The document discusses digital image processing and two-dimensional transforms. It provides an agenda that covers two-dimensional mathematical preliminaries and two transforms: the discrete Fourier transform (DFT) and discrete cosine transform (DCT). It then discusses the DFT and DCT in more detail over several pages, covering properties, examples, and applications such as image compression.

1. Digital Image Processing
Day 4 : 25/6/2020
Agenda :
Two Dimensional Mathematical Preliminaries
2D transforms
1) DFT- Discrete Fourier Transform
2) DCT - Discrete Cosine Transform

2. Two Dimensional
Mathematical Preliminaries
Image Transforms
Many times, image processing tasks are best performed in a domain other than
the spatial domain.
Key steps:
(1) Transform the image
(2)Carry the task(s) in the transformed domain.
(3)Apply inverse transform to return to the spatial domain
First Video

3. Fourier Series Theorem
 Anyperiodic function f(t) can beexpressed as aweighted sum (infinite) of sine
and cosine functions of varying frequency
is called the “fundamentalfrequency

4. Discrete Fourier Transform
(DFT)

5. Discrete Cosine Transform
 A discrete cosine transform (DCT) expresses a finite sequence of
data
points in terms of a sum of cosine functions oscillating at different frequencies.
 DCT is a Fourier-related transform similar to the discrete Fourier transform
(DFT), but using only real numbers. DCTs are equivalent to DFTs of roughly
twice the length, operating on real data with even symmetry. Types of DCT
listed below with 11 samples.

7. Step Function
The unit step function, is a discontinuous function whose value
is zero for negative argument and one for positive argument.
A-A

t  0
0
t  0
H(t) 
0
Rectangle Function
The unit step function, is a discontinuous function whose value
is zero for negative argument and one for positive argument. 0 t  A
 A t  A
Rect(t)  
1
A-A
 t A0
Impulse Function
The Dirac delta function, or δ function, is zero for all values of
the parameter except when the parameter is zero, and its integral
over the parameter from −∞ to ∞ is equal to one

0
t 0
t 0
(t) 

(t)dt 1

© 1992–2008 R. C. Gonzalez & R. E. Woods

8. Fourier Transform
For a give function g(x) of a real variable x, the
Fourier transformation of g(x) which is denoted
as

 j2ux
-
{g(x)} G(u) g(x)e dx


-
1
{g(x)} G(u) g(x)e dxj2ux
Second Video

9. Since
e j2t
 sin(2t) jcos(2t)
For that reason the Fourier transform F(u)
F(u) = R(u) + jI(u)
Which often appear as
F(u) | F(u) | e j(u)
| F(u) | R(u)2
 I (u)2
R(u)
© 1992–2008 R. C. Gonzalez & R. E. Woods
tan( (u)) 
I(u)

10. .
Spatial Domain Frequency Domain

11. 2D Fourier Transform
For a give function g(x) of a real variable x, the
Fourier transformation of g(x) which is denoted
as

{g(x, y)}  G(u, v)    g(x, y)e j 2 (uxvy )
dxdy
and

1
{G(u,v)}  g(x, y)  G(u,v)ej2 (uxvy)
dudv

© 1992–2008 R. C. Gonzalez & R. E. Woods

12.  


AA
X
0
 1]
  0 
vY

14. Discrete Fourier Transform
•Let us discretize a continuous function f(x) into the N uniform
samples that generate the sequence
f(x0), f(x0+Δx), f(x0+2Δx), f(x0+3Δx), …, f(x0+[N-1]Δx)
•Hence f(x) = f(x0+i Δx)
•We could denote the samples as f(0), f(1), f(2), …,f(N-1).
x0
and
0,1,2,..., N 1
N 1
1
 f(x)e j 2ux / N
;u
N
• The Fourier Transform is
F (u)
N1
Third Video - Examples
f (x)  F(u)e j2ux/ N
u0

15. NM
2D Discrete Fourier Transform
1 M 1N 1
F(u,v)   f (x, y)e j2 (ux/ M vy / N )
x0 y 0
u  0,1,2,...,M 1;v  0,1,2,...,N 1
and
M 1N1
1 1
f (x, y)   F(u,v)ej2 (ux/ M vy / N )
u 0 v0
Mx Ny
u  ; v

16. .

18. Examples

19. Properties of 2D FourierTransform
• Spatial and Frequency Domain
f(t, z) sampled from f(x, y) using the separation between samples as T and Z
NZ
v 
MT
1
u 
1
• Translation and Rotation
Multiplying f(x,y) by the exponential shifts the original of DFT to (u0,v0).
Multiplying F(u,v) by the exponential shifts the original of f(x, y) to (x0,y0);
f (x  x , y  y )  F (u, v)e j 2 ( x0u / M  y0v / N )
0 0
F (u  u , v v )  f (x, y)e j 2 (u0 x / M v0 y / N )
0 0

20. Properties of 2D Fourier Transform
Periodicity
The Fourier transform and inverse are
infinitely periodic on the u and v directions.
(k1 and k2 are integers).
F (u, v)  F (u  k1M , v)  F (u, v  k2 N )
 F (u  k1M , v  k2 N )
f (x, y)  f (x  k1M , y)  f (x, y  k2 N )
 f (x  k1M , y  k2 N )
To show the origin of F(u,v) at the center we
shift the data by M/2 and N/2
f (x, y)(1)x y
 F (u  M / 2, v  N / 2)

21. Properties of 2D Fourier Transform Symmetry
Any real or complex function can be expressed as the sum of even and odd part
w(x, y)  we (x, y)  wo (x, y)
2
w(x, y)  w(x,y)
ew (x, y)

2
e
w(x, y)  w(x,y)
w (x, y) 

Which shows that even functions are symmetric and odd functions are
antisymmetric
we (x, y)  we (x, y)
wo (x, y)  wo (x, y)

22. Properties of 2D Fourier Transform
Symmetry
The Fourier transform of a real function f(x,y) is conjugate symmetric
F*
(u,v)  F(u,v)
The Fourier transform of a imaginary function f(x,y) is conjugate anti-symmetric
F*
(u,v)  F(u,v)
1 
*
x0 y 0NM 

F*
(u,v)  
M 1N1
 f (x, y)e j2 (ux / M vy/ N )
Proof
NM x0 y 0
1 N1

1 M
 f *
(x, y)ej2 (ux/ M vy/ N )
1 N1

1 M
 f (x, y)e j2 ([u]x/ M [v]y / N )
NM
 F(u,v)
x0 y 0

23. Digital Image Processing, 3rd ed.
© 1992–2008 R. C.
Gonzalez & R. E. Woods

25. Digital Image Processing, 3rd ed.
© 1992–2008 R. C.
Gonzalez & R. E. Woods

27. Digital Image Processing, 3rd ed.
© 1992–2008 R. C.
Gonzalez & R. E. Woods

28. Sampling and Fourier Transform
1.Converting continuous function/signal into a discrete one.
2.The sampling is uniform at T intervals The sampled function
and the value of each Sample are :
~
f (t)f (t)  f (t)s (t)  

(t  nT)T
n

fk   f (t) (t  kT )dt  f (kT)


29. Sampling Theorem
Band limited-A function f(t) whose Fourier transform is zero out
of the interval [-max , max] is called band limited
~
We can recover a function f(t) from its sampled
representation if we can isolate a copy of F() from
the periodic sequence of copies.
Extracting from F() a single period that represents
F() is possible in the separation between copies is
sufficient , which is guaranteed if ½T > max
max 2
1
T
Which is call the Nyquist Rate
© 1992–2008 R. C. Gonzalez & R. E. Woods

30. Digital Image Processing, 3rd ed.
© 1992–2008 R. C.
Gonzalez & R. E. Woods

31. .
Application of Filters H(U,V)

33. Digital Image Processing, 3rd ed.
© 1992–2008 R. C.
Gonzalez & R. E. Woods
Zero Paddding

34. Digital Image Processing, 3rd ed.
© 1992–2008 R. C.
Gonzalez & R. E. Woods

35. Digital Image Processing, 3rd ed.
© 1992–2008 R. C.
Gonzalez & R. E. Woods

36. Digital Image Processing, 3rd ed.
© 1992–2008 R. C.
Gonzalez & R. E. Woods
Laplacian Filter

37. Digital Image Processing, 3rd ed.
© 1992–2008 R. C.
Gonzalez & R. E. Woods
Gaussian Filter

39. Discrete Fourier Transform
2
• we have studied the DFT
• due to its computational efficiency the DFT is very
popular
• however, it has strong disadvantages for some
applications
– it is complex
– it has poor energy compaction
• energy compaction
– is the ability to pack the energy of the spatial sequence into as
few frequency coefficients as possible
– this is very important for image compression
– we represent the signal in the frequency domain
– if compaction is high, we only have to transmit a few coefficients
– instead of the whole set of pixels

40. Discrete Cosine Transform
• A much better transform,
from this point of view, is the DCT
– in this example we see the
amplitude spectra of the image above
– under the DFT and DCT
– note the much more concentrated
histogram obtained with the DCT
• why is energy compaction
important?
– the main reason is image
compression
– turns out to be beneficial in other
applications
3

42. Image compression
• An image compression system has three main blocks
– a transform (usually DCT on 8x8 blocks)
– a quantizer
– a lossless (entropy) coder
• each tries to throw away information which is
not essential to understand the image, but
costs bits

45. • The transform throws away correlations
– if you make a plot of the value of a pixel as a function of one of its
neighbors
– you will see that the pixels are highly correlated (i.e. most of the
time they are very similar)
– this is just a consequence of the fact that surfaces are smooth
Image compression

46. Image compression
• a second advantage of working in the
frequency domain
– is that our visual system is less sensitive
to distortion around edges
– the transition associated with the edge
masks our ability to perceive the noise
– e.g. if you blow up a compressed picture,
it is likely to look like this
– in general, the compression errors are
more annoying in the smooth image
regions
4
6

47. Image compression
• three JPEG examples
36KB 5.7KB 1.7KB
– note that the blockiness is more visible in the torso

48. Image compression
• important point:
– does not save any bits
– does not introduce any distortion
• both of these happen when we throw away information
• this is called “lossy compression” and implemented by
the quantizer
• what is a quantizer?
– think of the round() function, that rounds to the nearest integer
– round(1) = 1; round(0.55543) = 1; round (0.0000005) = 0
– instead of an infinite range between 0 and 1 (infinite number of
bits to transmit)
– the output is zero or one (1 bit)
– we threw away all the stuff in between, but saved a lot of bits
– a quantizer does this less drastically

49. Quantizer
• it is a function of this type
– inputs in a given range are mapped
to the same output
• to implement this, we
– 1) define a quantizer step size Q
– 2) apply a rounding function
 x



Q
x  roundq
– the larger the Q, the less reconstruction levels we have
– more compression at the cost of larger distortion
– e.g. for x in [0,255], we need 8 bits and have 256 color values
– 4 levels and only need 2 bits

50. Quantizer
• note that we can quantize some frequency coefficients
more heavily than others by simply increasing Q
• this leads to the idea of a quantization matrix
• we start with an image block (e.g. 8x8 pixels)

51. Quantizer
• next we apply a transform (e.g. 8x8 DCT)
DCT

52. Quantizer
• and quantize with a varying Q
DCT
Q mtx

53. Quantizer
• note that higher frequencies are quantized more heavily
Q mtx
increasing frequency
– in result, many high frequency coefficients are simply wiped out
DCT quantized DCT

54. Quantizer
• this saves a lot of bits, but we no longer have an exact
replica of original image block
DCT quantized DCT
inverse DCT original pixels


55. Quantizer
• note, however, that visually the blocks are not very
different
original decompressed
– we have saved lots of bits without much “perceptual” loss
– this is the reason why JPEG and MPEG work

56. Image compression
• three JPEG examples
36KB 5.7KB 1.7KB
– note that the two images on the left look identical
– JPEG requires 6x less bits

57. Energy compaction
• The two extensions are
DFT DCT
– note that in the DFT case the extension introduces
discontinuities
– this does not happen for the DCT, due to the symmetry of y[n]
– the elimination of this artificial discontinuity, which contains a lot
of high frequencies,
– is the reason why the DCT is much more efficient

58. 2D-DCT
n2n2
1D-DCT
• 1) create
intermediate
sequence by
computing
1D-DCT of
rows
• 2) compute
k1
f [ k1 , n 2 ]
n1
x[ n1 , n 2 ]
1D-DCT of
columns
n2
k2
1D-DCT
k1
f [ k 1 , n 2 ]
5
8
k1
C x [ k 1 , k 2 ]
• the extension to 2D is trivial
• the procedure is the same