Incoherent single-lens, single-image, passive range estimation systems code object range information into unambiguous spatial information at the image. A practical system should require no a priori information about the spatial intensity characteristics of the particular object. A noiseless sampled image model of this system is
where
the system point spread function (PSF) or impulse response is given by
.
This PSF is characterized by the misfocus or normalized range
parameter 
. The symbol 
denotes convolution.
The matrix 
is a convolution matrix containing as
elements.
The unknown object is given by 
.
The unknown parameters of the noiseless sampled data are the
normalized range and the object . The system, described by
and , is
assumed to be known. The unknown parameters
can be grouped as
where the desired parameter is object range , while the nuisance
parameter for this example is the unknown object .
With this partitioning of the parameters, the sensitivity matrices from (5) are found to be
From (6) the Cramer-Rao bound on estimating the normalized
range with unknown is given by
where 
is a projection matrix projecting onto the
subspace orthogonal to 
. From (9) one can conclude that
for single-image, single-lens, passive range estimation to be possible must not
be a rank-zero projection, or equivalently must not be full rank. The Cramer-Rao
bound of (9) also shows that the variation of with
, in the intersection of the rank-one subspace 
and the
subspace orthogonal to , must be large for accurate
range estimation.
Since the matrix can be approximated by a circulant matrix,
the eigenvalues of are approximately the
DFT values of PSF [5], here given by . The denominator of
(9) can then be approximated as
where
and where 
and 
are the DFTs of the
sampled vectors and respectively. Notice that
is the optical transfer function (OTF) of the sampled system with misfocus ,
while 
is the spatial frequency spectrum of the sampled
object.
From (9) and (10), the Cramer-Rao bound on object range in terms of spatial frequencies can be approximated as
From (12), a necessary condition for a passive range estimation
systems is that the OTF must contain
zeros that are a function of misfocus or normalized range .
It is impossible to build a single-lens, single-image, system that passively ranges
to unknown objects whose OTF does
not contain zeros as a function of misfocus.
These range-dependent
zeros of the OTF also imply that
it is impossible for an incoherent single-lens, single-image, passive system to
reliably measure simultaneous range and spatial intensity of an unknown object.
From (12), it is clear that the estimation of object range is dependent on the unknown object. If there is a priori object information, such as Bayesian information, then a desired condition on the system is that the magnitude squared variation of the OTF should be a matched filter for the expected object spatial frequency power spectrum.
With spatially incoherent optical systems the sampled OTF is indirectly controlled through modification of the pupil function. For quasi-monochromatic systems, the OTF is equivalent to the autocorrelation of the pupil function, including any associated aberrations such as misfocus or normalized range [8][7][6]. Optical mask functions that modify spatially incoherent systems to form these passive range estimation OTFs can be either magnitude apodizers, non-absorbing phase masks, or a combination of both. Examples of passive range estimation OTFs are shown in [10][9].
