In Chapter 6, the stabilizing-functional approach to regularizing the solution to the inverse problem was described. In the following we lay down the framework for presenting the generalized mapping function (GMF). The signal space is symbolized by S. In the absence of any constraint on the solution, the estimated signal is the vector x* ∈ S that minimizes the stabilizing functional. The addition of a constraint(s) effectively forces the solution to belong to a subspace S1 ∈ S. The search for the true solution is now restricted to the subspace S1; the more constraints imposed, the smaller the space over which the search for the true solution occurs. Imposing additional constraints will speed up the algorithm, but unless the constraints are well grounded in physics, their use could lead to spurious solutions. We will use the noise norm constraint, which uses the noise variance. That is, SNR information is the only information about the noise assumed to be available.
Since we are concerned with finding solutions to problems involving measurements, it is reasonable to assume that the signals are time limited and that the spectra are bandlimited. By choice we will restrict our discussion to the space of nonnegative functions.