An inner product framework for image reconstruction

This paper presents a new framework for image reconstruction from line integral measurements. By defining inner products for images and for sets of line integrals, we formulate a generalized least-squares problem and then apply standard optimization techniques to derive optimal image reconstruction algorithms. The inner products, and their associated norms, are defined for infinite-dimensional functions and linear operators, thereby avoiding problems associated with discrete finite-dimensional representations. From the inner product framework, we derive new image reconstruction algorithms, and unify several known algorithms by showing them to be optimal under different inner products. Under one special inner product-which we call the rho inner product-we prove that the line integral transform is a norm preserving linear operator. This result enables us to prove that the well known convolution-backprojection algorithm, which has never before been understood to be optimal in any sense, is in fact an optimal least-squares algorithm under the rho norm.