Fundamentals of Inverse Problems: Background chapter from Electric Field Imaging Thesis

Inverse problems involve estimating an underlying continuous function from a set of measurements of some aspect of the function. The mapping from the underlying continuous function to the measurements is known as the forward or direct problem. The forward problem must be a well-posed physical problem. If x is an underlying function and y is a set of measurements, then x and y are related by a forward mapping operator F : F (x) = y. Note that x and y are not members of the same space|often x is in nite dimensional and y is nite dimensional. The inverse problem is to recover x from a measured y using the inverse operator F 1 and (as we will see) regularization, or extra information about the underlying function. The reason we need the extra information about x is that inverse problems are typically ill-posed. Strictly speaking, to be well-posed, an inverse problem would have to have the following properties: (1) solutions x must be unique, (2) for any data y a solution must exist, and (3) solutions must be stable, so that small perturbations in the data do not lead to large perturbations in the reconstruction.[Isa89] The second, existence condition is often ignored, since in practical situations the data will have been generated from an actual forward problem, guaranteeing the existence of an inverse. (However, it is conceivable that measurement noise might complicate this.) The real di culties in practice have to do with the rst and third conditions: in some problems, the same data set may have multiple feasible explanations ( rst problem), or two very similar data sets may have very di erent explanations (third problem). In the latter case, the noise inherent