2017/02/02 by Andrew Knyazev, Knyazev, Andrew, Akshay Gadde +5
Engineering · Medicine · #46C07 #46N99 #94A08 #94A12 #94A20 #94A24 #E.4 #FOS: Computer and information sciences #FOS: Mathematics #Functional Analysis (math.FA) #H.1.1 #H.5.5 #I.4.2 #I.4.8 #Information Theory (cs.IT) #Machine Learning (stat.ML) #Medical Imaging Techniques and Applications #Photoacoustic and Ultrasonic Imaging #Sparse and Compressive Sensing Techniques
paper · pdf · doi:10.48550/arxiv.1702.00852
openalex publication_date 2017/02/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.