Frequency Domain Filtering 1.ppt

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Image Enhancement:
Filtering in the Frequency Domain

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54
=
Any function that periodically repeats itself can
be expressed as a sum of sines and cosines of
different frequencies each multiplied by a
different coefficient – a Fourier series
Images
taken
from
Gonzalez
&
Woods,
Digital
Image
Processing
(2002)

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54
A function that is not periodic but 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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










du
e
u
F
x
f
dx
e
x
f
u
F
ux
j
ux
j


2
2
)
(
)
(
)
(
)
(
Continuous One-Dimensional Fourier Transform
and Its Inverse
Where 1


j
• (u) is the frequency variable.
• F(u) is composed of an infinite sum of sine and
cosine terms and…
• Each value of u determines the frequency of its
corresponding sine-cosine pair.

5
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54 Continuous One-Dimensional Fourier Transform
and Its Inverse

6
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54 Discrete One-Dimensional Fourier Transform
and Its Inverse
• A continuous function f(x) is discretized into a
sequence:
)}
]
1
[
(
),...,
2
(
),
(
),
(
{ 0
0
0
0 x
N
x
f
x
x
f
x
x
f
x
f 






by taking N or M samples x units apart.

7
of
and Its Inverse
• Where x assumes the discrete values (0,1,2,3,…,M-1)
then
)
(
)
( 0 x
x
x
f
x
f 


• The sequence {f(0),f(1),f(2),…f(M-1)} denotes any M
uniformly spaced samples from a corresponding
continuous function.

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















1
0
1
0
2
2
sin
2
cos
)
(
1
)
(
)
(
1
)
(
M
x
M
x
M
x
u
j
M
x
u
j
M
x
u
x
f
M
u
F
e
x
f
M
u
F







1
0
2
)
(
)
(
M
u
x
M
u
j
e
u
F
x
f

u =[0,1,2, …, M-1]
x =[0,1,2, …, M-1]
Discrete One-Dimensional Fourier Transform
and Its Inverse

9
of
and Its Inverse
• The values u = 0, 1, 2, …, M-1 correspond to samples
of the continuous transform at values 0, u, 2u, …,
(M-1)u.
i.e. F(u) represents F(uu), where:

u 
1
Mx
• Each term of the FT (F(u) for every u) is composed of the
sum of all values of f(x)

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of
and Its Inverse
• The Fourier transform of a real function is generally
complex and we use polar coordinates:













)
(
)
(
tan
)
(
)]
(
)
(
[
)
(
)
(
)
(
)
(
)
(
)
(
1
2
/
1
2
2
)
(
u
R
u
I
u
u
I
u
R
u
F
e
u
F
u
F
u
jI
u
R
u
F
u
j



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|F(u)| (magnitude function) is the Fourier spectrum of
f(x) and (u) its phase angle.
• The square of the spectrum
)
(
)
(
)
(
)
( 2
2
2
u
I
u
R
u
F
u
P 


is referred to as the Power Spectrum of f(x) (spectral
density).
Discrete One-Dimensional Fourier Transform
and Its Inverse

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• Fourier spectrum:   2
/
1
2
2
)
,
(
)
,
(
)
,
( v
u
I
v
u
R
v
u
F 

• Phase: 





 
)
,
(
)
,
(
tan
)
,
( 1
v
u
R
v
u
I
v
u

• Power spectrum: )
,
(
)
,
(
)
,
(
)
,
( 2
2
2
v
u
I
v
u
R
v
u
F
v
u
P 


Discrete 2-Dimensional Fourier Transform

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  2
/
1
2
2
1
0
1
0
)
(
2
1
0
1
0
)
(
2
)
,
(
)
,
(
)
,
(
)
,
(
)
,
(
)
,
(
1
)
,
(
v
u
I
v
u
R
v
u
F
e
v
u
F
y
x
f
e
y
x
f
MN
v
u
F
M
u
N
v
y
N
v
x
M
u
j
M
x
N
y
N
y
v
M
x
u
j



















Fourier Spectrum
Discrete Two-Dimensional Fourier Transform
and Its Inverse
for x = 0, 1, 2…M-1 and y = 0, 1, 2…N-1.
for u = 0, 1, 2…M-1 and v = 0, 1, 2…N-1.

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





1
0
1
0
)
,
(
1
)
0
,
0
(
M
x
N
y
y
x
f
MN
F
F(0,0) is the average intensity of an image
Discrete Two-Dimensional Fourier Transform and
Its Inverse

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54 Associations in frequency domain
Each term of F(u,v) contains all values of f(x,y)
modified by the values of the exponential terms
Frequencies in FT shows the intensity variations in
an image
 Low frequency-smooth regions
 High frequency -faster grey level changes (edges
and noise)

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DFT & Images
The DFT of a two dimensional image can be
visualised by showing the spectrum of the
images component frequencies
DFT

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DFT & Images (cont…)
DFT
Scanning electron microscope
image of an integrated circuit
magnified ~2500 times
Fourier spectrum of the image

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The Inverse DFT
It is really important to note that the Fourier
transform is completely reversible
The inverse DFT is given by:
for x = 0, 1, 2…M-1 and y = 0, 1, 2…N-1







1
0
1
0
)
/
/
(
2
)
,
(
1
)
,
(
M
u
N
v
N
vy
M
ux
j
e
v
u
F
MN
y
x
f 

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)
,
(
)
,
(
)
(
)
(
then
)
(
)
(
If
0
0
)
(
2
0
0
0
0
v
v
u
u
F
e
y
x
f
G
e
t
g
G
t
g
N
y
v
M
x
u
j
t
j












Frequency Shifting Property of the Fourier
Transform

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Transform

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The DFT and Image Processing
To filter an image in the frequency domain:
1. Compute F(u,v) the DFT of the image
2. Multiply F(u,v) by a filter function H(u,v)
3. Compute the inverse DFT of the result

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Basics of filtering in the frequency domain
1. Multiply the input image by (-1)x+y to center the
transform
2. Compute F(u,v), the DFT of the image
3. Multiply F(u,v) by a filter function H(u,v)
4. Compute the inverse DFT of (3)
5. Obtain the real part of the result
6. Multiply the result in (5) by (-1)x+y

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Some basic filters
• F(0,0) is the dc component or the average
value of the image
• Notch filter – sets F(0,0) to zero and leaves all
other frequency components of the FT
untouched.
H(u,v)= 0 if (u,v) = (M/2,N/2)
1 otherwise

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Some Basic Frequency Domain Filters
Low Pass Filter
High Pass Filter

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Smoothing Frequency Domain Filters
Smoothing is achieved in the frequency domain
by dropping out the high frequency components
The basic model for filtering is:
G(u,v) = H(u,v)F(u,v)
where F(u,v) is the Fourier transform of the
image being filtered and H(u,v) is the filter
transform function
Low pass filters – only pass the low frequencies,
drop the high ones

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Ideal Low Pass Filter
Simply cut off all high frequency components that
are a specified distance D0 from the origin of the
transform
changing the distance changes the behaviour of
the filter

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Ideal Low Pass Filter (cont…)
The transfer function for the ideal low pass filter
can be given as:
where D(u,v) is given as:






0
0
)
,
(
if
0
)
,
(
if
1
)
,
(
D
v
u
D
D
v
u
D
v
u
H
2
/
1
2
2
]
)
2
/
(
)
2
/
[(
)
,
( N
v
M
u
v
u
D 




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Above we show an image, it’s Fourier spectrum
and a series of ideal low pass filters of radius 5,
15, 30, 80 and 230 superimposed on top of it

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














u v
T
M
u
N
v
T
P
v
u
F
v
u
F
P
/
)
,
(
100
)
,
(
1
0
1
0
2

Total Power
The remained percentage
power after filtration
Smoothing Frequency Domain, Ideal Low-pass
Filters

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Original
image
Result of filtering
with ideal low pass
filter of radius 5
Result of filtering
with ideal low pass
filter of radius 30
Result of filtering
with ideal low pass
filter of radius 230
Result of filtering
with ideal low pass
filter of radius 80
Result of filtering
with ideal low pass
filter of radius 15

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Result of filtering
with ideal low pass
filter of radius 5

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Result of filtering
with ideal low pass
filter of radius 15

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Butterworth Lowpass Filters
The transfer function of a Butterworth lowpass
filter of order n with cutoff frequency at distance
D0 from the origin is defined as:
n
D
v
u
D
v
u
H 2
0 ]
/
)
,
(
[
1
1
)
,
(



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Butterworth Lowpass Filter (cont…)
Original
image
Result of filtering
with Butterworth filter
of order 2 and cutoff
radius 5
Result of filtering
with Butterworth
filter of order 2 and
cutoff radius 30
Result of filtering
with Butterworth
filter of order 2 and
cutoff radius 230
Result of filtering with
Butterworth filter of
order 2 and cutoff
radius 80
order 2 and cutoff
radius 15

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Original
image
Result of filtering
with Butterworth filter
of order 2 and cutoff
radius 5

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order 2 and cutoff
radius 15

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Gaussian Lowpass Filters
The transfer function of a Gaussian lowpass
filter is defined as:
2
0
2
2
/
)
,
(
)
,
( D
v
u
D
e
v
u
H 


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Gaussian Lowpass Filters (cont…)
Original
image
Result of filtering
with Gaussian filter
with cutoff radius 5
Result of filtering
with cutoff radius 30
Result of filtering
with cutoff radius
230
Result of filtering
with Gaussian
filter with cutoff
radius 85
Result of filtering
with Gaussian
filter with cutoff
radius 15

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Lowpass Filters Compared
Result of filtering
with ideal low pass
filter of radius 15
Result of filtering
with Butterworth
filter of order 2
and cutoff radius
15
Result of filtering
with Gaussian
filter with cutoff
radius 15

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Lowpass Filtering Examples
A low pass Gaussian filter is used to connect
broken text

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Lowpass Filtering Examples

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Lowpass Filtering Examples (cont…)
Original
image
Gaussian lowpass
filter
Processed
image
Spectrum of
original image

49
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54 Low-pass and High-pass Filters

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Sharpening in the Frequency Domain
Edges and fine detail in images are associated
with high frequency components
High pass filters – only pass the high
frequencies, drop the low ones
High pass frequencies are precisely the reverse
of low pass filters, so:
Hhp(u, v) = 1 – Hlp(u, v)

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Ideal High Pass Filters
The ideal high pass filter is given as:
where D0 is the cut off distance as before






0
0
)
,
(
if
1
)
,
(
if
0
)
,
(
D
v
u
D
D
v
u
D
v
u
H

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Ideal High Pass Filters (cont…)
Results of ideal
high pass filtering
with D0 = 15
Results of ideal
high pass filtering
with D0 = 30
Results of ideal
high pass filtering
with D0 = 80

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Butterworth High Pass Filters
The Butterworth high pass filter is given as:
where n is the order and D0 is the cut off
distance as before
n
v
u
D
D
v
u
H 2
0 )]
,
(
/
[
1
1
)
,
(



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Butterworth High Pass Filters (cont…)
Results of
Butterworth
high pass
filtering of
order 2 with
D0 = 15
Results of
Butterworth
high pass
filtering of
order 2 with
D0 = 80
Results of Butterworth high pass
filtering of order 2 with D0 = 30

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Gaussian High Pass Filters
The Gaussian high pass filter is given as:
where D0 is the cut off distance as before
2
0
2
2
/
)
,
(
1
)
,
( D
v
u
D
e
v
u
H 



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Gaussian High Pass Filters (cont…)
Results of
Gaussian
high pass
filtering with
D0 = 15
Results of
Gaussian
high pass
filtering with
D0 = 80
Results of Gaussian high pass
filtering with D0 = 30

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Highpass Filter Comparison
Results of ideal
high pass filtering
with D0 = 15

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Results of Butterworth
high pass filtering of order
2 with D0 = 15

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Results of Gaussian
high pass filtering with
D0 = 15

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Results of ideal
high pass filtering
with D0 = 15
Results of Gaussian
D0 = 15
2 with D0 = 15

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Results of ideal
high pass filtering
with D0 = 15

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2 with D0 = 15

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Results of Gaussian
D0 = 15

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Highpass Filtering Example
Original
image
Highpass
filtering
result
High
frequency
emphasis
result
After
histogram
equalisation

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Laplacian In The Frequency Domain
• Origin of F(u,v) is centered using f(x,y)(-1)x+y, before taking
the transform. Center of filter function is shifted:
• H(u,v)= -[(u-M/2)2+(v-N/2)2]1/2
• Laplacian filtered image in the spatial domain is obtained by
computed the IFT of H(u,v)F(u,v)
• Computing Laplacian in spatial domain and computing FT of the
result is equivalent to multiplying F(u,v) by H(u,v)
∇2 f (x, y) = ℑ−1 {-[(u-M/2)2+(v-N/2)2 ]1/2F(u,v)}
∇2 f (x, y) ⇔ {-[(u-M/2)2+(v-N/2)2 ]1/2F(u,v)}

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Laplacian In The Frequency Domain
Laplacian
in
the
frequency
domain
2-D
image
of
Laplacian
in
the
frequency
domain
Inverse
DFT
of
Laplacian
in
the
frequency
domain
Zoomed section of
the image on the
left compared to
spatial filter

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Frequency Domain Laplacian Example
Original
image
Laplacian
filtered
image
Laplacian
image scaled
Enhanced
image

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Fast Fourier Transform
The reason that Fourier based techniques have
become so popular is the development of the
Fast Fourier Transform (FFT) algorithm
Allows the Fourier transform to be carried out in
a reasonable amount of time
Reduces the amount of time required to perform
a Fourier transform by a factor of 100 – 600
times!

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Frequency Domain Filtering & Spatial
Domain Filtering
Similar jobs can be done in the spatial and
frequency domains
Filtering in the spatial domain can be easier to
understand
Filtering in the frequency domain can be much
faster – especially for large images

Frequency Domain Filtering 1.ppt

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