Adaptive Optics for Vision Science

Chapter 10.6.3 - Linear Deconvolution

10.6.3   Linear Deconvolution

The simplest form of deconvolution makes use of the Fourier properties of
convolution. The Fourier transform of the convolution operation is simply the
product of Fourier transforms of the object irradiance and PSF. If Fourier
transforms are denoted in upper case and (u, v) represents the spatial-frequency
domain, then

 

where O(u, v) and P(u, v) are the Fourier transforms of the object irradiance
and the PSF, respectively. The Fourier transform of the object irradiance can
be obtained by simply taking the quotient of the measurement Fourier transform
with the PSF Fourier transform,

 

and the object irradiance, o(x, y), is obtained by inverse Fourier transforming
the quotient. However, there is a problem related to the bandpass sampling
discussed above. This quotient is only defined for spatial frequencies less than
the cutoff frequency and inversion of the quotient will produce nonphysical
ringing artifacts because of the hard edge, similar to the structure in the PSF
as illustrated in Figure 10.14. One way to get around this is to restore the
object irradiance as viewed through a perfect aperture, that is,

 

where H(u, v) is the transfer function of the perfect optical system, that is,
the Fourier transform of the perfect PSF. However, the presence of noise in
the measured data [i.e., i(x, y) = i′(x, y) + n(x, y), where i′(x, y) is the noiseless
image] can still introduce problems in the Fourier inversion. This generally
occurs at high spatial frequencies where there could well be small number
divisions in the above quotient. This effect is usually minimized by using a
Wiener filter, Θ(u, v),

 

where

 

where N(u, v) is the Fourier transform of the noise. A Wiener filter is a signal-to-
noise filter. When the signal power spectrum, ½I′(u, v)½2, is much greater
than the noise, Θ(u, v) @ 1, and when the signal is much less than the noise,
Θ(u, v) @ 0. This can be a very powerful filtering technique and reduces (but
does not necessarily remove) the unphysical results in the form of negativity.
There is no guarantee that the transform of the Weiner filter is positive. The
presence of negativity is illustrated in the inverse filter reconstruction (Fig.
10.15) and is shown here to illustrate that linear inversion filters should be
used with caution. However, the benefit of this approach is that it is simple
and straightforward.

In practice, the Wiener filter is built from measurements/estimates of the
measurement noise in the image. In many cases, this takes the form of white
noise (i.e., has a uniform spatial-frequency distribution), and measurements
of the noise obtained at spatial frequencies greater than the cutoff frequency
(fc) can be interpolated to the lower spatial frequencies, permitting an estimate
of the noise term for the Weiner filter. Algorithmically, this is a very
straightforward setup and any mathematical-based package can easily be
used.

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