Digital image transforms
Digital image processing
4. DIGITAL IMAGE TRANSFORMS
4.1. Introduction
4.2. Unitary orthogonal two-dimensional transforms
Separable unitary transforms
4.3. Properties of the unitary transforms
Energy conservation
Energy compaction; the variance of coefficients
De-correlation
Basis functions and basis images
4.4. Sinusoidal transforms
The 1-D discrete Fourier transform (1-D DFT)
Properties of the 1-D DFT
The 2-D discrete Fourier transform (2-D DFT)
Properties of the 2-D DFT
The discrete cosine transform (DCT)
The discrete sine transform (DST)
The Hartley transform
4.5. Rectangular transforms
The Hadamard transform = the Walsh transform
The Slant transform
The Haar transform
4.6. Eigenvectors-based transforms
The Karhunen-Loeve transform (KLT)
The fast KLT
The SVD
4.7. Image filtering in the transform domain
4.8. Conclusions
Digital image transforms
Digital image processing
4.1 INTRODUCTION
Definition: Image transform = operation to change the default representation
space of a digital image (spatial domain -> another domain) so that:
(1) all the information present in the image is preserved in the
transformed domain, but represented differently;
(2) the transform is reversible, i.e., we can revert to the spatial domain
Generally: in the transformed domain -> image information is represented in a
more compact form => main goal of the transforms: image compression.
Other usage: image analysis - a new type of representation of different types
of information present in the image.
Note: Most image transforms = “generalizations” of frequency transforms => the
representation of the image by a DC component and several AC components.
Definition: “original representation space” of the image U[M×N] = a MNdimensional space:
- each coordinate of the space = a spatial location (m,n) in the digital
image;
- the value of the coordinate of U on an axis = the grey level in U in this
spatial location (m,n).
x1=(0,0); x2=(0,1); x3=(0,2); ... xMN=(M-1,N-1).
=> A unitary transform of the image U = a rotation of the MN-dimensional space,
defined by a rotation matrix A in MN-dimensions.
Digital image transforms
Digital image processing
0  n  N  1} ; A – unitary matrix, A1  A*T
{u(n),
v  Au , or
u  A*T v
N 1
v(k)   a(k,n)u(n), 0  k  N  1
n 0
N 1
*
(4.1)
or u(n)  a (k,n)v(k), 0  n  N  1
k 0
*
(4.2)
a*k  {a (k, n) , 0  n  N  1 } ,
4.2 UNITARY ORTHOGONAL TWO-DIMENSIONAL TRANSFORMS
N 1 N 1
v(k,l)    ak,l (m,n) u(m,n), 0  k,l  N  1
(4.3)
m0 n 0
N 1 N 1
u(m,n)    a*k,l (m,n)  v(k,l), 0  k,l  N  1
(4.4)
k 0 l 0
 orthonormality:
 completeness:
N 1 N 1
*
  a k,l (m, n) a k ',l' (m, n)  (k  k' , l  l' )
m 0 n 0
N 1 N 1
a
k ,l
( m, n )  ak*,l ( m', n' )   ( m  m', n  n' )
(4.5)
(4.6)
k 0 l 0
P 1 Q 1
uP,Q (m,n)    a*k,l (m,n)  v(k,l), P  N,Q  N
(4.7)
k 0 l 0
N 1 N 1
 2e    [u(m,n)  uP,Q (m,n)] 2
m 0 n  0
 e2  0 if
PQ N
(4.8)
Digital image transforms
Digital image processing
 Unitary separable transforms
ak ,l ( m, n)  ak ( m) b1 ( n)  a( k , m) b( l , n)
(4.9)
where {ak (m),k  0,...,N  1} and {b1(n),l  0,...,N  1} are the
orthonormal sets of basis vectors.
AA*T  AT A*  I, BB*T  BTB*  I
N 1 N 1
v(k,l)    a(k,m)u(m,n) a(n,l),or V  AUAT
m0 n 0
N 1N 1
u(m,n)    a* (k,m) v(k,l) a* (l,n), or U  A*TVA*
n 0 l 0
(4.10)
(4.11)
(4.12)
V  A M UA N
(4.13)
*T
U  A*T
M VA N
(4.14)
V  AUA T , V T  A[AU]T
(4.15)
Digital image transforms
Digital image processing
4.3 PROPERTIES OF UNITARY TRANSFORMS
 Energy conservation
N 1
N 1
v   v(k)  v  v  u A  A  u  u  u   u( n )  u
2
2
*T
*T
*T
2
*T
k 0
2
(4.16)
n 0
N 1
  u( m, n)
2
N 1
  v( k , l )

m, n  0
2
(4.17)
k ,l  0
 Energy compaction and the variance of coefficients
 v  E[v]  E[Au]  A E[u]  A u
(4.18)
R v  E[( v   v )( v   v )*T ]  A( E[( u   u )( u   u )*T ])A*T  A  R u  A*T
 2v ( k )  [R v ]k ,k  [AR u A*T ]k ,k
W 1
  (k )
v
k 0
N 1

k 0
2
v
2
    v    A  A u    u ( n )
k 0
*T
u
*T
(4.20)
2
(4.21)
n 0
N 1
( k )  Tr[AR u A *T ]  Tr[R u ]    2u ( n )
 E v( k )
N 1
N 1
*T
v
(4.19)
(4.22)
n 0
2
   E u(n) 
N 1
n 0
2
(4.23)
Digital image transforms
Digital image processing
Energy compaction and the variance of coefficients
 v ( k , l )    a( k , m)  a( l , n)   u ( m, n)
m
n

(4.24)

 v2 (k , l )  E v(k , l )  v (k , l )   a(k , m)a(l, n) r (m, n; m' , n' )  a* (k , m' )a* (l, n' )
2
m
n
m'
n'
(4.25)
r( m, n; m', n' )  r1 ( m, m' )  r2 ( n, n' )
(4.26)

 2v ( k , l )   12 ( k )   22 ( l )  AR1A *T
 AR A 
*T
2
k ,k
l ,l
where R1r1(m,m') and R2r2(n,n').
 Decorrelation
v
1 3

2   1
1 
u,
3 
1
where R u  


,0    1 ,
1 
 1  3 / 2
 / 2 
Rv  
 / 2
1  3 / 2 

 v (0,1) 
E[v(0) v(1)]

 v (0) v (1)

2(1 
3 2 1/ 2
 )
4
,
A
1 1 1 


2  1 1
Digital image transforms
Digital image processing
 Basis functions and basis images
KLT
Haar
Walsh
Slant
DCT
Basis functions (basis vectors)
Basis images (e.g.): DCT, Haar, ….
Imagine originala
V(1,1)
V(9,9)
=
+
Imagine originala
V(1,5)
V(1,3)
=
V(1,13)
+
+
+
+
+
+
V(5,8)
+
V(3,5)
V(3,1)
V(2,9)
V(5,6)
V(5,2)
V(5,1)
+
+
V(1,9)
+
V(2,1)
V(1,15)
+
+
V(1,7)
+
V(16,15)
…
+
Imagine aproximata
Keeping only 50% of coefficients
Digital image transforms
Digital image processing
4.4
SINUSOIDAL TRANSFORMS
 The discrete Fourier transform (DFT)
1-D DFT of a sequence u(n), n0,..., N-1 is defined as:
N 1
v(k)   u(n) WNkn , k = 0,1,...N - 1
(4.28)
n 0
where:
 2 
WN  exp  j 

N
(4.29)
The inverse DFT (IDFT):
1
N
u(n) 
v(k) 
u(n) 
1
N
N 1
 v(k)W
 kn
N
, n  0,1,...N  1
(4.30)
k 0
1
N
N 1
 u(n) W
kn
N
, k = 0,1,...N - 1
(4.31)
n 0
N 1
 v(k) W
 kn
N
, n  0,1,...N  1
(4.32)
k 0
 1

F
WNkn , 0  k,n  N  1
 N

(4.33)
Digital image transforms
Digital image processing
 DFT properties
N 1
N 1
v* (N  k)   u* (n)WN( Nk)n   u(n)WNkn  v(k)
n0
n0
N
N

N

v  k   v*   k  , k  0,...,  1
2

2

2
(4.34)
(4.35)
N

N

v  k   v  k 
2

2

(4.36)
v(0), Re v(k), k1,...,N/2 - 1, imv(k), k1,...,N/2 - 1, v(N/2)
(4.37)
(Conjugate symmetry – the DFT of a real sequence is conjugate
symmetric about N/2).
 The 2-D DFT:
N 1 N 1
v(k,l)    u(m,n) WNkm WNln , 0  k,l  N  1
(4.38)
m0 n 0
u(m,n) 
1
N2
v(k,l) 
N 1 N 1
 W
 km
N
 WN ln  v(k,l), 0  m,n  N  1
(4.39)
k 0 l 0
1 N 1 N 1 km ln
  WN WN u(m, n) , 0  k,l  N  1
N m0 n0
u(m,n) 
1
N
N 1 N 1
  v(k,l) W
k 0 l 0
 km
N
WN ln  v(k,l), 0  m,n  N  1
(4.40)
(4.41)
Digital image transforms
Digital image processing
Properties of 2-D DFT
 Symmetry:
F T  F , F 1  F *
 Periodicity:
v(k  N, l  N)  v(k, l)k, l
u( m  N , n  N )  u( m, n) m, n
(4.42)
(4.43)
(4.44)
 The sampled Fourier spectrum:
If u~(m,n)  u(m,n),0  m,n  N  1 si u~(m,n)  0 otherwise, =>:
~ 2k 2l 
U
,
  DFT u( m, n )  v ( k , l )
 N N 
(4.45)
~
where U ( w1 , w2 ) is the Fourier transform of u(m,n).
 Fast Fourier transform (FFT): since 2-D DFT is separable => equations (4.40)
and (4.41) are equivalent to 2N 1-D DFTs; each of them can be
computed in Nlog2N operations through FFT.
=> The total number of operations for 2-D DFT: N2log2N.
Digital image transforms
Digital image processing
Properties of 2-D DFT (continued)
 Conjugate symmetry: the 2-D DFT and unitary 2-D DFT of a real image
exhibit conjugate symmetry:
N 
N 
N
N
N
v  k,  l   v*   k,  l  , 0  k,l   1
2
2
2 
2 
2
or
(4.46)
v(k,l)  v* (N  k,N  l), 0  k,l  N  1
(4.47)
l
1
(N/2 -1)
N/2
N-1
0
k
(N/2)-1
N/2
N-1
N/2
Fig. 4.2 The conjugate symmetry of the 2-D DFT coefficients
Digital image processing
Digital image transforms
Digital image transforms
Digital image processing
 The discrete Cosine transform (DCT)
FDCT:
N 1 N 1
(2m  1)k  (2n  1)l 
v(k,l)  (k) (l)   u(m,n) cos
cos 2N 
2N

 

m0 n 0
(4.47)
where k, l  0, 1, ... N-1.
IDCT:
 (2m  1)k 
 (2n  1)l 
u(m,n)     (k)   (l)  v(k,l)  cos
cos


2N


 2N

k=0 l=0
N 1 N 1
where m, n  0,
 (0) 
(4.48)
... N-1, and the coefficients are:
1
N
and  (k) 
2
for 1  k  N
N
V  CuCT
(2m  1)k 
ck,m  (k)  cos

2N


(4.49)
(4.50)
(4.51)
Digital image processing
Digital image transforms
Digital image transforms
Digital image processing
 The discrete Sine transform (DST):
2 N 1 N 1
  (m  1)(k  1)    (n  1)(l  1) 
v(k,l) 
u(m,n) sin 


 sin 

N  1 m0 n 0
N 1
N 1

u(m,n) 
2 N 1 N 1
  (m  1)(k  1)    (n  1)(l  1) 
v(k,l)  sin 


 sin 

N  1 k 0 l 0
N 1
N 1

sm,k 
2
  (m  1)(k  1) 
sin 

N 1
N 1

(4.52)
(4.53)
(4.54)
 The Hartley transform:
1
v(k,l) 
N
u( m, n ) 
N 1 N 1
 2

 2

  u(m,n)cas  N (mk  nl)
(4.55)
m 0 n  0
1
N
N 1 N 1
  v(k,l)cas  N (mk  nl)
(4.56)
k 0 l 0
cas( )  cos( )  sin( )  2cos(   / 4)
hm,k 
1
N
  mk  
cas 2 N  


(4.57)
(4.58)
Digital image transforms
Digital image processing
4.5 RECTANGULAR TRANSFORMS
 The Hadamard transform
Walsh-Hadamard transform)
H2 
HN 
1
1

1

1 1
H8 
2 21

1
1

 1
1
1
1
1
1
1
1
1
1
1
1
1
1 1
1 1
1 1
1 1
1

1
1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 0
 1 7

 1 3

1 4
 1 1

1 6
1 2

 1 5
(4. 61)
(=
the
Walsh
1 1 1 


2 1  1
1 H N / 2

2 H N / 2
H 8,ord
transform;
the
(4.59)
HN / 2 
 H N / 2 
1
1

1

1 1

2 21

1
1

1
1
1
1
1
-1
1
-1
1
(4.60)
1
1
1
1
-1
-1
1
1
1
1
1
-1
1
1
1
-1
1
-1
-1
1
1
1
1
1
1
1
-1
1
1
1
1
-1
1
1
1
1
1
1
-1
1
1 0
 1 1

1 2

- 1 3
1 4

- 1 5
1 6

 1 7
(4. 62)
Digital image processing
Basis vectors for the
Walsh-Hadamard transform
Digital image transforms
Digital image transforms
Digital image processing
Original image
Ordered Hadamard
Non-ordered Hadamard
Digital image transforms
Digital image processing
 The Slant transform
S2 
 1 0
 a b
 n n
1 
0
Sn 
0
1
2

 bn an
0

1
0
1
2
0
I n -1
(4.63)
0 
 S n 1

0 




I n -1  




0 


I n -1   0
S n 1 

0
 an bn
0
0 1
bn an
0
I n -1
1 1 
1 1


3N 2
N2 1
and
b

n 1
4N 2  1
4N 2  1
an1 
(4.64)
(4.65)
 The Haar transform
(4.66)
k  2p  q 1
h0 (x) 
x  i / N,
1
N
q 1
q  1/2
 p/2
 2 , if 2 p  x  2 p
1 
q  1/2
q

p/2
hk (x) 
x p
  2 , if
p
2
2
N
0, otherwise



and
(4.67)
i=0,1,...,N-1
Hr 
1
8












1
1
1
1
1
1
1
1
2
1
2
1
 2
1
 2
1
0
1
0
1
0
0
2
0
0
0
0
2
0
0
0
0
0
2
0
0
0
0
2
0
0
2
0
0
2
0
2
0
0
2
0
 2
0
0
0
2
1 

1 
0 

 2
0 

0 
0 
2 
0
1
2
2
2
2
2
2
(4.68)
Digital image processing
Digital image transforms
Digital image transforms
Digital image processing
Applying the Haar transform at block level (e.g. 2×2 pixels blocks => Hr[2×2]):
Block
transform:
Rearrange
coefficients:
Applying the Haar transform at block level for a 4×4 pixels blocks => Hr[4×4]:
Block
transform:
Rearrange
coefficients:
Digital image transforms
Digital image processing
4.6 EIGENVECTOR BASED TRANSFORMS
A  k    0
(4.69)
Av k   k v k
(4.70)
mx 
1 L
 xl
L l 1


C x  E ( x  m x )( x  m x )T 
(4.71)
1 L
x l x tl  m x m tx

L l 1
(4.72)
y  A (x  m x )
(4.73)
Cy  ACx AT
(4.74)
 1

Cy  

 0
0 



 N 
x  A 1 y  m x  A T y  m x
(4.76)
~
y  Bx
(4.77)
~
x  BT  ~
y
 2e 
(4.75)
N

k  M 1
k
(4.78)
(4.79)
Digital image transforms
Digital image processing
 The Karhunen – Loeve transform (KLT = PCA)
 The fast KLT
 The SVD transform (singular values decomposition)
A  U  VT
(4.80)
  U T AV
0
1

A  2

1
0
1
3
4
3
2
4
5
4
1
3
4
3
1
2
1
0
1

2

1
0 
6
14

AA T  18

14
 6
0,186 0,638
0,476 0,058

U   0,691 0,422

0,476 0,058
0,186 0,638
0
0
0
12,58
 0
1,142
0
0

T
0
0,557 0
  U AU   0

0
0
0
 0
 0
0
0
0
(4.81)






0 
0
0
0
0
14
36
48
36
18
48
65
48
14
18
14 6 
36 14 

48 18 

36 14 
14 6 
147,07 
 1,872 


;    0,058 


 0 
 0 
0,241 0,695 0,695
0,52 0,133 0,128

0,587
0
0 

0,52 0,133 0,128 
0,241 0,695 0,695 
A  U  UT
0
1

 2

1
0
1
3
4
3
2
4
6
4
1
3
4
3
1
2
1
0
1

2

1
0 
 KLT (PCA)
Eigenimages – examples:
Facial
image
set
Corresponding
“eigenfaces”
3 eigenimages and the individual variations on those components
Face aproximation,
from rough to
detailed, as more
coefficients are
added
Digital image transforms
Digital image processing
4.7 FILTERING IN THE TRANSFORM DOMAIN
1
1

0

0
0

0
0

0
1 1 1 1 1 1 1  2
1 1 1 1 1 1 1  2
 
0 0 0 0 0 0 0  2,83
 
0 0 0 0 0 0 0  0

0 0 0 0 0 0 0  0
 
0 0 0 0 0 0 0  0
0 0 0 0 0 0 0  0
 
0 0 0 0 0 0 0  0
Im age
0 0 0 0 0 0 0
0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0
0 0 0 0 0 0 0

0 0 0 0 0 0 0
0 0 0 0 0 0 0

0 0 0 0 0 0 0
Its Haar transform
DFT
Original image = (white square, grey background)
+ aditive noise
DFT = sinc 2-D for the square + cst. (for noise)
LPF 2-D
IDFT
The 2-D spectrum of the image and the filters applied:
Noisy image; periodic noise as vertical lines In the regions corresponding to the vertical lines frequencies
Image restoration through filtering
Digital image transforms
Digital image processing
4.8 CONCLUSIONS
 DFT
- Fast transform; very useful in digital signal processing, convolution, filtering, image
analysis
- Good energy compaction; however – requires complex computations
 DCT
- Fast transform and requires only real number operations
- The optimal alternative to the KLT for highly correlated images
- Used in compression and image restoration by Wiener filtering
- Excellent energy compaction
 Hadamard
- Faster than sinusoidal transforms since it only implies sums and subtractions
- Used for hardware implementation of some digital image processing algorithms
- Applied in image compression, filtering, coding
- Good energy compaction
 Haar
- Very fast transform
- Useful for feature extraction (like horizontal or vertical lines), image coding, image
analysis
- Average energy compaction performance
 KLT
- Optimal transform as: energy compaction; decorrelation
- Does not have a fast algorithm
- Generally used for small sized vectors and to evaluate the performances of other
transforms, but also for image analysis and recognition (PCA)