Ill-posedness and precision in object-field reconstruction problems

The inherent stability or instability in reconstructing an object field, in the presence of observation noise, for a class of ill-posed problems is investigated for situations in which constraints are imposed on the object fields. The class of ill-posed problems includes inversion of truncated Fourier transforms. Two kinds of constraint are considered. It is shown that if the object field is restricted to a subset of L2 space over Rn that is bounded, closed, convex, and has nonempty interior, then a (nonlinear) least-squares estimate always exists but is unstable. It is also shown that if one is primarily concerned with the situation in which the object field belongs to a compact parallelepiped in L2, aligned in a natural way, there is a satisfactory, stable linear estimate that is optimal according to a min–max criterion. This also leads to a nonlinear modification for the case in which the object field is actually restricted to the parallelepiped. A summary of some relevant mathematical background is included.