Adaptive Optics for Vision Science

Chapter 10.6 - Image Postprocessing

10.6.1   Introduction

Adaptive optics can make significant improvements in image quality, producing
images with much finer detail than otherwise possible. However, AO
systems do not produce perfect images in that there are residual aberrations
that remain uncorrected. Although these cannot be corrected in real time,
they can be corrected for after the fact, that is, with postprocessing. In order
to do this, it is important to understand the physics of an imaging system.


10.6.2   Convolution

The principles of Fourier optics say that the image formed through an optical
system is the convolution of the object irradiance with the point spread function
(PSF) of the optical system [54]. If the measured image distribution is
denoted as i(x, y), the object irradiance as o(x, y), and the PSF as psf(x, y),
then the image distribution is expressed as:

 

where denotes the convolution operator given by the superposition integral.
This can be visualized as taking the PSF distribution shifted by a vector (x,
y) relative to the object distribution and then taking the volume of the product
of the two. Essentially this is a smoothing operation, as illustrated in Figure
10.13. This figure shows the resulting image (left) obtained by the convolution
of the true object irradiance (center) with a PSF (right).

The PSF is the response of the optical system to a point source, similar
to an impulse response function in signal processing. Thus, every point in
the object is blurred by the PSF. This equation makes the assumption that
the optical system is isoplanatic, that is, that the PSF is the same for all
points in the image. From Fourier optics, it is important to note that the
PSF is the modulus squared of the Fourier transform of the complex wave-


FIGURE 10.13 Convolution illustrated. The observed image (left) is the convolution of the true object (center) with the PSF (right).


front at the exit pupil of the imaging system, or the power spectrum. This
is illustrated in Figure 10.14 and the ringing of the PSF is caused by the hard
edge of the pupil. Because of the finite size of the aperture, the PSF is band
limited. This means that only specific spatial frequencies are passed by the
aperture. [This is also illustrated in Fig. 10.14 that shows the modulation
transfer function
(MTF) corresponding to this pupil.] The MTF is the
autocorrelation of the aperture or the modulus of the Fourier transform of
the PSF. The hard edge of the aperture means that the autocorrelation also
goes to zero at the band-limited spatial frequency of the optical system. This
band limit is analogous to a band-limited electronic signal that only passes
specific temporal frequencies. The highest spatial frequency sampled, fc, is
determined from the ratio of the aperture size to the wavelength of light
(i.e., d/λ). For a 6-mm aperture at visible wavelengths (e.g., 550 nm), this
critical spatial frequency is 0.01 cycles/μrad corresponding to a spatial scale
of 92 μrad.

As shown in Eq. (10.14), i(x, y) is the measured image irradiance, o(x, y)
is the true object irradiance, and psf(x, y) is the PSF of the AO system, where
the blurring is caused by residual aberrations after compensation. In order to
remove the effect of this residual blurring, the image needs to be deconvolved,

FIGURE 10.14 Examples of Fourier optics. The power spectrum of the aperture (left) produces the PSF (center). The modulation transfer function (MTF) (right) is the modulus of the Fourier transform of the PSF and represents the transmission of spatial frequencies by the aperture.

that is, we need to invert the convolution above. Assuming that an estimate
of the PSF is known, then the inversion is possible but not trivial.

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