This paper considers the problem of optimum reconstruction in generalized sampling-reconstruction processes (GSRPs). We propose constrained GSRP, a novel framework that minimizes the reconstruction error for inputs in a subspace, subject to a constraint on the maximum regret-error for any other signal in the entire signal space. This framework addresses the primary limitation of existing GSRPs (consistent, subspace and minimax regret), namely, the assumption that the a priori subspace is either fully known or fully ignored. We formulate constrained GSRP as a constrained optimization problem, the solution to which turns out to be a convex combination of the subspace and the minimax regret samplings. Detailed theoretical analysis on the reconstruction error shows that constrained sampling achieves a reconstruction that is 1) (sub)optimal for signals in the input subspace, 2) robust for signals around the input subspace, and 3) reasonably bounded for any other signals with a simple choice of the constraint parameter. Experimental results on sampling-reconstruction of a Gaussian input and a speech signal demonstrate the effectiveness of the proposed scheme.
Assume that the following direct-sum condition holds Then, the correction filter provides an error-free reconstruction for input signals in @math @cite_5 .
@cite_5
2013180153
This article introduces a general framework for sampling and reconstruction pro- cedures based on a consistency requirement, introduced by Unser and Aldroubi in (29). The procedures we develop allow for almost arbitrary sampling and reconstruction spaces, as well as arbitrary input signals. We first derive a nonredundant sampling procedure. We then introduce the new concept of oblique dual frame vectors, that lead to frame expansions in which the analysis and synthesis frame vectors are not constrained to lie in the same space. Based on this notion, we develop a redundant sampling procedure that can be used to reduce the quantization error when quantizing the measurements prior to reconstruction.