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The noise contained in images collected by a charge coupled device (CCD) camera is predominantly of Poisson type. This motivates the use of the negative logarithm of the Poisson likelihood in place of the ubiquitous least squares fit-to-data. However, if the underlying mathematical model is assumed to have the form z = Au, where A is a linear, compact operator, the problem of minimizing the negative log-Poisson likelihood function is ill-posed, and hence some form of regularization is required. This work involves solving a variational problem: minimizing the sum of the negative log Poisson likelihood and a regularizing functional. The main result of this book is a theoretical analysis of this variational problem for various regularization functionals. In addition, this work presents an efficient computational method for its solution, and the demonstration of the effectiveness of this approach in practice by applying the algorithm to simulated astronomical imaging data corrupted by the CCD camera noise model mentioned above.