Transcript Filtering in Freq. Domain

Image Processing
Image Filtering in the
Frequency Domain
•
Low Pass Filter
•
High Pass Filter
•
Band pass Filter
•
Blurring
•
Sharpening
Frequency Bands
Image
Fourier Spectrum
Percentage of image power enclosed in circles
(small to large) :
90, 95, 98, 99, 99.5, 99.9
Blurring - Ideal Low pass Filter
90%
95%
98%
99%
99.5%
99.9%
The Power Law of Natural Images
• The power in a disk of radii r=sqrt(u2+v2 )
follows:
P(r)=Ar- where 2
Images from: Millane, Alzaidi & Hsiao - 2003
Image Filtering
•
•
•
•
•
Low pass
High pass
Band pass
Local pass
Usages
Recall:
The Convolution Theorem
g=f*h
g = fh
implies
implies
G = FH
G=F*H
Low pass Filter
spatial domain
f(x,y)
frequency domain
F(u,v)
filter
G(u,v) = F(u,v) • H(u,v)
g(x,y)
f(x,y)
G(u,v)
F(u,v)
•
g(x,y)
H(u,v)
H(u,v) - Ideal Low Pass Filter
H(u,v) =
1
D(u,v)  D0
0
D(u,v) > D0
D(u,v) =  u2 + v2
D0 = cut off frequency
H(u,v)
H(u,v)
v
u
1
0
D0
D(u,v)
Blurring - Ideal Low pass Filter
99.7%
99.37%
98.65%
Blurring - Ideal Low pass Filter
96.6%
98.0%
99.0%
99.4%
99.6%
99.7%
The Ringing Problem
G(u,v) = F(u,v) • H(u,v)
Convolution Theorem
g(x,y) = f(x,y) * h(x,y)
IFFT
H(u,v)
 D0
sinc(x)
h(x,y)
 Ringing radius +  blur
The Ringing Problem
Freq. domain
Spatial domain
250
200
150
100
50
0
0
50
100
150
200
250
H(u,v) - Gaussian Filter
H(u,v)
H(u,v)
1
1/ e
v
u
0
D0
-D2(u,v)/(2D20)
H(u,v) = e
D(u,v) =  u2 + v2
Softer Blurring + no Ringing
D(u,v)
Blurring - Gaussain Lowpass Filter
99.11%
98.74%
96.44%
The Gaussian Lowpass Filter
Freq. domain
Spatial domain
300
250
200
150
100
50
0
0
50
100
150
200
250
300
Blurring in the Spatial Domain:
1 1
1 1
Averaging = convolution with
= point multiplication of the transform with sinc:
Gaussian Averaging = convolution with
1 2 1
2 4 2
1 2 1
= point multiplication of the transform with a gaussian.
0.15
1
0.1
0.8
0.6
0.05
0.4
0.2
0
0
50
Image Domain
100
0
-50
0
Frequency Domain
50
Image Sharpening - High Pass Filter
H(u,v) - Ideal Filter
H(u,v) =
0
D(u,v)  D0
1
D(u,v) > D0
D(u,v) =  u2 + v2
D0 = cut off frequency
H(u,v)
H(u,v)
v
u
1
0
D0
D(u,v)
High Pass Gaussian Filter
H(u,v)
H(u,v)
1
v
1 1/ e
u
0
H(u,v) = 1 - e
D0
-D2(u,v)/(2D20)
D(u,v) =  u2 + v2
D(u,v)
High Pass Filtering
Original
High Pass Filtered
High Frequency Emphasis
Original
High Pass Filtered
+
High Frequency Emphasis
Emphasize High Frequency.
Maintain Low frequencies and Mean.
H'(u,v) = K0 + H(u,v)
(Typically K0 =1)
H'(u,v)
1
0
D0
D(u,v)
High Frequency Emphasis - Example
Original
High Frequency Emphasis
Original
High Frequency Emphasis
High Pass Filtering - Examples
Original
High pass Emphasis
High Frequency Emphasis
+
Histogram Equalization
Band Pass Filtering
H(u,v) =
0
D(u,v)  D0 -
1
D 0-
0
D(u,v) > D0 +
w
2
w
2
 D(u,v)  D0 +
w
2
w
2
D(u,v) =  u2 + v2
D0 = cut off frequency
w = band width
H(u,v)
H(u,v)
v
u
1
D(u,v)
0
D0- w
2
D0 D0+w
2
Local Frequency Filtering
H(u,v)
v
u
H(u,v)
1
0
-u0,-v0
H(u,v) =
D0
u0,v0
D(u,v)
1
D1(u,v)  D0 or D2(u,v)  D0
0
otherwise
D1(u,v) =  (u-u0)2 + (v-v0)2
D2(u,v) =  (u+u0)2 + (v+v0)2
D0 = local frequency radius
u0,v0 = local frequency coordinates
Band Rejection Filtering
H(u,v)
v
u
H(u,v)
1
0
-u0,-v0
H(u,v) =
D0
u0,v0
D(u,v)
0
D1(u,v)  D0 or D2(u,v)  D0
1
otherwise
D1(u,v) =  (u-u0)2 + (v-v0)2
D2(u,v) =  (u+u0)2 + (v+v0)2
D0 = local frequency radius
u0,v0 = local frequency coordinates
Demo