Module II(8 Hrs)
Review of matrix theory, row and column ordering- Toeplitz,
Circulant and block matrix,
2D transforms - DFT, its properties, Walsh transform, Hadamard
transform, Haar transform,
DCT, KL transform and Singular Value Decomposition.
Matrix theory
Vectors and matrices:
- Both 1 d and 2 D can be represented as vectors
and matrices
- U(n) = u(1)
u(2)
.
.
- The nth element of the vector u is denoted by
u(n)
- A column vector of size N is also called an Nx1
vector
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-
a matrix A of size MxN has
M rows and N columns.
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..... TaE
(1, N ) 
O
a (1,2)
 a(1,1)
A = a(m,n)= a(2,1) a(2,2) ...... a (2, N ) 

..
...
...
... 


a
(
M
,
1
)
a
(
M
,
2
)
.....
a
(
M
,
N
)


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Row and column ordering:
- xT
T
  x(1,1) x(1,2).....x(1, N ) x(2,1)....x(2, N ).....x( M ,1)....x( M , N )
-for 1 to 1 mapping this is called as a lexicographic or dictionary ordering
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- X is row vector got by stacking each
row
one after another
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- Column ordered vector is got by stacking column by column.
T
xT
  x(1,1) x(2,1).....x( M ,1) x(1,2)....x( M ,2)....x(1, M ).....x( M , N )
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T
Transposition and conjugate rules:
-
 
T *
A  A
*T
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T T OTE
 AB KBTUAN
A   A 
T 1
1 T
 AB
*
AB
*
*
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Toeplitz and circulant matrix
- A Toeplitz matrix T is a matrix that has constant
elements along the diagonal and the sub
diagonals.
- A Toeplitz matrix is an n xn matrix Tn = [tk,j; k, j =
0, 1, . . . , n − 1]where tk,j = tk−j , i.e., a matrix of
the form
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T
Kt0 t1
t1
t2
t n 1
t0
t1
t  2 ...............t ( n 1)
t 1
t0
t0
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- circulant matrix is a special kind of Toeplitz matrix where
each row vector is rotated one element to the right relative to
the preceding row vector.
-
Each of its rows is a circular shift of the previous row
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Show each of the following:
a. A circulant matrix is Toeplitz but the converse is not true
b. The product of 2 circulant matrices is a circulant matrix
c. The product of 2 Toeplitz matrices need not be Toeplitz.
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Orthognal and unitary matrix:
An orthognal matrix is such that its inverse is equal to its
transpose
A is orthognal if
A-1 =A T
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Or
ATA=AAT=I
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• A matrix is called unitary matrix if ts inverse is equal to its
conjugate transpose
A-1= A*T
OR
AA*T-A*TA=I
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A real orthognal matrix is also unitary but a unitary
matrix need not be orthognal
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Problem
1. consider matrices below and check for orthognality and unitary
1
1 1
A
[
]IN
S
1.
2 1OT
E
N
U
T
K2
j
B [
]
 j
2
j
1 1
c 
[
]
1
2 j
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Block matrix
-
A matrix whose elements are matrices are called block
matrix
 P11
P
 P21
 1 1
P11  

1
1


P12 
P22 
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mxn
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K P  2
12
pxq
2
 2 2


 3 3
P 21  

3
3


 4 4
P 22  

 4 4
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- Therefore P is a mxn block matrix with basic dimension pxq.
- If block structure is toeplitz or circulant it is called a block
toeplitz or block circulant.
- if each bolck itself is toeplitz or circulant then we have the
doubly block toeplitz or doubly block circulant.
Kronecker Product:
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Problem
A= 1 1
1 -1
B= 1 3
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K product of A and B as well
Find kronecker
as B and A , Are they equal
Ans: They are not equal .A kronecker B is
not equal to B kronecker A
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



Image Transforms
Image Transforms are used for image processing and image
analysis.
Transform is a mathematical tool for moving from one domain
to another for performing task in a easy manner.
Image transforms are useful for fast computation of
convolution and correlation.
By using transforms the signals are represented as a set of
basis function
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Need for transforms
 Mathematical convenience:
convolution in time domain is multiplication in frequency
domain
 To Extract more information:
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• What does image transform do
– It represents a given image as a series
summation of a set of unitary matrices
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– A matrix A is a unitary
matrix
if
.
A A
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O
N
KTU
1
1
*T
A A
*T
– A* is the conjugate of A
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Walsh transform
• The basic kernel function of 1D walsh transform is given by
1 N 1
g (m, k )  ( )  (1) bi ( m )bn 1i ( k )
N i 0
IN
.
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NOT
• WT[x(m)] =
KTU 1
X (K ) 
N 1
x(m) g (m, k )

N
m 0
• n represents no of bits to represent a number and bi(n) rep the i th
bit of the binary value(from LSB) and n=log 2N
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• Inverse walsh transform
N 1
WT-1[X(K)] =x(m)=
 X ( K )h(m, k )
K 0
2D walsh Transform
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K N 1 N 1
1
X ( K , L) 
N
  x(m, n) g (m, k , n, l )
n 0
m 0
1 N 1
bi ( m ) bn 1 i ( k ) 
g (m, k , n, l )  ( )  (1)
N i 0
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bi ( n ) bn 1 i ( l )
Inverse 2D walsh Transform
N 1 N 1
x(m, n)   X ( K )Eh(Sm.I,Nn, k , l )
T
O
N
K  0K
LT
 0U
N 1
h(m, n, k , l )   (1)
bi ( m ) bn 1 i ( k )  bi ( n ) bn 1 i ( l )
i 0
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Hadamard matrix
• Hadamard matrices seem such simple matrix structures: they
are square, have entries +1 or −1 and have orthogonal row
vectors and orthogonal column vectors.
• A square matrix with elements ±1 and size h, whose distinct
row vectors are mutually orthogonal, is referred to as an
Hadamard matrix of order h.
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Hadamard Matrices
• The order N=2 hadamard matrix is
given by
1 1
H2  

1

1


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K matrix of order 2N
• The Hadamard
can be generated by kronecker
product operation
 HN HN 
H 2N  

 HN  HN 
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• Substituting N=2
 H 2 H 2
H4  

H
2
H
2


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Haar Transform
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• Develop hadamard for f(x)={1,2,0,3}------1D
• HT for 1D : F=H.f
=
1 1 1 1
 1  1 1  1


 1 1  1  1


1

1

1
1


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K
=
 1
 2
 
 0
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OT E
 3
 6
  4


 0


 2
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• Develop Hadamard for 2D
 2
1

 2

1
1 2 1
2 3 2
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3 4 3
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
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2
3
2
T

K
• 2D HT : F=H.f.HT =H.f.H
Ans:
2  6  6
2 2
2 
 6 2 2
2



6
2
2
2


 34
 2

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Discrete Cosine Transform
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Karhunen –Loeve Transform
 This transform also called as Hotelling Transform.
 KL transform is a reversible linear transform hat exploits statistical
properties of a vector representation
 The basic functions of KL transforms are orthogonal eigen vectors
of the covariance matrix of a data set
 After a KL transform , most of the energy of the transform
coefficients are concentrated within the first few components.
Drawbacks of KL transform:
 It is input dependent and the basic function has to be calculated for
each signal model in which it is operated.
 KL transform require O(m2) multiply / add operations .The DFT and
DCT require O(log2m) multiplications.
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Applications of KL transform:
 Clustering Analysis: to determine the new coordinates for the
sample data where largest covariance lies on the first axis and
next largest on the next axis and so on, and therefore used for
dimensionality reduction.
 Image compression
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Singular value decomposition(SVD)
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