2. Chapter 13 Discrete Image
Transforms
13.1 Introduction
13.2 Linear transformations

One-dimensional discrete linear transformations
Definition. If x is an N-by-1 vector and T is an N-by-N
matrix, then y = Tx is a linear transformation of the
vector x, I.e.,
N−
1
yi = ∑ ij x j
t
j =0
T is called the kernel matrix of the transformation

3. Linear Transformations
Rotation and scaling are examples of linear
transformations
 y1  cos(θ)
y  = sin(θ)
 2 
−sin(θ) x1 
cos(θ) x2 
 
If T is a nonsingular matrix, the inverse
linear transformation is
x = T −1y

4. Linear Transformations
Unitary Transforms


If the kernel matrix of a linear system is a
unitary matrix, the linear system is called a
unitary transformation.
A matrix T is unitary if
T −1 = (T∗ ) t , i.e., T(T∗ ) t = (T∗ ) t T = I

If T is unitary and real, then T is an orthogonal
matrix.
T −1 = Tt , i.e., TT t = Tt T = I

5. Unitary Transforms
The rows ( also the columns ) of an orthogonal
matrix are a set of orthogonal vectors.
One-dimensional DFT is an example of a unitary
transform
1
i
F =
k
∑f exp(−j 2π N )
N
or
N−
1
k
i=
0
i
F = Wf
where the matrix W is a unitary matrix.

6. Unitary Transforms
Interpretation




The rows (or columns) of a unitary matrix form a
orthonormal basis for the N-dimensional vector space.
A unitary transform can be viewed as a coordinate
transformation, rotating the vector in N-space without
changing its length.
A unitary linear transformation converts a Ndimensional vector to a N-dimensional vector of
transform coefficients, each of which is computed as
the inner product of the input vector x with one of the
rows of the transform matrix T.
The forward transformation is referred to as analysis,
and the backward transformation is referred to as
synthesis.

7. Two-dimensional discrete linear
transformations
A two-dimensional linear transformation is
N− N−
1
1
Gmn =∑ Fik t (i, k ; m, n)
∑
i= k =
0
0
where t (i, k ; m, n) forms an N2-by-N2 block
matrix having N-by-N blocks, each of
which is an N-by-N matrix
If t (i, k ; m, n) = t (i, m)t (k , n) , the transformation
is called separable.
r
Gmn
c
N 1
 −

=∑∑ ik t c ( k , n)  r (i , m)
F
t

i=  =
0
k 0

N−
1

8. 2-D separable symmetric unitary
transforms
If t (i, k ; m, n) = t (i, m)t (k , n) , the transformation is
symmetric,
Gmn
N 1
 −

=∑∑ ik t ( k , n)  (i , m)
F
t

i=  =
0
k 0

N−
1
or
G =
TFT
The inverse transform is
F = T −1GT −1 = (T∗) t G (T∗) t
Example: The two-dimensional DFT
G = WFW ,
F = ( W ∗) t G ( W ∗ ) t

9. Orthogonal transformations
If the matrix T is real, the linear
transformation is called an orthogonal
transformation. Its inverse transformation is
F = T′GT′
If T is a symmetric matrix, the forward and
inverse transformation are identical,
G = TFT and F = TGT

10. Basis functions and basis images
Basis functions

The rows of a unitary form a orthonormal basis
for a N-dimensional vector space. There are
different unitary transformations with different
choice of basis vectors. A unitary
transformation corresponds to a rotation of a
vector in a N-dimensional (N2 for twodimensional case) vector space.

11. Basis Images
The inverse unitary transform an be viewed as the
weighted sum of N2 basis image, which is the
} as
inverse transformation of a matrix G = {δ
p,q
Fmn
i − p , j −q
1
N −

=∑(i, m) ∑ i −p , k −q t ( k , n)  =t ( p, m)t ( q, n)
t
δ
i=
0
k =0

n−
1
This means that each basis image is the outer
product of two rows of the transform matrix.
Any image can be decomposed into a set of basis
image by the forward transformation, and the
inverse transformation reconstitute the image by
summing the basis images.

12. 13.4 Sinusoidal Transforms
13.4.1 The discrete Fourier transform

The forward and inverse DFT’s are
F = Wf and f = ( W ∗ ) t F
where
 w0, 0  w0, N −1 


W= 

 
 wN −1,0  wN −1, N −1 


ik
1 − j 2π N
wi ,k =
e
N

13. 13.4 Sinusoidal Transforms
The spectrum vector

The frequency corresponding to the ith element
of F is
 2i
 N fN

si = 
 2( N − i ) f
N
 N


0≤ i ≤ N /2
N / 2 +1≤ i ≤ N −1
The frequency components are arranged as
0
fN
N /2
0
1
i
fN
N /2
i
fN
N/2
− ( N − i)
fN
N /2
−
fN
N /2
N-1

14. 13.4 Sinusoidal Transforms

The frequencies are symmetric about the
highest frequency component. Using a circular
right shift by the amount N/2, we can place the
zero-frequency at N/2 and frequency increases
in both directions from there. The Nyquist
frequency (the highest frequency) at F0. This
can be done by changing the signs of the oddnumbered elements of f(x) prior to computing
the DFT. This is because

15. 13.4 Sinusoidal Transforms
F (u ) ⇔ ( x) ⇒ F (u − N / 2) ⇔ exp( j 2πx
N /2
) f ( x)
N
= exp( jπx) f ( x) = (−1) x f ( x)

The two-dimensional DFT
 For the two-dimensional DFT, changing the sign of
half the elements of the image matrix shifts its zero
frequency component to the center of the spectrum
F (u , v) ⇔ f ( x, y ) ⇒ F (u − N / 2, v − N / 2) ⇔ (− 1) x+ y f ( x, y )
1
4
3
2
2
3
4
1

17. 13.4.2 Discrete Cosine
Transform
The two-dimensional discrete cosine
transform (DCT) is defined as
 π (2i + 1)m   π (2k + 1)n 
Gc (m, n) = α (m)α (n)∑ ∑ g (i, k ) cos 
 cos  2 N 
 2N  

i=0 k =0
N −1 N −1
and its inverse
 π (2i + 1)m   π (2k + 1)n 
g (i, k ) = ∑ ∑ α (m)α (n)Gc (m, n) cos 
 cos  2 N 
 2N  

m= 0 n= 0
N −1 N −1
where
α ( 0) =
1
N
and α ( m) =
2
N
for 1 ≤ m ≤ N

18. The discrete cosine transform
DCT can be expressed as a unitary matrix form
G c = CgC
Where the kernel matrix has elements
Ci , m
π (2i + 1)m 
= α ( m) cos 

2N


The DCT is useful in image compression.

20. The sine transform
The discrete sine transform (DST) is defined as
2 N −1 N −1
 π (i + 1)(m + 1)   π (k + 1)(n + 1) 
Gs (m, n) =
g (i, k ) sin 
∑ ∑0
 sin 

N + 1 i=0 k =
N +1
N +1

 

and
2 N −1 N −1
 π (i + 1)(m + 1)   π (k + 1)(n + 1) 
g (i, k ) =
Gs (m, n) sin 
∑0 ∑0
 sin 

N + 1 m= n=
N +1
N +1

 

The DST has unitary kernel
ti , k =
2
π (i + 1)(k + 1) 
sin 

N +1
N +1



21. The Hartley transform
The forward two-dimensional discrete
Hartley transform1
N −1 N −
Gm ,n
1
=
N
 2π

g i ,k cas 
(im + kn)
∑∑
N

i =0 k =0
The inverse DHT
g i ,k
1
=
N
 2π

Gm , n cas 
(im + kn)
∑∑
N

m =0 n =0
N −1 N −1
where the basis function
cas(θ ) = cos(θ ) + sin(θ ) = 2 cos(θ − π / 4)

22. The Hartley transform
The unitary kernel matrix of the Hartley transform
has elements
ti , k
1
=
N
 
ik 
cas 2π N 

 
The Hartley transform is the real part minus the
imaginary part of the corresponding Fourier
transform, and the Fourier transform is the even
part minus j times the odd part of the Hartley
transform.

23. 13.5 Rectangular wave
transforms
13.5.1 The Hadamard Transform
 Also called the Walsh transform.
 The Hadamard transform is a symmetric, separable
orthogonal transformation that has only +1 and –1
as elements in its kernel matrix. It exists only for N = 2 n
 For the two-by-two case
1
H2 =
2
1 1 
1 −1


 And for general cases
1
1 H N / 2
HN =

N
N H N / 2
H N /2 
− HN /2 


24. The Hadamard Transform
H8
1
1

1

1 1
H8 =
2 2 1

1
1

1

1
1

1

1 1
H8 =
2 2 1

1
1

1

1 1 1 1 1 1 1 0
− 1 1 − 1 1 − 1 1 − 1 7

1 − 1 − 1 1 1 − 1 − 1 3

−1 −1 1 1 −1 −1 1  4
1 1 1 − 1 − 1 − 1 − 1 1

−1 1 −1 −1 1 −1 1  6
1 −1 −1 −1 −1 1 1  2

− 1 − 1 1 − 1 1 1 − 1 5

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 1 − 1

0
1
2
3
4
5
Ordered Hadamard transform
−1
−1
6
7

25. 13.5.3 The Slant Transform
The orthogonal kernel matrix for the slant
transform is obtained iteratively as
1
S2 =
2
 1
 2
1  5
S4 =

2 0
− 1

5

0
1
5
1
2
5
1
2
−
5
0
1
5

0 
1 

5 
−1 
2 

5 



1
2
1
2
1
2
1
−
2
0
0
0
0
1 1 
1 −1


0
0
1
2
1
2
 1

 2

 3



0
 = 1  10
1 
2 1


2 
 2
1 
 1
−
 10
2


0
1
2
1
10
1
−
2
3
−
10
1
2
1
−
10
1
−
2
3
10
1 
2 
3 

−
10 
1 

2 
1 
−
10 


26. 13.5.3 The Slant Transform

And
 1 0
a b
 N N
1  0
SN =
2 0 1
− b a
 N0 N

0
I
0
I
1 0

0
− aN bN
 S
0
I  N /2 0 
0 −1
  0 S N / 2 
0
bN aN
0
− I 
where I is the identity matrix of order N/2-2 and
a2 N
3N 2
N2 −1
=
, b2 N =
2
4N − 1
4N 2 − 1

27. 13.5.3 The Slant Transform
The basis function for N=8
0
4
1
5
2
6
3
7

28. 13.5.4 The Haar Transform
The Basis functions of Haar transform

For any integer
0 ≤ k ≤ N −1
, let
k = 2p + q −1
p ≥ 0, q ≥ 1

where
and
is the largest power of
2 that 2 p ≤ k , and q − 1 is the remainder.
The Haar function is defined by
h0 ( x) =
2p
1
N

29. 13.5.4 The Haar Transform

And

The 8-by-8 Haar orthogonal kernel matrix is
 p/2
2

1  −p/2
hk ( x) =
2
N 
0







1 
Hr =
2 2






q −1
q −1 / 2
≤x<
2p
2p
q −1 / 2
q
≤x< p
2p
2
otherwise
1 1
1
1
1 1
1
1 
1 1
1
1
−1 −1 −1
−1 

2 2 − 2 − 2 0 0
0
0 

0 0
0
0
2 2 − 2 − 2
2 −2 0
0
0 0
0
0 

0 0
2
−2 0 0
0
0 
0 0
0
0
2 −2 0
0 

1 0
0
0
0 0
2
−2 


30. 13.5.4 The Haar Transform
Basis Functions for N=8

31. Basis images of Haar

32. Basis image of Hadamard

33. Basis images of DCT

34. 13.6 Eigenvector-based
transforms
Eigenvalues and eigenvectors

For an N-by-N matrix A, λ is a scalar, if
| A −λ |=0
I
then λ is called an eigenvalue of A. The vector v
that satisfies
Av =λ
v
is called an eigenvectors of A.

35. 13.6 Eigenvector-based
transforms
13.6.2 Principal-Component Analysis

Suppose x is an N-by-1 random vector, The
mean vector can be estimated from its L
samples as
L
1
m x ≈ ∑x l
L l =1
and its covariance matrix can be estimated by
1
C = ε {( x − m )(x − m )′} ≈ ∑ x x − m m
L
L
x

x
x
l =1
l
t
l
x
t
x
The matrix C x is a real and symmetric matrix.

36. 13.6 Eigenvector-based
transforms

Let A be a matrix whose rows are eigenvectors
of C x , then C y = AC x A′ is a diagonal matrix
having the eigenvalues of C x along its
diagonal, I.e.,
0
λ1

Cy = 



0
λN 



Let the matrix A define a linear transformation
by
y = A ( x −m x )

37. 13.6 Eigenvector-based
transforms

It can be shown that the covariance matrix of
the vector y is C y = AC x A′ . Since the matrix
Cy

y
is a diagonal matrix, its off-diagonal
elements are zero, the element of are
uncorrelated. Thus the linear transformation
remove the correlation among the variables.
The reverse transform = A′y +m
x = A −1y +m
can reconstruct x from y.

38. 13.6 Dimension Reduction
We can reduce the dimensionality of the y
vector by ignoring one or more of the
eigenvectors that have small eigenvalues.
Let B be the M-by-N matrix (M<N) formed
by discarding the lower N-M rows of A, and
let mx=0 for simplicity, then the transformed
vector has smaller dimension
ˆ
y = Bx

39. 13.6 Dimension Reduction
The vector x can be
reconstructed(approximately) by
ˆ
ˆ
x = B′y
The mean square error is
MSE =
N
∑λ
k =M +
1
k
ˆ
The vector y is called the principal
component of the vector x.

40. 13.6.3 The Karhunen-Loeve
Transform
The K-L transform is defined as
y = A(x − m x )
The dimension-reducing capability of the K-L
transform makes it quite useful for image
compression.
When the image is a first-order Markov process,
where the correlation between pixels decreases
linearly with their separation distance, the basis
images for the K-L transform can be written
explicitly.

41. 13.6.3 The Karhunen-Loeve
Transform
When the correlation between adjacent
pixels approaches unity, the K-L basis
functions approach those of discrete cosine
transform. Thus, DCT is a good
approximation for the K-L transform.

42. 13.6.4 The SVD Transform
Singular value decomposition
Any N-by-N matrix A can be decomposed as
A = UΛV t
t
where the columns of U and V are the eigenvectors of AA
andA
, respectively.
is an N-by-N
At
Λ
diagonal matrix containing the singular values of A.
 The forward singular value decomposition(SVD)
transform
Λ = U t AV


The inverse SVD transform
A = UΛV t

43. 13.6.4 The SVD transform
For SVD transform, the kernel matrices are
image-dependent.
The SVD has a very high power of image
compression, we can get lossless
compression by at least a factor of N. and
even higher lossy compression ratio by
ignoring some small singular values may be
achieved.

44. The SVD transform
Illustration of figure 13-7

45. 13.7 Transform domain filtering
Like in the Fourier transform domain, filter can be
designed in other transform domain.
Transform domain filtering involves modification
of the weighting coefficients prior to
reconstruction of the image via the inverse
transform.
If either of the desired components or the
undesired components of the image resemble one
or a few of the basis image of a particular
transform, then that transform will be useful in
separating the two.

46. 13.7 Transform domain filtering
Haar transform is a good candidate for
detecting vertical and horizontal lines and
edges.

47. 13.7 Transform domain filtering
Illustration of fig. 13-8.

48. 13.7 Transform domain filtering
Figure 13-9