Doctoral Dissertation, University of California, San Diego, 2014. With modern optimization techniques it is generally possible to form some kind of image estimate using an arbitrary collection of views or sensor measurements of an unknown object. The question is what is the best result that one can get with that data and will it be acceptable? We use optimization theory to analyze inverse problems such as this and formulate strategies for understanding how much high resolution information is contained in the data and how to intelligently reconstruct the best image possible.