Nonlinear Combined
SpatialFrequency Domain
TwoStage Image Filtering (nonlinear image processing)
The purpose of the filtering operation is
assumed to be an effective elimination or attenuation of
the noise that is corrupting the signal.
A fundamental difference between the behavior
of linear and nonlinear image filters is the following. The impulse
response (that is, the output signal when the input signal
is equal to 1 at time instant 0 and zero otherwise) of a
nontrivial timeinvariant linear filter cannot be a zerovalued
sequence. This means that a linear filter always smoothes
the signal. A nonlinear image processing filter preserves the signal carefully,
and usually it is more effective for noise removal despite
the fact that it can be idle only for the partial signals.
At the same time, despite the fact that nonlinear image processing
spatial domain filters remove noise effectively and preserve
image boundaries with higher accuracy, the resulting image
always becomes smoothed after filtering. Of course, it may
not be so smoothed as after linear filtering, but the latter
case some important details (especially the smallest ones)
may be lost.
From the first point of view it is possible
to solve the problem by moving to frequency domain filtering.
But on the other hand, a classical frequency domain filtering
(e.g. Wiener filtering) is not a simple problem from the
computational point of view.
So an appropriate solution may be found using
spatial and frequency domain filtering in combination. This
means that a noisy image may be processed by an appropriate
spatial domain filter in order to reduce noise as strongly
as possible. Then a spectrum of the resulting image is corrected
in order to eliminate smoothing of image boundaries. This
procedure should be repeated several times, if necessary.
In this way filtering is organized as iterative process of
spatial domain nonlinear filtering and further correction
of the resulting image spectrum.
The simplest means of spectrum correction
from our point of view is correction of the spectrum amplitude
and preservation of the spectrum phase. We propose such a
solution precisely because distortion of the amplitude involves
image smoothing, and also because a major part of noise is
concentrated in the amplitude.
It is also necessary to note that a similar
problem exists in image sharpening. Actually, sharpening
is a kind of frequency correction or socalled antifiltering.
Sharpening effect is achieved by amplification of high frequencies.
This operation may be called high frequency correction. Nonlinear image processing
spatial domain filters (multivalued frequency correction
and median frequency correction) for solving the frequency
correction problem are effective for solving this problem,
but at the same time they have a common disadvantage: these
spatial domain filters shift image boundaries. A natural
way to overcome this disadvantage is to work in the frequency
domain. But as for noise reduction, solving the problem in
the frequency domain is complicated. Thus we combine spatial
and frequency domain solutions.
Examples
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Speckle Noise Filtering



Original radar satellite noisy image 
Combined Filter With RankOrder KNV, image
boundaries are preserved with very high accuracy 
Enhanced difference of the filtered image from the
original image 



Original radar satellite noisy image 
Combined Filter With RankOrder EV 
Enhanced difference of the filtered image from the
original image 
