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 , here given by . The denominator of (9) can then be approximated as
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 . 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 .