2. 2
Contents
Fourier Transform and DFT
Walsh Transform
Hadamard Transform
Walsh-Hadamard Transform (WHT)
Discrete Cosine Transform (DCT)
Haar Transform
Slant Transform
Comparison of various Transforms
3. 3
Introduction
Although we discuss other transforms in
some detail in this chapter, we emphasize
the Fourier transform because of its wide
range of applications in image processing
problems.
12. 12
Discrete Fourier Transform
% Program written in Matlab for computing FFT of a given gray color image.
% Clear the memory.
clear;
% Getting the name and extension of the image file from the user.
name=input('Please write the name and address of the image : ','s');
% Reading the image file into variable 'a'.
a=imread(name);
% Computing the size of image. Assuming that image is squared.
N=length(a);
% Computing DFT of the image file by using fast Fourier algorithm.
F=fft2(double(a))/N;
MATLAB program page 1 from 3.
13. 13
Discrete Fourier Transform
% Shifting the Fourier spectrum to the center of the frequency square.
for i=1:N/2; for j=1:N/2
G(i+N/2,j+N/2)=F(i,j);
end;end
for i=N/2+1:N; for j=1:N/2
G(i-N/2,j+N/2)=F(i,j);
end;end
for i=1:N/2; for j=N/2+1:N
G(i+N/2,j-N/2)=F(i,j);
end;end
for i=N/2+1:N; for j=N/2+1:N
G(i-N/2,j-N/2)=F(i,j);
end;end
MATLAB program page 2 from 3.
14. 14
Discrete Fourier Transform
% Computing and scaling the logarithmic Fourier spectrum.
H=log(1+abs(G));
for i=1:N
H(i,:)=H(i,:)*255/abs(max(H(i,:)));
end
% Changing the color map to gray scale (8 bits).
colormap(gray(255));
% Showing the main image and its Fourier spectrum.
subplot(2,2,1),image(a),title('Main image');
subplot(2,2,2),image(abs(G)),title('Fourier spectrum');
subplot(2,2,3),image(H),title('Logarithmic scaled Fourier spectrum');
MATLAB program page 3 from 3.
15. 15
Discrete Fourier Transform
(Properties)
Separability
The discrete Fourier transform pair can be expressed in the seperable forms:
( ) [ ] ( ) [ ]
( ) [ ] ( ) [ ]∑∑
∑∑
−
=
−
=
−
=
−
=
ππ=
π−π−=
1N
0v
1N
0u
1N
0y
1N
0x
N/vy2jexpv,uFN/ux2jexp
N
1
y,xf
N/vy2jexpy,xfN/ux2jexp
N
1
v,uF
16. 16
Discrete Fourier Transform
(Properties)
Translation
( ) ( )[ ] ( )
( ) ( ) ( )[ ]N/vyux2jexpv,uFyy,xxf
and
vv,uuFN/yvxu2jexpy,xf
0000
0000
+π−⇔−−
−−⇔+π
The translation properties of the
Fourier transform pair are :
17. 17
Discrete Fourier Transform
(Properties)
Periodicity
The discrete Fourier transform and its
inverse are periodic with period N; that is,
F(u,v)=F(u+N,v)=F(u,v+N)=F(u+N,v+N)
If f(x,y) is real, the Fourier transform also
exhibits conjugate symmetry:
F(u,v)=F*
(-u,-v)
Or, more interestingly:
|F(u,v)|=|F(-u,-v)|
18. 18
Discrete Fourier Transform
(Properties)
Rotation
If we introduce the polar coordinates
ϕω=ϕω=
θ=θ=
sinvcosu
sinrycosrx
Then we can write:
( ) ( )00 ,F,rf θ+φω⇔θ+θ
In other words, rotating F(u,v)
rotates f(x,y) by the same angle.
19. 19
Discrete Fourier Transform
(Properties)
Convolution
The convolution theorem in
two dimensions is expressed
by the relations :
( ) ( ) ( )
( ) ( ) ( ) ( )v,uG*v,uFy,xgy,xf
and
v,uGv,uF)y,x(g*y,xf
⇔
⇔
Note :
( ) ( ) ( ) ( )∫ ∫
∞
∞−
∞
∞−
βαβ−α−βα= ddy,xg,fy,xg*y,xf
20. 20
Discrete Fourier Transform
(Properties)
Correlation
The correlation of two continuous
functions f(x) and g(x) is defined
by the relation
( ) ( ) ( ) ( ) αα+α= ∫
∞
∞−
dxgfxgxf *
So we can write:
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )v,uGv,uFy,xgy,xf
and
v,uGv,uFy,xgy,xf
*
*
⇔
⇔
21. 21
Discrete
Fourier
Transform
Sampling
(Properties)
1-D
The Fourier transform and the convolution theorem provide the
tools for a deeper analytic study of sampling problem. In particular, we
want to look at the question of how many samples should be taken so that
no information is lost in the sampling process. Expressed differently, the
problem is one of the establishing the sampling conditions under which a
continuous image can be recovered fully from a set of sampled values. We
begin the analysis with the 1-D case.
As a result, a function which is band-limited in frequency domain
must extend from negative infinity to positive infinity in time domain (or x
domain).
22. 22
Discrete
Fourier
Transform
Sampling
(Properties)
1-D
f(x) : a given function
F(u): Fourier Transform of f(x)
which is band-limited
s(x) : sampling function
S(u): Fourier Transform of s(x)
G(u): window for recovery of
the main function F(u) and f(x).
Recovered f(x) from sampled data
23. 23
Discrete
Fourier
Transform
Sampling
(Properties)
1-D
f(x) : a given function
F(u): Fourier Transform of f(x)
which is band-limited
s(x) : sampling function
S(u): Fourier Transform of s(x)
h(x): window for making f(x)
finited in time.
H(u): Fourier Transform of h(x)
25. 25
Discrete
Fourier
Transform
Sampling
(Properties)
2-D
The sampling process for 2-D
functions can be formulated
mathematically by making use
of the 2-D impulse function
δ(x,y), which is defined as
( ) ( ) ( )0000 y,xfdydxyy,xxy,xf =−−δ∫ ∫
∞
∞−
∞
∞−
A 2-D sampling function is
consisted of a train of impulses
separated Δx units in the x
direction and Δy units in the y
direction as shown in the figure.
26. 26
Discrete
Fourier
Transform
Sampling
(Properties)
2-D
If f(x,y) is band limited (that is, its
Fourier transform vanishes outside
some finite region R) the result of
covolving S(u,v) and F(u,v) might
look like the case in the case
shown in the figure. The function
shown is periodic in two
dimensions.
( )
=
0
1
v,uG
(u,v) inside one of the rectangles
enclosing R
elsewhere
The inverse Fourier transform of
G(u,v)[S(u,v)*F(u,v)] yields f(x,y).
28. 28
Other Seperable Image Transforms
For 2-D square arrays the forward and inverse transforms are
( ) ( ) ( )
( ) ( ) ( )∑∑
∑∑
−
=
−
=
−
=
−
=
=
=
1N
0u
1N
0v
1N
0x
1N
0y
v,u,y,xhv,uTy,xf
and
v,u,y,xgy,xfv,uT g(x,y,u,v) : forward transformation kernel
h(x,y,u,v) : inverse transformation kernel
The forward kernel is said to be seperable if
g(x,y,u,v)=g1(x,u)g2(y,v)
In addition, the kernel is symmetric if g1 is functionally equal to g2. In this case
we can write:
g(x,y,u,v)=g1(x,u)g1(y,v)
29. 29
Other Seperable Image Transforms
AFAT =
BTBF
AB
BAFABBTB
1
=⇒
=
=
−
Where F is the N×N image matrix,
A is an N×N symmetric transformation matrix
T is the resulting N×N transform.
If the kernel g(x,y,u,v) is seperable and symmetric,
( ) ( ) ( )∑∑
−
=
−
=
=
1N
0x
1N
0y
v,u,y,xgy,xfv,uT
also may be expressed in matrix form:
And for inverse transform we have:
30. 30
Walsh Transform
When N=2n
, the 2-D forward and inverse Walsh kernels are given by the relations
( ) ( ) ( ) ( ) ( ) ( )[ ]
( ) ( ) ( ) ( ) ( ) ( )[ ]
∏∏
−
=
+
−
=
+ −−−−−−−−
−=−=
1n
0i
vbybubxb
1n
0i
vbybubxb i1nii1nii1nii1ni
1
N
1
v,u,y,xhand1
N
1
v,u,y,xg
Where bk(z) is the kth bit in the binary representation of z.
So the forward and inverse Walsh transforms are equal in form; that is:
31. 31
Walsh Transform
“+” denotes for +1 and “-” denotes for -1.
( ) ( ) ( ) ( )
∏
−
=
−−
−=
1n
0i
ubxb i1ni
1
N
1
u,xgIn 1-D case we have :
In the following table N=8 so n=3 (23
=8).
1-D kernel
32. 32
Walsh Transform
This figure shows the basis functions (kernels) as
a function of u and v (excluding the 1/N constant
term) for computing the Walsh transform when
N=4. Each block corresponds to varying x and y
form 0 to 3 (that is, 0 to N-1), while keeping u
and v fixed at the values corresponding to that
block. Thus each block consists of an array of
4×4 binary elements (White means “+1” and
Black means “-1”). To use these basis functions
to compute the Walsh transform of an image of
size 4×4 simply requires obtaining W(0,0) by
multiplying the image array point-by-point with
the 4×4 basis block corresponding to u=0 and
v=0, adding the results, and dividing by 4, and
continue for other values of u and v.
34. 34
Walsh Transform (WT)
% Program written in Matlab for computing WT of a given gray color image.
clear;
% Getting the name and extension of the image file from the user.
name=input('Please write the name and address of the image : ','s');
a=imread(name);
N=length(a);
% Computing Walsh Transform of the image file.
n=log2(N);n=1+fix(n);f=ones(N,N);
for x=1:N; for u=1:N
p=dec2bin(x-1,n); q=dec2bin(u-1,n);
for i=1:n; f(x,u)=f(x,u)*((-1)^(p(n+1-i)*q(i)));
end;end;end
F=(1/N)*f*double(a)*f;
MATLAB program page 1 from 3.
35. 35
Walsh Transform (WT)
% Shifting the Fourier spectrum to the center of the frequency square.
for i=1:N/2; for j=1:N/2
G(i+N/2,j+N/2)=F(i,j);
end;end
for i=N/2+1:N; for j=1:N/2
G(i-N/2,j+N/2)=F(i,j);
end;end
for i=1:N/2; for j=N/2+1:N
G(i+N/2,j-N/2)=F(i,j);
end;end
for i=N/2+1:N; for j=N/2+1:N
G(i-N/2,j-N/2)=F(i,j);
end;end
MATLAB program page 2 from 3.
36. 36
Walsh Transform (WT)
% Computing and scaling the logarithmic Walsh spectrum.
H=log(1+abs(G));
for i=1:N
H(i,:)=H(i,:)*255/abs(max(H(i,:)));
end
% Changing the color map to gray scale (8 bits).
colormap(gray(255));
% Showing the main image and its Walsh spectrum.
subplot(2,2,1),image(a),title('Main image');
subplot(2,2,2),image(abs(G)),title('Walsh spectrum');
subplot(2,2,3),image(H),title('Logarithmic scaled Walsh spectrum');
MATLAB program page 3 from 3.
37. 37
Hadamard Transform
When N=2n
, the 2-D forward and inverse Hadamard kernels are given by the relations
( ) ( ) ( ) ( ) ( ) ( )[ ]
( ) ( ) ( ) ( ) ( ) ( )[ ]
∏∏
−
=
+
−
=
+
−=−=
1n
0i
vbybubxb
1n
0i
vbybubxb iiiiiiii
1
N
1
v,u,y,xhand1
N
1
v,u,y,xg
Where bk(z) is the kth bit in the binary representation of z.
So the forward and inverse Hadamard transforms are equal in form; that is:
38. 38
Hadamard Transform
( ) ( ) ( ) ( )
∏
−
=
−=
1n
0i
ubxb ii
1
N
1
u,xgIn 1-D case we have :
In the following table N=8 so n=3 (23
=8).
1-D kernel
“+” denotes for +1 and “-” denotes for -1.
39. 39
Hadamard Transform
This figure shows the basis functions
(kernels) as a function of u and v (excluding
the 1/N constant term) for computing the
Hadamard transform when N=4. Each block
corresponds to varying x and y form 0 to 3
(that is, 0 to N-1), while keeping u and v
fixed at the values corresponding to that
block. Thus each block consists of an array
of 4×4 binary elements (White means “+1”
and Black means “-1”) like Walsh
transform. If we compare these two
transforms we can see that they only differ
in the sense that the functions in Hadamard
transform are ordered in increasing sequency
and thus are more “natural” to interpret.
41. 41
Hadamard Transform (HT)
% Program written in Matlab for computing HT of a given gray color image.
clear;
% Getting the name and extension of the image file from the user.
name=input('Please write the name and address of the image : ','s');
a=imread(name);
N=length(a);
% Computing Hadamard Transform of the image file.
n=log2(N);n=1+fix(n);f=ones(N,N);
for x=1:N; for u=1:N
p=dec2bin(x-1,n); q=dec2bin(u-1,n);
for i=1:n; f(x,u)=f(x,u)*((-1)^(p(n+1-i)*q(n+1-i)));
end;end;end
F=(1/N)*f*double(a)*f;
MATLAB program page 1 from 3.
42. 42
Hadamard Transform (HT)
% Shifting the Fourier spectrum to the center of the frequency square.
for i=1:N/2; for j=1:N/2
G(i+N/2,j+N/2)=F(i,j);
end;end
for i=N/2+1:N; for j=1:N/2
G(i-N/2,j+N/2)=F(i,j);
end;end
for i=1:N/2; for j=N/2+1:N
G(i+N/2,j-N/2)=F(i,j);
end;end
for i=N/2+1:N; for j=N/2+1:N
G(i-N/2,j-N/2)=F(i,j);
end;end
MATLAB program page 2 from 3.
43. 43
Hadamard Transform (HT)
% Computing and scaling the logarithmic Hadamard spectrum.
H=log(1+abs(G));
for i=1:N
H(i,:)=H(i,:)*255/abs(max(H(i,:)));
end
% Changing the color map to gray scale (8 bits).
colormap(gray(255));
% Showing the main image and its Hadamard spectrum.
subplot(2,2,1),image(a),title('Main image');
subplot(2,2,2),image(abs(G)),title('Hadamard spectrum');
subplot(2,2,3),image(H),title('Logarithmic scaled Hadamard
spectrum');
MATLAB program page 3 from 3.
50. 50
Discrete Cosine Transform (DCT)
Each block consists
of 4×4 elements,
corresponding to x
and y varying from 0
to 3. The highest
value is shown in
white. Other values
are shown in grays,
with darker meaning
smaller.
52. 52
Discrete Cosine Transform
% Program written in Matlab for computing DCT of a given gray color image.
% Clear the memory.
clear;
% Getting the name and extension of the image file from the user.
name=input('Please write the name and address of the image : ','s');
% Reading the image file into variable 'a'.
a=imread(name);
% Computing the size of image. Assuming that image is squared.
N=length(a);
% Computing DCT of the image file.
F=dct2(double(a));
MATLAB program page 1 from 3.
53. 53
Discrete Cosine Transform
% Shifting the Fourier spectrum to the center of the frequency square.
for i=1:N/2; for j=1:N/2
G(i+N/2,j+N/2)=F(i,j);
end;end
for i=N/2+1:N; for j=1:N/2
G(i-N/2,j+N/2)=F(i,j);
end;end
for i=1:N/2; for j=N/2+1:N
G(i+N/2,j-N/2)=F(i,j);
end;end
for i=N/2+1:N; for j=N/2+1:N
G(i-N/2,j-N/2)=F(i,j);
end;end
MATLAB program page 2 from 3.
54. 54
Discrete Cosine Transform
% Computing and scaling the logarithmic Cosine spectrum.
H=log(1+abs(G));
for i=1:N
H(i,:)=H(i,:)*255/abs(max(H(i,:)));
end
% Changing the color map to gray scale (8 bits).
colormap(gray(255));
% Showing the main image and its Cosine spectrum.
subplot(2,2,1),image(a),title('Main image');
subplot(2,2,2),image(abs(G)),title('Cosine spectrum');
subplot(2,2,3),image(H),title('Logarithmic scaled Cosine spectrum');
MATLAB program page 3 from 3.
60. 60
Haar Transform
The Haar transform is based on the Haar functions, hk(z), which are defined over the
continuous, closed interval [0,1] for z, and for k=0,1,2,…,N-1, where N=2n
. The first
step in generating the Haar transform is to note that the integer k can be decomposed
uniquely as k=2p
+q-1
where 0≤p≤n-1, q=0 or 1 for p=0, and 1≤q≤2p
for p≠0.
With this background, the Haar functions are defined as
( ) ( ) [ ]
( ) ( )
[ ]
∈
≤
−
−
−
≤
−
==
∈==
∆
∆
1,0zforotherwise0
2
q
z
2
2/1q
2
2
2/1q
z
2
1q
2
N
1
zhzh
and
1,0zfor
N
1
zhzh
pp
2/p
pp
2/p
00k
000
64. 64
Haar Transform
MATLAB program page 1 from 1.
% Program written in Matlab for computing HT of a given gray color image.
% Getting the name and extension of the image file from the user.
name=input('Please write the name and address of the image : ','s');
a=imread(name); N=length(a);
% Computing Haar Transform of the image file.
for i=1:N p=fix(log2(i)); q=i-(2^p);
for j=1:N z=(j-1)/N;
if (z>=(q-1)/(2^p))&&(z<(q-1/2)/2^p) f(i,j)=(1/(sqrt(N)))*(2^(p/2));
elseif (z>=(q-1/2)/(2^p))&&(z<(q/2^p)) f(i,j)=(1/(sqrt(N)))*(-(2^(p/2)));
else f(i,j)=0;
end;end;end
F=f*double(a)*f;
% Changing the color map to gray scale (8 bits).
colormap(gray(255));
% Showing the main image and its Hadamard spectrum.
subplot(2,2,1),image(a),title('Main image');
subplot(2,2,2),image(abs(F)),title('Haar spectrum');
68. 68
Slant Transform
MATLAB program page 1 from 1.
% Program written in Matlab for computing ST of a given gray color image.
% Getting the name and extension of the image file from the user.
name=input('Please write the name and address of the image : ','s');
a=imread(name); N=length(a);
% Computing Slant Transform of the image file.
A=[ 1/(2^0.5) 1/(2^0.5);1/(2^0.5) -1/(2^0.5)];
for i=2:log2(N) N=2^i; aN=((3*(N^2))/(4*((N^2)-1)))^0.5;
bN=(((N^2)-4)/(4*((N^2)-1)))^0.5;
m=1/(2^0.5)*[1 0 zeros(1,(N/2)-2) 1 0 zeros(1,(N/2)-2)
aN bN zeros(1,(N/2)-2) -aN bN zeros(1,(N/2)-2)
zeros((N/2)-2,2) eye((N/2)-2) zeros((N/2)-2,2) eye((N/2)-2)
0 1 zeros(1,(N/2)-2) 0 -1 zeros(1,(N/2)-2)
-bN aN zeros(1,(N/2)-2) bN aN zeros(1,(N/2)-2)
zeros((N/2)-2,2) eye((N/2)-2) zeros((N/2)-2,2) -eye((N/2)-2)];
n=[A A-A;A-A A]; A=m*n;
end
F=A*double(a)*A;
% Changing the color map to gray scale (8 bits).
colormap(gray(255));
% Showing the main image and its Hadamard spectrum.
subplot(2,2,1),image(a),title('Main image');
subplot(2,2,2),image(abs(F)),title('Slant spectrum');
73. 73
References
“Digital image processing” by Rafael C.
Gonzalez and Richard E. Woods
“Digital image processing” by Jain
Image communication I by Bernd Girod
Lecture 3, DCS339/AMCM053 by Pengwei
Hao, University of London
