A class of ill-posed inverse problems that arises in astronomical imaging is considered. An iterative steepest descent method for regular least squares problems which constrains the solution to be nonnegative is presented. A careful consideration of the noise statistics that arise from the use of a CCD camera for data generation motivates the extension of this algorithm for use on weighted least squares problems. Preconditioning strategies are examined for both algorithms, and it is shown that in order to preserve noise statistics, preconditioners must be highly structured. Examples from astronomical imaging are used to illustrate behavior of the methods.