A Minimum Squared-Error Framework for Sampling and Reconstruction in Arbitrary Spaces

We consider non-ideal sampling and reconstruction schemes in which the sampling and reconstruction spaces as well as the input signal can be arbitrary. To obtain a good reconstruction of the signal in the reconstruction space from arbitrary samples, we suggest processing the samples prior to reconstruction with a linear transformation that is designed to minimize the worst-case squared-norm error between the reconstructed signal, and the best possible (but usually unattainable) approximation of the signal in the reconstruction space. If the input signal is known to lie in an appropriately chosen subspace, then we propose a linear transformation that achieves the minimal squared-error norm approximation. We show both theoretically and through simulations that if the input signal does not lie in the reconstruction space, then the suggested methods can outperform the consistent reconstruction method previously proposed for this problem.