Guided Signal Reconstruction Theory
This work addresses signal reconstruction challenges in fields like imaging and signal processing, offering a theoretical framework with practical algorithms, though it appears incremental by building on known consistent and regularized reconstructions.
The paper tackles the problem of signal reconstruction by proposing a new axiomatic approach that defines a reconstruction set as the shortest pathway between sample-consistent and guiding sets, and derives new stability and error bounds using the largest nontrivial angle between subspaces, with numerical illustration for image magnification.
An axiomatic approach to signal reconstruction is formulated, involving a sample consistent set and a guiding set, describing desired reconstructions. New frame-less reconstruction methods are proposed, based on a novel concept of a reconstruction set, defined as a shortest pathway between the sample consistent set and the guiding set. Existence and uniqueness of the reconstruction set are investigated in a Hilbert space, where the guiding set is a closed subspace and the sample consistent set is a closed plane, formed by a sampling subspace. Connections to earlier known consistent, generalized, and regularized reconstructions are clarified. New stability and reconstruction error bounds are derived, using the largest nontrivial angle between the sampling and guiding subspaces. Conjugate gradient iterative reconstruction algorithms are proposed and illustrated numerically for image magnification.