Multiplicative algorithms for maximum penalized likelihood inversion with nonnegative constraints and generalized error distributions

In many linear inverse problems the unknown function f (or its discrete approximation θp×1), which needs to be reconstructed, is subject to the nonnegative constraint(s); we call these problems the nonnegative linear inverse problems (NNLIPs). This paper considers NNLIPs. However, the error distribution is not confined to the traditional Gaussian or Poisson distributions. We adopt the exponential family of distributions where Gaussian and Poisson are special cases. We search for the nonnegative maximum penalized likelihood (NNMPL) estimate of θ. The size of θ often prohibits direct implementation of the traditional methods for constrained optimization. How to develop easy-to-implement algorithms for the NNMPL estimates is an interesting and challenging question. Given that the measurements and point-spread-function (PSF) values are all nonnegative, we propose a simple multiplicative iterative algorithm. We show that if there is no penalty, then this algorithm is almost sure to converge; otherwise a relaxation or line search is necessitated to assure