Generalized Sampling: A Variational Approach

We consider the problem of reconstructing a multidimensional and multivariate function ƒ: $ℜ ^{ m }$ \rightarrow $ℜ ^{ n }$ from the discretely and irregularly sampled responses of q linear shift-invariant filters. Unlike traditional approaches which reconstruct the function in some signal space V, our reconstruction is optimal in the sense of a plausibility criterion J. The reconstruction is either consistent with the measures, or minimizes the consistence error. There is no band-limiting restriction for the input signals. We show that important characteristics of the reconstruction process are induced by the properties of the criterion J. We give the reconstruction formula and apply it to several practical cases.