The Cramér-Rao (CR) bound is an analytical expression that can be used to assign a lower limit to the minimum obtainable mean square error (MSE) associated with an estimate, â, of an unknown object, a. In our formulation a is a vector of parameters and usually represents the unknown intensities of a 2-D object. The bound can be used for any restoration technique such as a Wiener filter estimate or an iterative algorithm like the conjugate gradient or Richardson-Lucy, provided certain statistics are known.
To calculate the bound one must know the probability density function (PDF) of
the observed data, d, given a, namely, , and
the bias of the estimate. Typically,
is Gaussian,
Poisson, or some combination of both. In this paper we present results only
for the Poisson case. The ``given a'' assumption and the calculation of
the bias are the source of two problems which we address in this paper.
Regarding the bias, we can usually calculate it for an estimate which is linear in the data, i.e., for estimates such as the inverse or Wiener filter. If the estimate is nonlinear in the data, the bias may be difficult to calculate.
Regarding the assumption that a is known, we assert that, of course, a is not known. Otherwise we would not want to estimate it. We provide calculations for the two obvious extremes: we calculate the bound as though a were indeed known, just to check the analyses; then we calculate the bound using the estimate â, as though it were the true value of a. The latter approach provides a recipe for calculation of the bound in real estimation problems: one makes the estimate, then puts that estimate into an algorithm which calculates the bound.