14 - Incomplete Problems

Abstract

There are situations in which overdetermining the inverse problem of radiation imaging, to attain a desired spatial resolution, is not practical or possible. One then has to deal with an incomplete inverse problem, which has potentially many possible solutions. This chapter presents methods for solving incomplete problems. When the number of measurements is less than the number of unknown image parameters, one has an inverse problem with multiple solutions. The inverse problem is then said to be incomplete or underdetermined. Even if the problem is well-determined, that is the number of measurements is exactly equal to the number of unknown parameters, the inverse problem can still in effect be incomplete, since statistical fluctuations, systematic measurement errors, modeling errors, and the discretization process itself inevitably produce inconsistencies. The inverse problem becomes an estimation problem, in which one seeks the most appropriate solution amongst all possible solutions. One can also compensate for the missing measurements with some a priori information or some physical constraints on the solution. To overcome these inconsistencies, one should always seek to overdetermine the problem, either by acquiring more measurements or by decreasing the number of problem unknowns. There is no rigid rule to determine the number of measurements required to arrive at an acceptable level of overdetermination, but it is usually dictated by practical considerations in terms of measurement time, available source strength, number and efficiency of detectors, scanning mechanisms, etc.