The goal of an orthogonal passive range detection system is the partitioning of the object volume into statistically independent range and azimuth ``bins''. This partitioning leads to very simple and fast processing to detect the presence of an object in these bins, or the determination of object range. This partitioning is reminiscent of the partitioning of object volume that occurs with coherent radar systems.
A single-image, single-lens, passive range detection system can be modelled as
The normalized range values 
are assumed known while the objects
at the different ranges, 
, are assumed unknown. The system,
given by 
and 
, is again assumed known.
For simplicity consider the imaging of objects at two ranges, 
. The
results can be generalized to any number of ranges.
Partition the parameter vector as
where the object at range 1 is the nuisance parameter when estimating object 2, and vis versa. The sensitivity matrices of (5) then become
The Cramer-Rao bound on the estimation of 
with unknown
is then
The projection matrix of (21) projects onto
the subspaces orthogonal to 
.
For reliable operation at distinct ranges 
and 
,
the system should possess orthogonal subspaces 
and . And,
the eigenvalues of 
, within
the subspace 
, should be maximized. Parallel statements to those above can be made for the estimation of object
.
Again, the denominator of this bound can be placed in terms of the OTF and spatial frequency spectrum of the object as
where
From (21) and (22), the Cramer-Rao bound on
estimation of object ,
in terms of spatial frequencies, can be approximated as
From (24), the necessary condition for orthogonal passive range detection systems is that the expected OTFs, as a function of misfocus or object range, should form an orthogonal set. A desired condition is that the total power the OTF related to each expected object range should be maximized and equalized among all expected ranges. Notice that the performance of this type of system is independent of the unknown objects.
The simplest orthogonal set is that of the sinusoids. With ambiguity function based design techniques [8], it can be shown that a spatially incoherent quasi-monochromatic optical system with a sinusoidal mask function approximates an ideal orthogonal passive range detection system. Such a system will yield a PSF that can be approximated by a biased sinusoid with a range-dependent period. Computationally efficient FFT processing of the resulting imagery can then separate object power into statistically independent range and azimuth bins. Constant false alarm rate (CFAR) detectors [13] can be used to statistically detect the presence or absence of an object in each bin.
