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# Splines, Noise, Fractals and Optimal Signal Reconstruction

## M. Unser

### Plenary talk, Seventh International Workshop on Sampling Theory and Applications (SampTA'07), Θεσσαλονίκη (Thessaloniki), Ελληνική Δημοκρατία (Hellenic Republic), June 1-5, 2007.

We consider the generalized sampling problem with non-ideal acquisition device. The task is to “optimally” reconstruct a continuously-varying signal from its discrete, noisy measurements in some integer-shift-invariant space.

We propose three formulations of the problem—variational/Tikhonov, minimax, and minimum mean square error estimation—and derive the corresponding solutions for a given reconstruction space. We prove that these solutions are also globally-optimal, provided that the reconstruction space is matched to the regularization operator (deterministic signal) or, alternatively, to the whitening operator of the process (stochastic modeling). Moreover, the three formulations lead to the same generalized smoothing spline reconstruction algorithm, but only if the reconstruction space is chosen optimally.

We then show that fractional splines and fractal processes (fBm) are solutions of the same type of differential equations, except that the context is different: deterministic versus stochastic. We use this link to provide a solid stochastic justification of spline-based reconstruction algorithms.

The presentation features joint work with Yonina Eldar (Technion) and Thierry Blu (EPFL).

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