Frequeny Domain Filtering
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Transcript of Frequeny Domain Filtering
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Image Enhancement in
Frequency Domain
ASHISH GHOSH
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Spatial Vs Frequency Domain
Spatial Domain Frequency Domain
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Any function thatperiodically
repeats itself can be expressed
as the sum of sines and/or
cosines of different frequencies,
each multiplied by a different
coefficients.
This sum is called aFourier
series.
Fourier Series
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+
+
+
=
=
=
=
++=
00
0
00
0
00
0
0
0
0
0
0
0
1
000
sin)(2
cos)(2
)(1
]sincos[)(
Tt
t
n
Tt
t
n
Tt
t
n
nn
dttnwtg
T
b
dttnwtg
T
a
dttg
T
a
tnwbtnwaatg
Fourier Series
Where T0 is the period.
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A function that isnot periodicbut the area under its
curve is finite can be expressed as the integral of
sines and/or cosines multiplied by a weighing
function. The formulation in this case is Fourier
transform.
Fourier Transform
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Fourier Transform
Fourier transform
Functions which are not periodic (but whose area under the curve is
finite) can be expressed as the integral of sines and/or cosines
multiplied by a weighting function
Its utility is greater than the Fourier series in most practical problems
A function, expressed in either as a Fourier series or a Fourier transform,
can be reconstructed (recovered) completely via an inverse process, with
no loss of information
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=
=
dueuFxf
dxexfuF
uxj
uxj
2
2
)()(
)()(
Continuous One-Dimensional Fourier
Transform and Its Inverse
Where 1=j
uis the frequency variable.
F (u)is composed of an infinite sum of sine and cosine terms
Each value of udetermines the frequency of its corresponding
sine-cosine pair.
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>
=
Smoothing Frequency Domain, Ideal Low-
pass Filters
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=
=
=
=
u v
c
M
u
N
v
c
fvuF
vuFf
/),(100
),(1
0
1
0
2
Total Power
The remained percentagepower after filtration
Smoothing Frequency Domain, Ideal Low-
pass Filters
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fc =5
= 92%
fc =15
= 94.6%
fc =30= 96.4%
fc =80
= 98%
fc =230
= 99.5%
Smoothing Frequency Domain, Ideal Low-
pass Filters
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Cause of Ringing
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[ ]n
DvuD
vuH2
0/),(1
1),(
+
=
Smoothing Frequency Domain, Butterworth
Low-pass Filters
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Smoothing Frequency Domain, Butterworth
Low-pass Filters
Radii= 5
Radii= 15 Radii= 30
Radii= 80 Radii= 230
Butterworth Low-passFilter: n=2
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Smoothing Frequency Domain, Butterworth
Low-pass Filters
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2
0
2
2/),(),( DvuDevuH =
Smoothing Frequency Domain, Gaussian
Low-pass Filters
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Radii= 5
Radii= 15 Radii= 30
Radii= 80 Radii= 230
Gaussian Low-pass
Smoothing Frequency Domain, Gaussian
Low-pass Filters
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Smoothing Frequency Domain, Gaussian
Low-pass Filters
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Smoothing Frequency Domain, Gaussian
Low-pass Filters
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Smoothing Frequency Domain, Gaussian
Low-pass Filters
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Hhp(u,v) = 1 -Hlp(u,v)
Sharpening Frequency Domain Filters
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Sharpening Frequency Domain Filters
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>
=
),(if1
),(if0),(
0
0
DvuD
DvuDvuH
Sharpening Frequency Domain, Ideal High-
pass Filters
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[ ]n
vuDD
vuH2
0 ),(/1
1),(
+
=
Sharpening Frequency Domain, Butterworth
High-pass Filters
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20
2 2/),(1),(
DvuDevuH
=
Sharpening Frequency Domain, Gaussian
High-pass Filters
