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Splines and Optimal Signal Reconstruction

M. Unser

Tutorial, Meeting on "Information, Signal, Images et ViSion: Thème A—Traitement Statistique de l'Information [Échantillonnage Irrégulier]" (ISIS-A-EI'08), Paris, French Republic, April 18, 2008.



We consider the generic problem of reconstructing a signal from its noisy samples. We argue that an “optimal” solution can be specified through the minimization of a hybrid cost function that is the sum of a discrete-domain data term, and a continuous-domain regularization functional that forces the reconstruction to be well behaved.

In order to derive a practical algorithm, we propose to represent the solution using compactly-supported B-spline basis functions, which has a number of computational advantages. In the case where the data is uniformly sampled and the regularization function quadratic (Tikhonov criterion), this yields a smoothing spline estimator that can be implemented efficiently by digital filtering. We also propose an alternative stochastic formulation (hybrid Wiener filter) that leads to the same type of algorithm. We prove that both solutions are globally-optimal and computationally equivalent, provided that the reconstruction space is matched to the regularization operator (deterministic signal) or, alternatively, to the whitening operator of the process (stochastic modeling). This suggests a unifying interpretation of the optimal reconstruction process in terms of generalized splines. Finally, we address the problem of the reconstruction of a multidimensional signal from non-uniform samples. We review the classical thin-plate spline solution, and present an alternative B-spline-based algorithm that is computationally much more efficient.

References


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