Frequency Domain
Filtering
Dr. Sk. Md. Masudul Ahsan
Dept. of CSE, KUET
Background

Any function that periodically repeats itself can be
expressed as the sum of sines and/or cosines of different
frequencies, each multiplied by a different coefficient
(Fourier series).

Fourier transform

even functions that 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
Background
Periodic function f
Frequency Weight
f1
w1
f2
w2
f3
w3
f4
w4
Jean Baptiste Joseph Fourier (1768-1830)
French mathematician
had crazy idea (1807):
Any univariate function can be
rewritten as a weighted sum of
sines and cosines of different
frequencies.
•
...the manner in which the author arrives at these
equations is not exempt of difficulties and...his
analysis to integrate them still leaves something
to be desired on the score of generality and
even rigour.
Don’t believe it?
–
–
Neither did Lagrange,
Laplace, Poisson and other
big wigs
Not translated into English
until 1878!
Laplace
But it’s (mostly) true!
•
–
–
called Fourier Series
there are some subtle
restrictions
Lagrange
Legendre
Background

Fourier transform

we will be dealing only with functions (images) of finite
duration, so the Fourier transform is the tool in which we
are interested.

A function, expressed in either a Fourier series or transform,
can be reconstructed (recovered) completely via an inverse
process, with no loss of information

The frequency domain refers to the plane of the two
dimensional discrete Fourier transform of an image.
Frequency Spectra

example : g(t) = sin(2πf t) + (1/3)sin(2π(3f) t)
=
+
Slides: Efros
Frequency Spectra
Frequency Spectra

1
= A sin(2 kt )
k 1 k
1D Fourier Transform

Fourier transform F(u) of a continuous function f(x) is:
e
Forward Fourier transform:
F (u ) 



f ( x )e
 j 2  ux
dx
j 2  ux
du
j
 cos   j sin 
Inverse transform:
f (x) 



F (u )e
j  1
• (u) is the frequency variable.
• F(u) is composed of an infinite sum of sine and cosine terms
• Each value of u determines the frequency of its corresponding
sine-cosine pair.
1D Discrete Fourier Transform (DFT)

A continuous function f(x) is discretized into a sequence:
{ f ( x0 ), f ( x0  x), f ( x0  2x),..., f ( x0  [ N  1]x)}
by taking N or M samples x units apart.
1D DFT

Where x assumes the discrete values (0,1,2,3,…,M-1)
then
f ( x)  f ( x0  xx)

The sequence {f(0),f(1),f(2),…f(M-1)} denotes any M
uniformly spaced samples from a corresponding
continuous function.
1D DFT


f(x), x=0,1,…,M-1. discrete function
F(u), u=0,1,…,M-1. DFT of f(x)
Forward discrete Fourier transform:
1
F (u ) 
M
M 1

f ( x )e
M 1

u
x
M
x0
Inverse transform:
f (x) 
 j2
F (u )e
j2
u
x
M
u0

The location of 1/M multiplier doesn’t matter
 in
front of the Fourier transform
 in
front of the inverse Fourier transform
1D DFT

Since e j  cos   j sin  and the fact
cos( )  cos 
then discrete Fourier transform can be written as
1
F (u ) 
M
M 1
 f ( x)[cos 2ux / M  j sin 2ux / M ]
x 0
for u  0,1,2,..., M  1
1D DFT
The Fourier transform of a real function is generally
complex
imaginary
 F(u) can be expressed in polar
m
coordinates:

real
j ( u )

F (u )  F (u ) e

2
2
where F (u )  R (u )  I (u )
1
2

 I (u ) 
 (u )  tan 

R
(
u
)


 R(u): the real part of F(u)
 I(u): the imaginary part of F(u)
Magnitude or spectrum
1

Power spectrum:
Phase angle or phase spectrum
2
P(u )  F (u )  R 2 (u )  I 2 (u )
1D – Fourier Transform-example
1D DFT - example
2D DFT

Forward transform
F (u , v) 



f ( x, y )e  j 2 ( ux  vy )
a-periodic signal
periodic transform
x   y  
1
F (u , v) 
MN
M 1 N 1

f ( x, y )e  j 2 (ux / M  vy / N )
x 0 y 0
for u  0,1,2,..., M  1, v  0,1,2,..., N  1

Inverse transform:
M 1 N 1
f ( x, y )   F (u , v)e j 2 ( ux / M  vy / N )
u 0 v 0
for x  0,1,2,..., M  1, y  0,1,2,..., N  1

u, v : the transform or frequency variables

x, y : the spatial or image variables
periodized signal
periodic and sampled
transform
2D DFT

Similarly, we define the Fourier spectrum, phase angle, and power
spectrum as follows:
F (u , v)  F (u , v) e j (u ,v )

F (u , v)  R 2 (u, v)  I 2 (u , v)
1
2

( spectrum)
 I (u , v) 
 (u, v)  tan 
(phase angle)

 R(u , v) 
1
2
P(u,v)  F (u , v)  R 2 (u , v)  I 2 (u, v) (power spectrum)

R(u,v): the real part of F(u,v)

I(u,v): the imaginary part of F(u,v)
2D DFT – Properties: Separability

The 2D FT can be implemented as two consecutive 1D FTs:
first in the x direction, then in the y direction (or vice versa).
1
F1 (u, y ) 
M
1
F (u , v) 
N

Proof: DIY
M 1

f ( x, y )e  j 2 (ux / M )
x 0
M 1
 j 2 ( vy / N )
F
(
u
,
y
)
e
 1
y 0
2D DFT – Properties: Separability

Separable: The 2D DFT F(u,v) can be obtained by
1.
taking the 1D DFT of every row of image f(x,y),  F(u,y),
2.
taking the 1D DFT of every column of F(u,y),  F(u,v),
(a)f(x,y)
(b)F(u,y)
(c)F(u,v)
2D DFT – Properties: Shifting

The FT of a shifted function is unaltered except for a linearly
varying phase factor.
f ( x  x 0 , y  y 0 )  F ( u , v ) e  j 2  ( ux / M  vy / N )
0
0
Time / spatial shift

and
f ( x, y ) e
j 2 (u 0 x / M  v0 y / N )
 F (u  u 0 , v  v 0 )
frequency
shift
2D DFT – Properties: Shifting

The FT of a shifted function is unaltered except for a linearly
varying phase factor.
f ( x  x 0 , y  y 0 )  F ( u , v ) e  j 2  ( ux / M  vy / N )
0
0
2D DFT – Properties: Shifting

The FT of a shifted function is unaltered except for a linearly
varying phase factor.
f ( x  x 0 , y  y 0 )  F ( u , v ) e  j 2  ( ux / M  vy / N )
0

0
Proof:
1
DFT [ f ( x  x 0 , y  y 0 )] 
MN
 e  j 2  [ ux 0 / M  vy 0 / N ]
1
MN
M 1 N 1

f (x  x 0 , y  y 0 )e  j 2  ( ux / M  vy / N )
x0 y 0
M 1 N 1
  f (x  x , y  y
0
x 0 y 0
 e  j 2  [ ux 0 / M  vy 0 / N ] F ( u , v )
0
)e
 j 2 [
u ( x  x0 ) v ( y  y 0 )

]
M
N
2D DFT – Properties: Shifting

If, u0 = M/2, v0 = N/2
f ( x, y ) e

j 2 (u0
x
y
 v0 )
M
N
 F (u  u 0 , v  v 0 )
Becomes
f ( x , y ) (  1) x  y  F ( u  M / 2 , v  N / 2 )
0
x
u
u
y
F
v
shift
v
2D DFT – Properties: Periodicity

A [M,N] point DFT is periodic with period [M,N]
F(u,v) = F(u+M, v ) = F(u, v+N)
= F(u+M, v+N) = F(u+pM, v + qN)

1
Proof: F ( u , v ) 
MN
M  1 N 1

x0 y 0
1
F (u  M , v  N ) 
MN
1

MN
f ( x , y )e  j 2  ( ux / M  vy / N )
M 1 N  1

x0 y 0
M  1 N 1

f ( x , y )e
v N 
 uM
 j 2 
x
y
N
 M

f ( x , y )e
v 
 u
 j 2 
x y 
N 
M
.e
N 
M
 j 2 
x y 
N 
M
x0 y 0
 e  j 2 ( x  y ) F (u , v )
 F (u , v )
 cos 2 n 
 1, sin 2 n   0 
2D DFT – Properties: Periodicity
2D DFT – Properties: Rotation

if a function is rotated, then its Fourier transform rotates
an equal amount.
DFT
DFT
2D DFT – Properties: Rotation
y
y
x
x   r cos(    ) ; y   r sin(    )

r

x
x  r cos 
y  r sin 
Define a new coord system (x’, y’), where
x   r cos  cos   r sin  sin 
=>
y   r sin  cos   r cos  sin 
x   x cos   y sin 
=>
y    x sin   y cos 
We can also express (x, y) in terms of the new coordinate system,
where
x  x  cos   y  sin 
; y  x  sin   y  cos 
2D DFT – Properties: Rotation
v
v
u



u
Similarly, in Fourier domain
u   cos 
v   sin 
u    cos(    ) ; v    sin(    )
u   u cos   v sin 
; v    u sin   v cos 
in terms of the new coordinate system, where
u  u  cos   v  sin 
; v  u  sin   v  cos 
2D DFT – Properties: Rotation

Proof:
DFT  f ( x , y )   F ( u , v )
DFT  f ( r cos  , r sin  )   F ( cos  , w sin  )
DFT  f ( x , y )   DFT  f ( r cos(    ), r sin(    )) 



 
f ( x , y )e  j 2  ( xu  yv )
x   y  



 
f ( x , y )e  j 2  ( x  cos   y  sin  ) u  ( x  sin   y  cos  ) v 
x   y  




x   y  
 j 2   x  ( u cos   v sin  )  y  (  u sin   v cos  ) 


f ( x , y )e
2D DFT – Properties: Rotation



 
f ( x , y )e
 j 2   x u   y v  
x   y  
 F ( u , v  )
 F ( cos(    ),  sin(    ))
2D DFT – Properties: Convolution theorem

~ states that convolution in time domain corresponds to
multiplication in frequency domain and vice versa

The discrete convolution of two functions f(x,y) and h(x,y) of size
MXN is defined as
M 1 N 1
f ( x, y )  h( x, y )   f (a, b)h( x  a, y  b)
a 0 a 0

Let F(u,v) and H(u,v) denote the Fourier transforms of f(x,y) and h(x,y),
then
f ( x, y )  h( x, y )  F (u , v) H (u , v) Eq. (4.2-31)
f ( x , y ) h ( x , y )  F (u , v )  H ( u , v )
Eq. (4.2-32)
2D DFT – Properties: Convolution theorem

Proof:
M 1 N 1 M 1 N 1

  j 2 ( ux / M  vy / N )
DFT  f ( x, y )  h( x, y )     f (a, b)h( x  a, y  b)e
x 0 y 0  a 0 b 0

M 1 N 1 M 1 N 1
   f (a, b)h( x  a, y  b)e
 u ( x  a  a ) v ( y b  b ) 
 j 2 


M
N


x 0 y 0 a 0 b 0
M 1 N 1
  f (a, b)e
 ua vb 
 j 2    M 1 N 1
M N 
a 0 b 0
 F (u , v) H (u , v)
 h( x  a, y  b)e
x 0 y 0
 u ( x  a ) v ( y b ) 
 j 2 


M
N 

2D DFT

Some other properties of Fourier transform:
1
F ( 0 ,0 ) 
MN

M 1 N 1

f ( x, y)
x0 y0
F(0,0) is the average intensity of an image
2-D DFT – importance of phase angle
Properties in the frequency domain

Each term of F(u,v) contains all values of (x,y), modified
by the values of the exponential terms
 DC
component: the slowest varying frequency component
(u=v=0) corresponds to the average gray level of an image
 The
higher frequencies begin to correspond to faster and
faster gray level changes in the image
 Edges
of objects
 Components of an image characterized by abrupt changes in
gray level, such as noise
Properties in the frequency domain

Fourier transform works globally
 No
direct relationship between a specific components in an
image and frequencies

Intuition about frequency
 Frequency
 Rate
content
of change of gray levels in an image
2D DFT - example
2D DFT - example

Strong edges that run approximately at ±45. And the two
white oxide protrusions resulting from thermally induced failure


the Fourier spectrum shows prominent components along the ±45
directions that correspond to the edges
Vertical component that is off-axis slightly to the left
 this
component was caused by the edges of the oxide
protrusions
 the
off-axis angle of the frequency component corresponds to
the inclination off horizontal of the long white element
 zeros
in the vertical frequency component, corresponding to
the narrow vertical span of the oxide protrusions
Basics of Filtering in the Frequency Domain
To filter an image in the frequency domain:
1.
2.
3.
Compute F(u,v) the DFT of the image
Multiply F(u,v) by a filter function H(u,v)
Compute the inverse DFT of the result
Steps for Filtering in the Frequency Domain
Steps for Filtering in the Frequency Domain
Steps for Filtering in the Frequency Domain

In general, the components of F are complex quantities,
but the filters with which we deal typically are real
 zero-phase-shift
filters: do not change the phase of the
transform
 if
the input image and the filter function are real, the
imaginary components of the inverse transform should all be
zero
 in
practice, the inverse DFT generally has parasitic imaginary
components due to computational round-off errors
Notch Filter

Suppose that we wish to force the average value of an
image to zero
 set
F(0,0) to zero & leave all other frequency components of
the Fourier transform untouched
 0 if (u , v)  ( M / 2, N / 2)
H (u , v)  
otherwise.
1
a
constant function with a hole (notch) at the origin
 Note
 the
drop in overall average gray level resulting from forcing the
average value to zero
 make prominent edges stand out
Notch Filter

In reality the average of the displayed image cannot be
zero

display the most negative value as 0, with all other values scaled up from that
Smoothing Frequency-Domain Filters

The basic model for filtering in the frequency domain
G (u , v)  H (u, v) F (u , v)
where F(u,v): the Fourier transform of the image to be smoothed
H(u,v): a filter transfer function



Smoothing (blurring) is achieved in the frequency domain by
attenuating a specified range of high-frequency components in the
transform of a given image
Smoothing is a low pass operation in the frequency domain.
There are several standard forms of low pass filters (LPF).



Ideal low pass filter
Butterworth low pass filter
Gaussian low pass filter
Ideal Low pass Filter (ILPF)

The simplest low pass filter is a filter that “cuts off” all high-frequency
components of the Fourier transform that are at a distance greater
than a specified distance D0 from the origin of the transform.

The transfer function of an ideal low pass filter
1
H (u , v)  
0
if D(u , v)  D0
if D(u , v)  D0
where D(u,v) : the distance from point (u,v) to the center of their
frequency rectangle

D(u , v)  (u  M / 2) 2  (v  N / 2)
1
2 2

ILPF
ILPF
ILPF - example
ILPF - example

As the filter radius increases, less &
less power is removed, resulting in
less server blurring

The ringing effect is a characteristic
of ideal filters

For α=99.95, only slight blurring in
the noisy squares are shown

the little edge information is contained in
the upper 0.5% of the spectrum power in
this case
ILPF – ringing effect

It has a central dominant
circular component (providing
the blurring)

It has concentric circular
components (rings) giving rise
to the ringing effect
spatial
Figure 4.13 (a) A frequency-domain
ILPF of radius 5. (b) Corresponding
spatial filter. (c) Five impulses in the
spatial domain, simulating the values
of five pixels. (d) Convolution of (b)
and (c) in the spatial domain.
f ( x, y )  h( x, y )  F (u , v) H (u, v)
spatial
Butterworth Low pass Filters (BLPFs)
With order n
1
H (u, v) 
2n
1  D(u , v) / D0 

Filter function

for high values of “filter order” the Butterworth filter approaches the
form of the ideal lowpass filter

for lower-order values, the Butterworth filter has a smooth form
similar to the Gaussian filter
BLPF

BLPF does not have a sharp discontinuity that establishes a clear
cutoff between passed and filtered frequencies

H(u,v)=0.5 (down 50% from its maximum value of 1) when D(u,v)=D0
BLPF - example

A smooth transition in blurring as
a function of increasing cutoff
frequency

No ringing is visible due to the
smooth transition between low &
high frequencies
n=2
D0=5,15,30,80,and 230
BLPF - Spatial Representation

Ringing becomes a significant factor in filters of higher order

Order 2: a good compromise between effective lowpass filtering &
acceptable ringing characteristics
n=1
n=2
n=5
n=20
Gaussian Lowpass Filters (GLPF)
GLPF


The inverse Fourier transform of the GLPF also is
Gaussian
Guarantees no ringing
GLPF - example

As in the case of the BLPF, a smooth
transition in blurring as a function of
increasing cutoff frequency

No ringing

The GLPF did not achieve as much
smoothing as the BLPF of order 2 for
the same value of cutoff frequency

BLPF is a more suitable choice if: tight
control of the transition between low &
high frequencies about the cutoff
frequency are needed

the price: ringing
D0=5,15,30,80,and 230
Additional Examples of Lowpass Filtering

Examples: fax transmissions, duplicated material & historical records

The characters have distorted shapes due to lack of resolution, and
many of the characters are broken


solution: bridge small gaps by blurring
D0: 80
Additional Examples of Lowpass Filtering

Printing and publishing industry: “cosmetic” processing

Different lowpass Gaussian filters used to remove blemishes in a
photograph
Sharpening Frequency Domain
H hp (u , v)  1  H lp (u , v)
Ideal highpass filter
0
H (u , v)  
1
if D (u , v)  D0
if D (u, v)  D0
Butterworth highpass filter
H (u , v ) 
1
2n
1  D0 / D(u , v)
Gaussian highpass filter
H (u , v )  1  e
 D 2 ( u ,v ) / 2 D02
Highpass Filters
Spatial Representations
A dot
in the
center
Ideal Highpass Filters
0
H (u, v)  
1
if D (u, v)  D0
if D (u, v)  D0
Highpass Filters
Butterworth
1
H (u, v) 
2n
1  D0 / D (u, v)
Gaussian
H (u , v)  1  e
 D 2 ( u ,v ) / 2 D02
Text orientation finding - Example


Text recognition using image
processing techniques is simplified if
we can assume that the text lines are
in a predefined direction.
Here we show how the Fourier
Transform can be used to find the
initial orientation of the text and
then a rotation can be applied to
correct the error.
Text orientation finding - Example
The logarithm of the magnitude of its
Fourier transform are
We can see that the main
values lie on a vertical
line, indicating that the
text lines in the input
image are horizontal.