2. Frequency Domain Filtering
• Filtering in the frequency domain consists of
modifying the Fourier transform of an image
and then computing the inverse transform to
obtain the processed result
where H(u,v) is a filter function
2
( )( )
( )
1 1
( , )
( , ) ( , )ˆ( , ) ( , ) ( , )
F f x y
f x y F F f x y F FH u v H u v u v− −
=
3. Periodic Noise
• Periodic Noise is a sinusoidal wave with the frequency
r (period 1/r) added to a signal
• In the spatial domain, this noise corrupts the entire
signal
• In the frequency domain, it corrupts only a few spectral
coefficients corresponding to the frequency of the
noisy wave. This kind of noise leads to unusually high
magnitude of the corresponding spectral coefficients
• Thus, in the frequency domain, this noise can be
reduced or even completely removed if the
corresponding spectral coefficients have been
corrected
3
10. “Ideal” Mask Generation
( )
( )
( )
≥
<
=
Tk,lC
Tk,lC
lkM
if,0
if,1
,
MCC ⊗=
~
To filter, we do element-wise multiplication of C and M:
Moreover if C(k,l) is larger than T for some (k,l), we not just
put one 0 in the mask, we put a circle of zeroes with center
at (k,l) to improve the result.
13. Mean Filter in Frequency Domain
( ) ( ) ( )
( )
( )
≤
=
otherwise,/,
,
,
if,,
,*
dlkC
T
lkS
lkC
lkC
lkC
S(k,l) is the local mean in the moving window around
the coefficient C(k,l).
T is the threshold
C*(k,l) is the filtered value of C(k,l)
d - is the parameter specifying the strength of peak
reduction
20. Spectral Median Filter
( ) ( ){ } ( )
( ){ }
( )
≥
=
otherwise,,
,
,
if,,
,*
lkC
T
lkCMED
lkC
lkCMED
lkC
The peak is substituted by median:
21. Gaussian Median Notch Filter
The m x n vicinity of the peak is multiplied by the
following surface:
( ) ( ) ( )[ ]
1,...,0;1,...,0
1,
2
2
12
2
1
−=−=
−=
−− −+−−
mynx
AeyxG
mn yxB
⊗ =
Peak Gaussian-like Surface Filtered Peak
26. Filtering With Overlapping Windows
• If the image sizes are not the power of 2 (the
limitation of FFT) it is better to filter it by
blocks instead of extending to the closest
power of 2.
• When the noise is non-uniform (quasi-
periodic) there’s a chance that in smaller
blocks it should be uniform.
