Sampling with Arbitrary Sampling and Reconstruction Spaces and Oblique Dual Frame Vectors

This article introduces a general framework for sampling and reconstruction procedures 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.