Singular Value Analyses of Inversion of Laplace and Optical Imaging Transforms

Abstract

The use of the eigenvalues and eigenfunctions of the first order Fredholm equations describing optical imaging has long been known. The eigenvalues and eigenfunctions of the first order Fredholm equation of the Laplace transform have only recently been discovered and similarly used (McWhirter and Pike 1978). Let us consider for simplicity the one dimensional case with magnification unity. For an eigenfunction to be defined the linear mapping A:f → g of the "object" f into its "image" g must define a compact bijective operator A of, say, L2 (−1,+1) into itself.