A Brief Theory of Guided Signal Reconstruction

An axiomatic approach to signal reconstruction is formulated, involving a sample consistent set, defined as a set of signals sample-consistent with the original signal, and a given guiding set, describing desired reconstructions. New frame-less reconstruction methods are proposed, based on a reconstruction set, defined as a shortest pathway between the sample consistent set and the guiding set, where the guiding set is a closed subspace and the sample consistent set is a closed plane in a Hilbert space. Existence and uniqueness of the reconstruction set are investigated. Connections to earlier known consistent, generalized, and regularized reconstructions are clarified, and new and improved reconstruction error bounds are derived.