UC San Diego

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. We start by formulating the reconstruction problem for a variety of different imaging systems, which we can view as special cases or extensions of tomography. In order to understand the range of potential solutions to such inverse problems, we formulate the optimization problems of bounding the elements of the unknown. We proceed to develop a novel approach to investigating inverse problems by defining uniqueness as the case where the upper and lower bounds on possible values each element can take are equal. This allows us both to express uniqueness on an element-wise basis, e.g. some elements may be uniquely-reconstructed while others may not, and to formulate conditions for uniqueness. These conditions provide a generalization of known conditions for unique solutions to systems with sparse or non-negative unknowns. We use these uniqueness conditions to pose a novel approach to estimation of system resolution and to thereby find low-resolution approximations which provide as much resolution as possible while retaining the uniqueness of the solution. Finally we demonstrate the extension to noisy and regularized problems using a Bayesian formulation which allows us to form trade-offs between resolution and uncertainty