Theory
The inverse filtering is a restoration technique
for deconvolution, i.e.,
when the image is blurred by a known lowpass filter,
it is possible to recover the image by inverse filtering
or generalized inverse filtering. However, inverse
filtering is very sensitive to additive noise.
The approach of reducing one degradation at
a time allows us to develop a restoration algorithm for each type
of degradation and simply combine them.
The Wiener filtering executes an optimal tradeoff between inverse
filtering and noise smoothing. It removes the additive noise and
inverts the blurring simultaneously.
The Wiener filtering is optimal in terms of the mean square error.
In other words, it minimizes the overall mean square error
in the process of inverse filtering and noise smoothing.
The Wiener filtering is a linear estimation of the original
image.
The approach is based on a stochastic framework.
The orthogonality principle implies that the Wiener filter in
Fourier domain can be expressed as follows:
where
are respectively
power spectra of the original image and the
additive noise, and
is the blurring filter.
It is easy to see that the Wiener filter has two separate
part, an inverse filtering part and a noise smoothing part.
It not only performs the deconvolution by inverse filtering
(highpass filtering)
but also removes the noise with a compression operation (lowpass
filtering).
Implementation
To implement the Wiener filter in practice we have to estimate
the power spectra of the original image and the additive noise.
For white additive noise the power spectrum is equal to the
variance of the noise. To estimate the power spectrum of the
original image many methods can be used. A direct estimate is
the periodogram estimate of the power spectrum computed from
the observation:
where Y(k,l) is the DFT of the observation. The advantage
of the estimate is that it can be implemented very easily
without worrying about the singularity of the inverse filtering.
Another estimate which leads to a cascade implementation
of the inverse filtering and the noise smoothing is
which is a straightforward result of the fact:
The
power spectrum
can be estimated directly
from the observation using the periodogram estimate.
This estimate results in a cascade implementation
of inverse filtering and noise smoothing:
The disadvantage of this implementation is that when
the inverse filter is singular, we have to use the generalized
inverse filtering.
People also suggest the power spectrum of the original image
can be estimated based on a model such as the
model.
Experimental Result
To illustrate the Wiener filtering in image restoration
we use the standard 256x256 Lena test image.
We blur the image with the lowpass filter
then put into the blurred image the
additive white Gaussian noise of variance 100.
The Wiener filtering is applied to the image with a cascade
implementation of the noise smoothing and inverse filtering.
The images are listed as follows together with the
PSNRs and MSEs. Notice that the restored image
is improved in terms of the visual performance, but the
MSEs don't indicate this, the reason of which is that
MSE is not a good metric for deconvolution.
Test Image Lena and Blurred Image Lena



Standard Lena Image PSNR = Infinity, MSE = Zero 
Blurred Lena Image PSNR = 23.2993, MSE = 304.1938 
Restored Lena Image PSNR = 19.1447, MSE = 791.7906 
Matlab Programs