Frequency Domain
Nonlinear MedianLike Filter
for QuasiPeriodic and
Periodic Noise Image Filtering
The theory of linear filtering gives optimal
noise reduction methods, when the corruption can be modeled
as a Gaussian process and the mean square error is the criterion
for accuracy. These assumptions are reasonable in most applications.
But in digital systems, errors are often caused by bit changes,
and the resulting distribution is far from Gaussian. Also
in many cases, where visual quality is the ultimate criterion,
the mean square error is not a realistic criterion, and images
with large mean square error can be visually much better
than ones with a small mean square error. This is where nonlinear
methods prove most useful.
Removal of halftone patterns from photographs
is often an important problem to address, which occupies
many news group discussions (usually with misinformation).
Processing in the frequency domain is a much better solution
than operations such as blurring that can hide the dots but
also reduce edge sharpness. The resolution of the scanner
used to input the image also affects the high frequency noise
pattern in the acquired image and can produce additional
moire patterns. This artifact is also characteristic of gray
scale images obtained from singlechip video cameras.
Usually such quasiperiodic and periodic
noise in image results peaks in its spectrum. In case of
periodic nose, these peaks are even visible to the eye, while
peaks corresponding to quasiperiodic noise are hidden and
it’s quite hard to detect them. A good idea is to find some
mean to detect such peaks. To filter images corrupted by
the quasiperiodic noise, for example, Wiener filter can
be used, but it’s slow from the computational point of view.
It works with a whole spectrum at once and it needs a mask,
which we cannot supply without detecting the needed coefficients.
Thus we again come to the problem of peak detection.
We utilize a new filter for the detection,
reduction and even removal (filter) of the quasiperiodic and periodic
noise. It is quite fast and it allows achieving a very good
result.
The main idea of the filter is to remove
peaks from the image spectrum, which are replaced by the
median calculated in the local window. Fourier, Cosine and
Walsh (with Walsh ordering) spectra can be used to achieve
the result. The Fourier basis gives the best results, while
correction of the Cosine and Walsh spectra for removing quasiperiodic
noise adds the white one (Cosine and Walsh spectra can be
used for truly periodic noise filtering). To be more specific,
the correction of the spectrum using the proposed filter
is concentrated in the correction of the spectrum amplitude.
It is analyzed using a local window of NxN size. To decide
whether the current spectral coefficient needs to be modified,
the coefficients that belong to the local window around it
are taken. A median value is calculated within this window
and then the central coefficient is compared to the median
value. Depending on the comparison result, the central value
is replaced by the median one. Mutual dependence of the central
and median values is stable for the noncorrupted spectral
coefficients independently of the frequencies to which they
correspond. So it is independent of the position of the peaks
in the spectrum amplitude.
Using this filter, it is possible to filter
periodic noise, and to reach a significant reduction of quasiperiodic
noise, completely preserving the image boundaries.
Examples
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Periodic Noise Filtering
Quasiperiodic and Impulsive noise filtering

Input infrared image 


Quasiperiodic noise filtering and impulsive
noise filtering after the noise detection 
Difference between the input and the resulting
image: image boundaries are preserved completely 
