Department of Physics and Astronomy
DIGITAL IMAGE PROCESSING
Course 3624
Topic 8 - Image Filtering - II
Professor Bob Warwick
8. Image Filtering II - Fourier Domain Filtering
Filtering is the suppression or enhancement of particular spatial
frequency components in a dataset.
The spatial domain process is a convolution:
Filtered image
Original Image
Mask Function
g(x, y) = f (x, y)*h(x, y)
From the Convolution Theorem we have the equivalent Fourier
Domain Process:
FT of Filtered image
FT of Original Image
Filter Function
G(u, v) = F(u, v)´ H(u, v)
Thus the Fourier Domain process is simply one of
MULTIPLICATION by the filter function, where
Filter Function
FT
Mask Function
H(u, v) Û h(x, y)
Types Of Fourier Domain Filter
(i) LOW-PASS FILTERS -- Noise Suppression
(high freq. suppression, low freq. enhancement)
(ii) HIGH-PASS FILTERS - Edge enhancement/detection
(low freq. suppression, high freq. enhancement)
(iii) BAND-STOP FILTERS - Interference Suppression
(mid-frequency suppression)
8.1 Low-Pass Filters
In many applications the noise has a much wider frequency spread than
the signal. For example in 1-d:
F(u)
Signal
White noise
u
In these circumstances an effective method of noise suppression is to
apply a low-pass Fourier domain filter:
F(u)
´
u
H(u)
=
u
G(u)
u
Types of Low-Pass Filters
(a) Rectangular or "Top-Hat" Filter
H(u)
|h(x)|
H(u) Û h(x)
umax
x
u
But this type of filter introduces both: BLURRING and RINGING!
(b) Trapezoidal Low-Pass Filter
H(u)
umax
2-d forms
u
(c) Gaussian Low-Pass Filter
In (b) & (c)
BLURRING remains
but RINGING is
reduced because
the filter edges are
less sharp
H(u)
u
The Top-Hat Filter
The Gaussian Filter
8.2 High-Pass Filters
Abrupt edges and/or sharp boundaries of objects generally produce most of the
high frequency signal in the image. It follows that to sharpen an image it is
necessary to enhance the high frequency components.
Types of HIGH-PASS FILTER
H(u)
(a) Rectangular
H(u)
umin
u
(b) Trapezoidal
u
(c) Gaussian
H(u)
u
The High-Pass Filter
High-Pass Filtering Example
8.3 Band-Stop Filtering
Band-stop filters are used to suppress interference signals in a specific
frequency range:
H(u)
u
Band-Stop Filtering Example I
Band-Stop Filtering Example II
8.4 Equivalent Filters
The effect of a spatial domain filter can be investigated by considering the
equivalent Fourier domain filter (and vica-versa).
hxy Û Huv
Consider the 1-d case of an N element image spatially filtered with 1 x 3 mask.
The first step is to “pad out” the mask with N-3 zeros.
Example: 1-d Smoothing Filter
1 1 1  1 1 1 0 0 0 0 0 …..
Then:
In the Fourier domain
j 2 p ux
N-1
1
the smoothing filter has
Hu = åhx e N
u = 0 ® N -1
N x=0
a low-pass response
j 2 p ux
2
(rather as expected):
1
N
Hu =
Hu =
N
1
N
åh
e
x
x=0
æ
ç1+ e
è
1 Hu = e
N
1 Hu = e
N
j 2pu
N
+e
j 2 p u2
N
ö
÷
ø
j 2pu
N
j 2pu ö
æ j 2p u
N
+1+ e N ÷
çe
è
ø
j 2pu
N
æ
2p u ö
ç1+ 2 cos
÷
è
N ø
Scale Phase
Factor Term
Amplitude
Response
Hu
0
N
2
u
Equivalent Filters cont.
Example 2: 1-d gradient operator:
1 0 -1  1 0 -1 0 0 0 0 0 …..
Then:
1 N-1
Hu = åhx e
N x=0
j 2 p ux
N
1 2
Hu = åhx e
N x=0
j 2 p ux
N
u = 0 ® N -1
j 2p u
N
Scale Phase
Factor Terms
1 N-1
Hu = åhx e
N x=0
j 2 p ux
N
1 2
Hu = åhx e
N x=0
j 2 p ux
N
æ 2p u ö
ç sin
÷
è
N ø
1 Hu = e
N
Amplitude
Response
u = 0 ® N -1
j 2p u
j 2 p u2 ö
1 æ
N
H u = ç -1+ 2e
-e N ÷
Nè
ø
j 2pu
j 2pu
j 2pu ö
1 - N æ
N
Hu = e
+2-e N ÷
ç -e
N
è
ø
j 2 p u2 ö
1 æ
H u = ç1+ 0 - e N ÷
Nè
ø
j 2pu
j 2p u
j 2pu ö
1 - N æ N
Hu = e
-e N ÷
çe
N
è
ø
2 Hu =
je
N
Example 3: 1-d Laplacian operator:
-1 2 -1  -1 2 -1 0 0 0 0 0 …..
Then:
A high-pass
response:
Hu
j 2pu
N
æ
2p u ö
ç 2 - 2 cos
÷
è
N ø
Scale Phase
Factor Terms
Amplitude
Response
A high-pass
response:
Hu
0
N
2
u
0
N
2
u
2-D Equivalent Filters
For a 2-D spatial filter, the equivalent Fourier domain filter can be calculated as:
H uv
1
=
N
N -1 N -1
å åh
x=0
xy e
-
j 2 p (ux+vy)
N
u = 0 ® N -1
v = 0 ® N -1
y=0
If the 2-d filter is “separable”, then the same result can be achieved by the
application of two 1-d filters:
hxy º h x * hy Û Hu ´ Hv º Huv
Example 1:
1 1 1
1
h xy = 1 1 1 º 1 * 1 1 1 Û
1 1 1
1
Hu
Hv
´
0
Example 2:
1 0 -1
1
h xy = 1 0 -1 º 1 * 1 0 -1 Û
1 0 -1
1
N
2
u
Hu
N
2
0
v
Hv
´
0
N
2
u
0
N
2
v