Non-linear optimal signal models and stability of sampling-reconstruction

Abstract

This dissertation has two main goals: To study the stability of sampling-reconstruction models, and to study the existence of optimal non-linear signal models. In the ¯rst part, we describe and quantify admissible perturbation of the sampling set X(Jitter), or the measuring devices (the way the sampling is performed, that is, we average a signal by ¯nite Borel measures), or the generator of the shift-invariant space (class to which belongs the signal to be sampled and reconstructed from its samples on X). We also study the reconstruction of a signal belonging to a shift-invariant space from its samples using an iterative algorithm, and we show that the sequence de¯ned by the algorithm converges to the original signal geometrically fast. Furthermore, we show that the reconstruction error due to perturbations we describe is controlled continuously by the perturbation errors. In the second part, we guarantee the existence of a signal model M = S i2I Ci from observed data F = ff1; : : : ; fmg ½ H that minimize the quantity e(F; fC1; : : : ;Clg) = Pm j=1 min1·i·l d2(fj ;Ci), where H is a separable Hilbert space (Problem 1). Su±cient conditions are given over the class C = fCigi2I in terms of the weak operator topology in order to guarantee the existence of a minimizer to Problem 1 for any set of data F. Moreover, we consider the problem when the class C is de¯ned in terms of a collection of unitary operators applied to a convex subset of H, and we obtain an algorithm for constructing collections of closed subspaces for which we a priori know that Problem 1 can be solved. As a consequence, we obtain the well-known qualitative version of the Eckart- Young Theorem.