Image restoration

J.B.Jeeva

Based on
Digital image processing by Gonzalez and Woods

J.B.JEEVA 1
 Introduction
• Image enhancement : subjective process
• Image restoration : objective process
• Restoration: recover an image that has been
degraded by using a priori knowledge of the degradation
phenomenon
• Process: modelling the degradation and applying the
inverse process to recover the original image
e.g.: “de-blurring”
Some techniques are best formulated in the spatial
domain (e.g. additive noise only), others in the
frequency domain (e.g. de-blurring)
J.B.JEEVA 2
Image Restoration and Reconstruction

1. A Model of the Image Degradation/Restoration

Process
2. Noise Models
3. Restoration in the Presence of Noise Only - Spatial
Filtering
4. Periodic Noise Reduction by Frequency Domain
Filtering
5. Estimating the Degradation Function
6. Inverse Filtering
7. Minimum Mean Square Error (Wiener) Filtering
8. Geometric Mean Filter
J.B.JEEVA 3
 A Model of the Image
Degradation/Restoration Process

If H is a linear, position-invariant process

J.B.JEEVA 4
J.B.JEEVA 5
J.B.JEEVA 6
2.2 Some Important Noise Probability Density
Functions

J.B.JEEVA 7
J.B.JEEVA 8
J.B.JEEVA 9
J.B.JEEVA 10
J.B.JEEVA 11
J.B.JEEVA 12
J.B.JEEVA 13
J.B.JEEVA 14
J.B.JEEVA 15
J.B.JEEVA 16
J.B.JEEVA 17
J.B.JEEVA 18
J.B.JEEVA 19
J.B.JEEVA 20
J.B.JEEVA 21
J.B.JEEVA 22
Adaptive, local noise reduction filter

– If  2 is zero, return simply the value of g ( x, y )

- If  2   2 , return a value close to g ( x, y )
 L

– If 2   L2 , return the arithmetic mean valuemL

 2
ˆf ( x, y )  g ( x, y )   g ( x, y )  m 
L2 L
• Adaptive median filter
– zmin = minimum gray level value in S xy

–
zmax= maximum gray level value in
S xy

– z med = median of gray levels in S xy

– z xy = gray level at coordinates ( x, y )
– S max = maximum allowed size of S xy
Algorithm:
– Level A: A1= z med  z min
– A2=
z med  z max

– If A1>0 AND A2<0, Go to

– level B
– Else increase the window size
– If window size  S max
– repeat level A
– Else output z med
– Level B: B1=
z xy  zmin
– B2= z xy  zmax
– If B1>0 AND B2<0, output z
xy
– Else output
zmed
– Purposes of the algorithm
• Remove salt-and-pepper (impulse) noise
• Provide smoothing
• Reduce distortion, such as excessive thinning or
thickening of object boundaries
Periodic Noise Reduction by Frequency Domain
Filtering
• Bandreject filters
– Ideal bandreject filter
 W
1 if D(u, v)  D 0 
2

 W W
H (u, v)  0 if D 0   D(u, v)  D 0 
 2 2
1 W
if D(u, v)  D 0 

 2

D(u, v)  (u  M / 2)  (v  N / 2)
2

2 1/ 2
– Butterworth bandreject filter of order n

1
H (u, v)  2n
 D(u, v)W 
1  2 2
 D (u, v)  D0 
– Gaussian bandreject filter
2
1  D 2 ( u ,v )  D02 
  
2  D ( u ,v )W 
H (u, v)  1  e
• Bandpass filters

H bp (u, v)  1  H br (u, v)
J.B.JEEVA 36
 Notch filters

– Ideal notch reject filter

0 if D1 (u, v)  D 0 or D 2 (u, v)  D 0
H (u , v)  
1 otherwise


D1 (u , v)  (u  M / 2  u0 )  (v  N / 2  v0 )
2

2 1/ 2

D (u, v)  (u  M / 2  u )
2 0
2
 (v  N / 2  v ) 
0
2 1/ 2
• Butterworth notch reject filter of order n

1
H (u, v)  n
 D 2

1  0

 D1 (u, v) D2 (u , v) 
• Gaussian notch reject filter

1  D1 ( u ,v ) D2 ( u ,v ) 
  2 
2  
H (u, v)  1  e
D0
J.B.JEEVA 41
J.B.JEEVA 42
• Notch pass filter

H np (u, v)  1  H nr (u, v)
• Optimum notch filtering
 – Interference noise pattern

N (u, v)  H (u, v)G(u, v)

– Interference noise pattern in the spatial

domain
 ( x, y )   {H (u, v)G(u, v)}
1

– Subtract fromg ( x, y ) a weighted portion of

 ( x, y) to obtain an estimate of f ( x, y)
fˆ ( x, y)  g ( x, y)  w( x, y) ( x, y)
– Minimize the local variance of fˆ ( x, y)
– The detailed steps are given in Page 363,364
– Result

g ( x, y ) ( x, y )  g ( x, y ) ( x, y )
w( x, y ) 
 2 ( x, y )   2 ( x, y )
 Image restoration
J.B.Jeeva

Based on
Digital image processing by Gonzalez and Woods

J.B.JEEVA 1
 Introduction
• Image enhancement : subjective process
• Image restoration : objective process
• Restoration: recover an image that has been
degraded by using a priori knowledge of the degradation
phenomenon
• Process: modelling the degradation and applying the
inverse process to recover the original image
e.g.: “de-blurring”
Some techniques are best formulated in the spatial
domain (e.g. additive noise only), others in the
frequency domain (e.g. de-blurring)
J.B.JEEVA 2
Image Restoration and Reconstruction

1. A Model of the Image Degradation/Restoration

Process
2. Noise Models
3. Restoration in the Presence of Noise Only - Spatial
Filtering
4. Periodic Noise Reduction by Frequency Domain
Filtering
5. Estimating the Degradation Function
6. Inverse Filtering
7. Minimum Mean Square Error (Wiener) Filtering
8. Geometric Mean Filter
J.B.JEEVA 3
 A Model of the Image
Degradation/Restoration Process

If H is a linear, position-invariant process

J.B.JEEVA 4
J.B.JEEVA 5
J.B.JEEVA 6
J.B.JEEVA 7
 n is an
J.B.JEEVA 8
integer
J.B.JEEVA 9
J.B.JEEVA 10
J.B.JEEVA 11
J.B.JEEVA 12
J.B.JEEVA 13
J.B.JEEVA 14
J.B.JEEVA 15
J.B.JEEVA 16
J.B.JEEVA 17
J.B.JEEVA 18
J.B.JEEVA 19
J.B.JEEVA 20
J.B.JEEVA 21
 Digital Image Processing, 3rd ed.
Gonzalez & Woods
www.ImageProcessingPlace.com

Chapter 1
Introduction

The need/motivation for image processing:

The enhancement/improvement of pictorial information for:
• human interpretation
• automatic management (identification, storage, transmission,
quantification, ...)

© 1992–2008 R. C. Gonzalez & R. E. Woods

 Digital Image Processing, 3rd ed.
Gonzalez & Woods
www.ImageProcessingPlace.com

Chapter 1
Introduction

What is digital image processing?

Processing of an image by means of digital
computers.

Image analysis - Image processing - Computer vision

© 1992–2008 R. C. Gonzalez & R. E. Woods

 Digital Image Processing, 3rd ed.
Gonzalez & Woods
www.ImageProcessingPlace.com

Chapter 1
Introduction

One of the first application areas of

digital images was newspapers Digital computers: 1940
industries (cable between London and 1st computer able to do digital
NY) image manipulations: early
1960
Important to reduce transfer time.
© 1992–2008 R. C. Gonzalez & R. E. Woods
 Digital Image Processing, 3rd ed.
Gonzalez & Woods
www.ImageProcessingPlace.com

Chapter 1
Introduction

© 1992–2008 R. C. Gonzalez & R. E. Woods

 Digital Image Processing, 3rd ed.
Gonzalez & Woods
www.ImageProcessingPlace.com

Chapter 1
Introduction

Principal energy source for images today:

electromagnetic energy spectrum.

© 1992–2008 R. C. Gonzalez & R. E. Woods

 Digital Image Processing, 3rd ed.
Gonzalez & Woods
www.ImageProcessingPlace.com

Chapter 1
Introduction

Gamma rays:

Nuclear
medicine
(injection of
radioactive
tracer)

Astronomical
observations
(object generate
gamma rays)

© 1992–2008 R. C. Gonzalez & R. E. Woods

 Digital Image Processing, 3rd ed.
Gonzalez & Woods
www.ImageProcessingPlace.com

Chapter 1
Introduction

PET=Positron Emission Tomography

imaging at molecular level

© 1992–2008 R. C. Gonzalez & R. E. Woods

 Digital Image Processing, 3rd ed.
Gonzalez & Woods
www.ImageProcessingPlace.com

Chapter 1
Introduction

X-rays (the oldest radiation-type imaging) An x-ray picture

(radiograph) taken by
Röntgen of Albert von
-Discovered in 1895 by german Kölliker's hand at a
physicist William Roentgen public lecture on 23
January 1896
(Nobel prize in physics, 1901)

-used in medicine/industry/astronomy

X-ray tube (catode/anode, controlled by

voltage), emitting ray, absorbption by
object, rest captured onto a film,
digitised.

C.A.T. (Computerized Axial

Tomography) uses X-rays.

© 1992–2008 R. C. Gonzalez & R. E. Woods

Copyright: Radiology Centennial, Inc.
 Digital Image Processing, 3rd ed.
Gonzalez & Woods
www.ImageProcessingPlace.com

Chapter 1
Introduction

Ultraviolet band:

microscopy (fluorescence)
the excited electron jumps to another energy
level emitting light as a low-energy photon
in the red region

lasers
biological imaging
astronomical imaging
industrial inspections

A fluorescent tracer
is bind to a
molecular target
© 1992–2008 R. C. Gonzalez & R. E. Woods
 Digital Image Processing, 3rd ed.
Gonzalez & Woods
www.ImageProcessingPlace.com

Chapter 1
Introduction

Visible and infrared band: the

most familiar to us….

light microscopy

infrared: remote sensing, weather

prediction,
satellite sensing/ night vision
© 1992–2008 R. C. Gonzalez & R. E. Woods
 Digital Image Processing, 3rd ed.
Gonzalez & Woods
www.ImageProcessingPlace.com

Chapter 1
Introduction

© 1992–2008 R. C. Gonzalez & R. E. Woods

 Digital Image Processing, 3rd ed.
Gonzalez & Woods
www.ImageProcessingPlace.com

Chapter 1
Introduction

Mount Everest
Nasa/Landsat

Mount Everest is the highest mountain on Earth, rising

29,029 feet above sea level. It is located on the border
of Nepal and Tibet in the Himalayan mountain range.
In Tibet the mountain is known as Chomolunga and in
Nepal it is called Sagarmatha.

This image of Mount Everest was taken from the

International Space Station on November 26, 2003.
In this image you can see Mount Everest covered in
white snow with Lhotse, the fourth highest mountain
on Earth connected via the South Col — the saddle
point between the two peaks. Vegetation appears green
and rock and soil appear brown in the image.
This natural color Landsat 5 image was collected on
June 11, 2005. It was created using bands 3, 2 and 1.
Mount Everest is found on Landsat WRS-2 Path 140
Row 41.

© 1992–2008 R. C. Gonzalez & R. E. Woods

 Digital Image Processing, 3rd ed.
Gonzalez & Woods
www.ImageProcessingPlace.com

Chapter 1
Introduction

Mono Lake, California

Nasa/Landsat

This Landsat 7 image of Mono Lake was acquired on July 27,

2000. This image is a false-color composite made from the
mid-infrared, near-infrared, and green spectral channels of the
Landsat 7 ETM+ sensor – it also includes the panchromatic
15-meter band for spatial sharpening purposes. In this image,
the waters of Mono Lake appear a bluish-black and
vegetation appears bright green. You will notice the
vegetation to the west of the lake and following the tributaries
that enter the lake.

© 1992–2008 R. C. Gonzalez & R. E. Woods

 Digital Image Processing, 3rd ed.
Gonzalez & Woods
www.ImageProcessingPlace.com

Chapter 1
Introduction

© 1992–2008 R. C. Gonzalez & R. E. Woods

 Digital Image Processing, 3rd ed.
Gonzalez & Woods
www.ImageProcessingPlace.com

Chapter 1
Introduction

Visible range:

automated inspection
tasks

© 1992–2008 R. C. Gonzalez & R. E. Woods

 Digital Image Processing, 3rd ed.
Gonzalez & Woods
www.ImageProcessingPlace.com

Chapter 1
Introduction

© 1992–2008 R. C. Gonzalez & R. E. Woods

 Digital Image Processing, 3rd ed.
Gonzalez & Woods
www.ImageProcessingPlace.com

Chapter 1
Introduction

radio band:

MRI - imaging
(Nobel prizes: Bloch 1952,
… , 2003)

A strong magnet passes radio waves

though short pulses which causes
a response pulse (echo)

© 1992–2008 R. C. Gonzalez & R. E. Woods

 Digital Image Processing, 3rd ed.
Gonzalez & Woods
www.ImageProcessingPlace.com

Chapter 1
Introduction

Other sources of energy

beside electromagnetic waves:

- acoustic waves
(seismic, marine/atmospheric,
sonar/radar, ultrasound)

- electron microscopy

- synthetic images

© 1992–2008 R. C. Gonzalez & R. E. Woods

 Digital Image Processing, 3rd ed.
Gonzalez & Woods
www.ImageProcessingPlace.com

Chapter 1
Introduction

© 1992–2008 R. C. Gonzalez & R. E. Woods

 Digital Image Processing, 3rd ed.
Gonzalez & Woods
www.ImageProcessingPlace.com

Chapter 1
Introduction

© 1992–2008 R. C. Gonzalez & R. E. Woods

 Digital Image Processing, 3rd ed.
Gonzalez & Woods
www.ImageProcessingPlace.com

Chapter 1
Introduction

+ extra stuff
© 1992–2008 R. C. Gonzalez & R. E. Woods
 Digital Image Processing, 3rd ed.
Gonzalez & Woods
www.ImageProcessingPlace.com

Chapter 2
Digital Image Fundamentals

Retina: consist of receptors

- cones: highly sensitive to colors. Photopic
or bright-light vision

-rods: give overall picture with reduced detail.

Scotopic or dim-light vision

© 1992–2008 R. C. Gonzalez & R. E. Woods

 Digital Image Processing, 3rd ed.
Gonzalez & Woods
www.ImageProcessingPlace.com

Chapter 2
Digital Image Fundamentals

This distance varies between 14-17 mm

depending on the lens’ focussing
Classical optical theory:
A ray passes through the centre C of the lens.

The two triangle are proportional:

h is the height of the object on the retina
(note that is located close to the fovea)

© 1992–2008 R. C. Gonzalez & R. E. Woods

 Digital Image Processing, 3rd ed.
Gonzalez & Woods
www.ImageProcessingPlace.com

Chapter 2
Digital Image Fundamentals

Perceived intensity is not a

simple function of actual
intensity.

- under/overshoot boundary of
regions of different intensity
(Mach bands)

- A region’s perceived
brightness does depend on
the background intensity as
well (simultaneous contrast)
© 1992–2008 R. C. Gonzalez & R. E. Woods
 Digital Image Processing, 3rd ed.
Gonzalez & Woods
www.ImageProcessingPlace.com

Chapter 2
Digital Image Fundamentals

Optical illusions and

perception:

The eye “fills in” non-

existing information or
wrongly perceives
geometrical properties of
objects.

© 1992–2008 R. C. Gonzalez & R. E. Woods

Intensity Transformations and Spatial
Filtering
Background

 Spatial domain process

g ( x, y) = T [ f ( x, y)]
⚫ where f ( x, y) is the input image, g ( x, y )
is the processed image, and T is an
operator on f, defined over some
neighborhood of ( x, y )
 Neighborhood about a point
 Gray-level transformation function
s = T (r )
⚫ where r is the gray level of f ( x, y) and
s is the gray level of g ( x, y ) at any
point ( x, y )
 Contrast enhancement
⚫ For example, a thresholding function
 Masks (filters, kernels, templates,
windows)
⚫ A small 2-D array in which the values
of the mask coefficients determine the
nature of the process
Some Basic Gray Level
Transformations
 Image negatives
s = L −1 − r
⚫ Enhance white or gray details
 Log transformations
s = c log(1 + r )
⚫ Compress the dynamic range of images
with large variations in pixel values
⚫ From the range 0-1.5 10 6 to the range
0 to 6.2
 Power-law transformations
 s = cr  or s = c(r +  )
⚫   1 maps a narrow range of dark
input values into a wider range of
output values, while   1 maps a
narrow range of bright input values into
a wider range of output values
⚫  : gamma, gamma correction
 Monitor,  = 2.5
 Piecewise-linear transformation
functions
⚫ The form of piecewise functions can be
arbitrarily complex
⚫ Contrast stretching
⚫ Gray-level slicing
⚫ Bit-plane slicing
Histogram Processing

 Histogram
h(rk ) = nk
⚫ where rk is the kth gray level and nkis
the number of pixels in the image
having gray level rk
⚫ Normalized histogram
p(rk ) = nk / n
 Histogram equalization
s = T (r ), 0  r  1
r = T −1 ( s), 0  s  1
 ⚫ Probability density functions (PDF)
dr
p s ( s ) = pr ( r )
ds
r
s = T (r ) = ( L − 1) pr (w)dw
0

ds dT (r ) d  r  = ( L − 1) p (r )
dr
=
dr
= ( L − 1)
dr 
 0
pr ( w) dw

 r

1
ps ( s ) =
L −1
 k k nj
sk = T (rk ) = ( L − 1) pr (rj ) =( L − 1) , k = 0,1,2,..., L − 1
j =0 j =0 n
 Local enhancement
⚫ Histogram using a local neighborhood,
for example 7*7 neighborhood
⚫ Histogram using a local 3*3
neighborhood
Fundamentals of Spatial Filtering

 The Mechanics of Spatial Filtering

R = w(−1,−1) f ( x − 1, y − 1) +
w(−1,0) f ( x − 1, y ) +  +
w(0,0) f ( x, y ) +  +
w(1,0) f ( x + 1, y ) +
w(1,1) f ( x + 1, y + 1)
⚫ Image size: M N
⚫ Mask size: m n
a b
g ( x, y ) =   w(s, t ) f ( x + s, y + t )
s = − at = − b

⚫ a = (m − 1) / 2 and b = (n − 1) / 2
⚫ x = 0,1,2,..., M − 1 and y = 0,1,2,..., N − 1
 Spatial Correlation and Convolution
 Vector Representation of Linear
Filtering
R = w1 z1 + w2 z 2 + ... + w9 z9
9
=  wi zi
i =1
Smoothing Spatial Filters

 Smoothing Linear Filters

⚫ Noise reduction
⚫ Smoothing of false contours
⚫ Reduction of irrelevant detail
 1 9
R =  zi
9 i =1
 a b

  w(s, t ) f ( x + s, y + t )
g ( x, y ) = s = − at = − b
a b

  w(s, t )
s = − at = − b
 Order-statistic filters
⚫ median filter: Replace the value of a
pixel by the median of the gray levels
in the neighborhood of that pixel
⚫ Noise-reduction
Sharpening Spatial Filters

 Foundation
⚫ The first-order derivative
f
= f ( x + 1) − f ( x)
x
⚫ The second-order derivative

 f
2
= f ( x + 1) + f ( x − 1) − 2 f ( x)
x 2
 Use of second derivatives for
enhancement-The Laplacian
⚫ Development of the method

 f  f
2 2
 f = 2 + 2
2

x y
 f
2
= f ( x + 1, y) + f ( x − 1, y) − 2 f ( x, y)
x 2

2 f
= f ( x, y + 1) + f ( x, y − 1) − 2 f ( x, y)
y 2
  2 f = [ f ( x + 1, y ) + f ( x − 1, y ) + f ( x, y + 1) +
f ( x, y − 1)] − 4 f ( x, y )
 if the center coefficien t
 f ( x, y ) −  2 f ( x, y ) of the Laplacian mask

 is negative
g ( x, y ) = 
if the center coefficien t

 f ( x, y ) +  2 f ( x, y ) of the Laplacian mask

 is positive
 ⚫ Simplifications

g ( x, y ) = f ( x, y ) − [ f ( x + 1, y ) + f ( x − 1, y) + f ( x, y + 1) +
f ( x, y − 1)] + 4 f ( x, y)
= 5 f ( x, y) − [ f ( x + 1, y ) + f ( x − 1, y ) + f ( x, y + 1) +
f ( x, y − 1)]
 Unsharp masking and highboost
filtering
⚫ Unsharp masking
 Substract a blurred version of an image
from the image itself

g mask ( x, y) = f ( x, y) − f ( x, y)

 f ( x, y) : The image, f ( x, y ) : The

blurred image
g ( x, y) = f ( x, y) + k * g mask ( x, y) ,k =1
 ⚫ High-boost filtering

g ( x, y) = f ( x, y) + k * g mask ( x, y) ,k 1
 Using first-order derivatives for
(nonlinear) image sharpening—The
gradient
 f 
Gx   x 
f =   =  f 
G y   
 y 
⚫ The magnitude is rotation invariant
(isotropic)


f = mag (f ) = G + G2
x
2
y 
1
2

1
 f  2  f  2  2

=   +   
 x   y  

f  G x + G y
⚫ Computing using cross differences,
Roberts cross-gradient operators
G x = ( z9 − z5 ) and Gy = ( z8 − z6 )


f = ( z9 − z5 ) + ( z8 − z6 )
2 2

1
2

f  z9 − z5 + z8 − z6
⚫ Sobel operators
 A weight value of 2 is to achieve some
smoothing by giving more importance to
the center point

f  ( z7 + 2 z8 + z9 ) − ( z1 + 2 z2 + z3 )
+ ( z3 + 2 z6 + z9 ) − ( z1 + 2 z4 + z7 )
Combining Spatial Enhancement
Methods

 An example
⚫ Laplacian to highlight fine detail
⚫ Gradient to enhance prominent edges
⚫ Smoothed version of the gradient
image used to mask the Laplacian
image
⚫ Increase the dynamic range of the gray
levels by using a gray-level
transformation
Image Enhancement in the
Frequency Domain

Dr. J.B.Jeeva
Associate Professor
Dept. of Sensor and Biomedical Technology
School of Electronics Engineering
VIT University, Vellore

Based on Digital Image Processing by Gonzalez

 Why Transform?
?
• Better image processing
o Take into account long-range correlations in space
o Conceptual insights in spatial-frequency information.
what it means to be “smooth, moderate change, fast
change, …”
• Fast computation: convolution vs. multiplication

• Alternative representation and sensing

o Obtain transformed data as measurement in radiology
images (medical and astrophysics), inverse transform to
recover image

• Efficient storage and transmission

o Energy compaction
o Pick a few “representatives” (basis)
o Just store/send the “contribution” from each basis
 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).
• 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 (Fourier transform).
Background

• The frequency domain

refers to the plane of
the two dimensional
discrete Fourier
transform of an image.
• The purpose of the
Fourier transform is to
represent a signal as a
linear combination of
sinusoidal signals of
various frequencies.
 1-D continuous FT
 1D – FT

real(g(x)) imag(g(x))

 1D – DFT of length N
 =0

 =7
 1-D DFT in as basis expansion

Forward
transform real(A) imag(A)
u=0

Inverse transform

basis

u=7

n n
1-D DFT in matrix notations

real(A) imag(A)
u=0

N=8

u=7

n n
 1-D DFT of different lengths

real(A) imag(A)
N=32
un

N=8

N=16 N=64
 Introduction to the Fourier Transform
and the Frequency Domain

 The one-dimensional Fourier transform and its

inverse
◦ Fourier transform (continuous case)

F (u)   f ( x)e j 2uxdx where j   1

◦ Inverse Fourier transform:

f ( x)   F (u)e j 2ux
du e j  cos  j sin 


 The two-dimensional Fourier transform and its

inverse
◦ Fourier transform (continuous case)
 
F (u, v)    f ( x, y)e j 2 (uxvy ) dxdy
 

Inverse Fourier transform:

 
f ( x, y)    F (u, v)e j 2 (uxvy ) dudv
 
 Introduction to the Fourier Transform
and the Frequency Domain

• The one-dimensional Fourier transform and its

inverse
o Fourier transform (discrete case) DTC
M 1
1
F (u ) 
M
 f ( x )e
x 0
 j 2ux / M
for u  0,1,2,..., M  1

o Inverse Fourier transform:

M 1
f ( x)   F (u )e j 2ux / M for x  0,1,2,..., M  1
u 0
 Introduction to the Fourier Transform
and the Frequency Domain

• Since e j  cos  j sin  and the fact cos( )  cos

then discrete Fourier transform can be redefined

M 1
1
F (u ) 
M
 f ( x)[cos 2ux / M  j sin 2ux / M ]
x 0
for u  0,1,2,..., M  1

o Frequency (time) domain: the domain (values of u)

over which the values of F(u) range; because u
determines the frequency of the components of
the transform.
o Frequency (time) component: each of the M
terms of F(u).
 Introduction to the Fourier Transform
and the Frequency Domain

• F(u) can be expressed in polar coordinates:

F (u )  F (u ) e j (u )

 
1
where F (u )  R (u )  I (u )
2 2 2 (magnitude or spectrum)
 I (u ) 
 (u )  tan  1
 (phase angle or phase spectrum)
 R(u ) 
o R(u): the real part of F(u)
o I(u): the imaginary part of F(u)
• Power spectrum:
P(u )  F (u )  R 2 (u )  I 2 (u )
2
The One-Dimensional Fourier Transform
Example
 The One-Dimensional Fourier Transform
Some Examples

• The transform of a constant function is a DC value

only.

• The transform of a delta function is a constant.

 The One-Dimensional Fourier Transform
Some Examples

• The transform of an infinite train of delta functions

spaced by T is an infinite train of delta functions
spaced by 1/T.

• The transform of a cosine function is a positive delta

at the appropriate positive and negative frequency.
 The One-Dimensional Fourier Transform
Some Examples
• The transform of a sine function is a negative
complex delta function at the appropriate positive
frequency and a negative complex delta at the
appropriate negative frequency.

• The transform of a square pulse is a sinc function.

 Introduction to the Fourier Transform
and the Frequency Domain

• The two-dimensional Fourier transform and its

inverse
o Fourier transform (discrete case) DTC
1 M 1 N 1
F (u , v)  
MN x 0 y 0
f ( x , y ) e  j 2 ( ux / M  vy / N )

for u  0,1,2,..., M  1, v  0,1,2,..., N  1

o Inverse Fourier 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
 Introduction to the Fourier Transform
and the Frequency Domain

• We define the Fourier spectrum, phase angle, and

power spectrum as follows:

 
1
F (u, v)  R (u, v)  I (u, v)
2 2 2 ( spectrum)
 I (u, v) 
 (u, v)  tan 1   (phase angle)
 R(u, v) 
P(u,v)  F (u, v)  R 2 (u, v)  I 2 (u, v) (power spectrum)
2

o R(u,v): the real part of F(u,v)

o I(u,v): the imaginary part of F(u,v)
 Introduction to the Fourier Transform
and the Frequency Domain

• Some properties of Fourier transform:


 f ( x, y )( 1) x y
 M N
 F (u  , v  ) (shift)
2 2
M 1 N 1
1
F (0,0) 
MN
 f ( x, y)
x 0 y 0
(average)

F (u, v)  F * (u,v) (conujgate symmetric)

F (u, v)  F (u,v) (symmetric )
 The Two-Dimensional DFT and Its Inverse

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)

(a)f(x,y) (b)F(u,y) (c)F(u,v)

The Two-Dimensional DFT and Its Inverse

shift
The Two-Dimensional DFT and Its Inverse
The Property of Two-Dimensional DFT
Rotation

DFT

DFT
 The Property of Two-Dimensional DFT
Linear Combination

A
DFT

B
DFT

0.25 * A
+ 0.75 * B
DFT
 The Property of Two-Dimensional DFT
Expansion

A
DFT

B DFT

Expanding the original image by a factor of n (n=2), filling

the empty new values with zeros, results in the same DFT.
 Two-Dimensional DFT with Different Functions

Sine wave Its DFT

Rectangle
Its DFT
 Two-Dimensional DFT with Different Functions

2D Gaussian Its DFT

function

Impulses
Its DFT
Summary of Some Important Properties
of the 2-D Fourier Transform
Summary of Some Important Properties
of the 2-D Fourier Transform