Night-sky reconstructions for linear digital imaging systems

Abstract

In tomographic and other digital imaging systems the goal is often to reconstruct an object function from a finite amount of noisy data generated by that function through a system operator. One way to determine the reconstructed function is to minimize the distance between the noiseless data vector it would generate via the system operator, and the data vector created through the system by the real object and noise. The former we will call the reconstructed data vector, and the latter the actual data vector. A reasonable constraint to place on this minimization problem is to require that the reconstructed function be non-negative everywhere. Different measures of distance in data space then result in different reconstruction methods. For example, the ordinary Euclidean distance results in a positively constrained least squares reconstruction, while the Kulback-Leibler distance results in a Poisson maximum likelihood reconstruction. In many cases though, if the reconstruction algorithm is continued until it converges, the end result is a reconstructed function that consists of many point-like structures and little else. These are called night-sky reconstructions, and they are usually avoided by stopping the reconstruction algorithm early or using regularization. The expectation-maximization algorithm for Poisson maximum likelihood reconstructions is an example of this situation.