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Interpolation in the Presence of Noise

Investigators: Michael Unser

Summary: We derive spline-fitting algorithms that are optimized for noisy data.

Introduction

Interpolation is a crucial operation in a variety of medical imaging tasks (e.g., image registration, tomographic reconstruction, and more). Presently, most techniques are optimized for a noise-free scenario. Our goal here is to optimize the prefilter that yields the B-spline expansion coefficients so that it can handle noisy data.

Main Contributions

We are investigating extensions of spline-type interpolation algorithms for the nonideal case where the signal samples are corrupted by noise.

The first aspect is that the spline space itself can be optimized depending on the smoothness properties or the statistics of the class of signal of interest. When the samples are corrupted by noise, it is appropriate to apply a smoothing-spline estimator with a regularization parameter that is set inversely proportional to the signal-to-noise ratio.

The second aspect is the design of the "optimal" prefilter that yields the expansion coefficient of the signal in a shift-invariant (or spline) subspace spanned by the integer shifts of a given generating function. In our formulation, we treat in a unified way the interpolation problem from ideal, noisy samples, and the deconvolution problem in which the signal is filtered prior to sampling. We have proposed several alternative approaches to designing the correction filter, which differ in their assumptions on the signal and noise. In particular, we have adapted the classical deconvolution solutions (least-squares, Tikhonov, and Wiener) to our particular situation and also proposed new methods that are optimal in a minimax sense. The solutions often have a similar structure and can be computed simply and efficiently by digital filtering.

Collaborations: Prof. Yonina Eldar (Technion)

Past Investigators: Sathish Ramani, Thierry Blu

Period: 2005-ongoing

Funding: Grant 200020-101821 from the Swiss Science Foundation, European HASSIP Network

Major Publications

[1] 

Y.C. Eldar, M. Unser, "Nonideal Sampling and Interpolation from Noisy Observations in Shift-Invariant Spaces," IEEE Transactions on Signal Processing, vol. 54, no. 7, pp. 2636-2651, July 2006.

[2] 

S. Ramani, D. Van De Ville, M. Unser, "Non-Ideal Sampling and Adapted Reconstruction Using the Stochastic Matérn Model," Best student paper award, Proceedings of the IEEE Thirty-First International Conference on Acoustics, Speech, and Signal Processing (ICASSP'06), Toulouse, France, May 14-19, 2006, pp. II-73-II-76.

[3] 

M. Unser, T. Blu, "Generalized Smoothing Splines and the Optimal Discretization of the Wiener Filter," IEEE Transactions on Signal Processing, vol. 53, no. 6, pp. 2146-2159, June 2005.

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