10.6.4   Nonlinear Deconvolution

Problems associated with straight inversion as described above can be overcome
by more sophisticated deconvolution approaches. These are nonlinear
approaches that solve for the object irradiance directly and not from its

Fourier transform. These algorithms are based upon statistical analyses, especially
Bayesian statistics, and use approaches known as maximum likelihood
(ML) and maximum a posteriori (MAP). It is beyond the scope of this chapter
to delve into the depths of statistical analysis and the reader is directed to the
reference [55].

A more descriptive outline of a typical algorithm is given below. In a least-squares
setting, the following metric, E, is used to solve for the object
irradiance:

This metric calculates the difference between the measurement i(x, y) and an
estimate of the measurement obtained by convolving the PSF with an estimate
(indicated by the ^Ù) of the object irradiance. In this analysis, the estimate
is updated so that convergence occurs, which minimizes the metric shown
above. How the estimate is updated depends strongly upon the algorithm/
statistical approach being used. This least-squares problem can be solved
using a conjugate-gradient technique that essentially varies all of the elements
of the object-irradiance map in order to minimize the expression. Thus for a
256 × 256 pixel image, there are approximately 65,000 variables being
simultaneously minimized. Therefore, these nonlinear approaches take signifi-
cantly longer to solve than the simple Fourier inversion discussed above.
Various algorithms are available, one of the most common being the Lucy–
Richardson maximum-likelihood approach [56, 57]. If the variables obey
Gaussian statistics, then the ML approach is equivalent to an error metric
minimization of Eq. (10.20). The nonnegativity constraint can be simply
applied by reparameterizing the object irradiance as the square of the variable
being solved for, that is, o(x, y) = α(x, y)².

In all of the above analysis it has been assumed that the PSF is well known.
In many cases this is not so, and a technique known as multiframe blind
deconvolution (MFBD) can be used to estimate both the object irradiance
and the PSF simultaneously [58]. MFBD assumes that the object irradiance
is stationary and that only the PSF changes from one exposure to the next,
that is,

For a single observation, this is a highly underdetermined problem in that
there is one observation and two unknowns. For multiple observations, the
degree of underdeterminism is decreased and prior knowledge of the object
irradiance and the PSFs make the solution tractable. The error metric can
now be written as:

where the k subscript refers to the number of frames and both o(x, y) and
psf_k(x, y) are now estimates. The prior information permits the solution space
to be reduced. We have already discussed prior information concerning the
object irradiance (i.e., the nonnegativity). An additional constraint is that the
object irradiance does not change so the set of equations can be solved for
simultaneously (i.e., one object irradiance and n PSFs). Similar constraints
can be applied to the PSFs. These are (1) nonnegativity because the PSF of
an incoherent optical system is the power spectrum of the complex wavefront
at the pupil; (2) the PSF cannot be better than that produced by the perfect
aperture (i.e., it is band limited) and the MTF cannot exceed the shape of the
perfect MTF; and (3) the PSF cannot be significantly worse than that produced
by the AO system being used. These physical constraints mean that we
are not working blindly. This approach is commonly known as myopic deconvolution.
A known PSF for each observation is used but is allowed to vary
within these constraints. It is important to note that the variability of the
PSF from frame to frame aids the solution. The commonality in the deconvolution
is assumed to be due to the object irradiance. In practice, however,
the PSF has common structure in all frames and this can affect the
solution.

An example of MFBD is shown in Figure 10.16. A synthetic, fully bleached
AO retinal image of a living retina is shown before and after deconvolution.
The before image is the average of 10 independent observations and the after
image is the common solution to the set of observations. The reconstructed
PSFs for the individual frames are shown in Figure 10.17, compared with the

true PSFs used to generate the images. The difference in AO compensation
between frames is clearly seen.

Figure 10.18 shows the application of the MFBD to actual retinal images
for three different bleach cases. Each bleach case comprised 36 individual
observations, and these were averaged to obtain the AO image shown in the
figure. All 36 frames were also reduced simultaneously; 36 PSFs and one
common object yielded 606,208 variables to be minimized. The contrast
enhancement is clearly seen.