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
where
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 [8][11]. 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 [12][8].
