In a spatially incoherent range-invariant imaging system, or an extended depth of field system, the spatial intensity of unknown objects is estimated without knowledge of object range or misfocus. One model for the noiseless sampled image is identical to the model of the passive range estimation system
The difference between the passive range estimation problem and the range-invariant imaging problem is that instead of estimating object range without knowledge of the object , the unknown object is estimated without knowledge of the range . For this problem the nuisance parameter becomes object range.
Partition the parameter vector as in the passive ranging problem. The sensitivity matrices and are then the same as in (8). The Cramer-Rao bound on the estimate of the unknown object from (6) is then
where is the projection matrix projecting onto the subspace orthogonal to . The relationship of (14) then implies that the variation of with , in the rank-one subspace , should be zero. Additionally, the eigenvalues of should be large.
The denominator of (14) can be placed in terms of the OTF and spatial frequency spectrum of , as in (10), by
From (14) and (15), the Cramer-Rao bound on estimation of the unknown object, in terms of spatial frequencies, can be approximated as
From (17), the necessary condition on a reliable extended depth of field system is that the variation of the OTF with misfocus should be zero over the spatial frequencies where the expected object has non-zero power. A desired condition is that the power of the OTF should be maximized where its variation with misfocus, or the object spatial frequency spectrum, is zero. From (16) and (17), the performance of this type of system is independent of the unknown object if the object spatial frequency spectrum contains no zero values in the passband of the system.
Woodward's ambiguity function is a practical representation of spatially incoherent range-invariant systems . In effect, the ambiguity function related to a specific pupil function is a polar display of the OTF for all values of misfocus or normalized range. Range-invariant systems can almost be seen by inspection of their respective ambiguity functions. An example of ambiguity function based design of a spatially incoherent range-invariant optical system is given in .