EPFL
 Biomedical Imaging GroupSTI
EPFL
  Publications
English only   BIG > Publications > Smoothing Splines


 CONTENTS
 Home Page
 News & Events
 People
 Publications
 Tutorials and Reviews
 Research
 Demos
 Download Algorithms

 DOWNLOAD
 PDF
 Postscript
 All BibTeX References

Generalized Smoothing Splines and the Optimal Discretization of the Wiener Filter

M. Unser, T. Blu

IEEE Transactions on Signal Processing, vol. 53, no. 6, pp. 2146-2159, June 2005.



We introduce an extended class of cardinal L*L-splines, where L is a pseudo-differential operator satisfying some admissibility conditions. We show that the L*L-spline signal interpolation problem is well posed and that its solution is the unique minimizer of the spline energy functional, ||L s||L22, 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*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*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.


@ARTICLE(http://bigwww.epfl.ch/publications/unser0506.html,
AUTHOR="Unser, M. and Blu, T.",
TITLE="Generalized Smoothing Splines and the Optimal Discretization of
        the {W}iener Filter",
JOURNAL="{IEEE} Transactions on Signal Processing",
YEAR="2005",
volume="53",
number="6",
pages="2146--2159",
month="June",
note="")

© 2005 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from IEEE.
This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.