establishing an equivalence between L*L smoothing splines and the MMSE (minimum mean square error) 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 such spline estimators.

References

  1. I.J. Schoenberg, "Contribution to the Problem of Approximation of Equidistant Data by Analytic Functions," Quarterly of Applied Mathematics, vol. 4, no. 2, pp. 45-99 & 112-141, 1946.

  2. A.N. Tikhonov, V.Y. Arsenin, Solutions of Ill-Posed Problems, W.H. Winston and Sons, Washington DC, USA, 258p., 1977.

  3. N. Wiener, Extrapolation, Interpolation and Smoothing of Stationary Time Series with Engineering Applications, MIT Press, Cambridge MA, USA, 163p., 1964.


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