Unit ii

Why Do Transforms?
 Fast computation
 E.g., convolution vs. multiplication for filter with
wide support
 Conceptual insights for various image processing
 E.g., spatial frequency info. (smooth, moderate
change, fast change, etc.)
 Obtain transformed data as measurement
 E.g., blurred images, radiology images (medical
and astrophysics)
 Often need inverse transform
 May need to get assistance from other transforms
 For efficient storage and transmission
 Pick a few “representatives” (basis)
 Just store/send the “contribution” from each basis

Introduction
 Image transforms are a class of
unitary matrices used for
representing images.
 An image can be expanded in terms
of a discrete set of basis arrays called
basis images.
 The basis images can be generated
by unitary matrices.

One dimensional orthogonal and unitary
transforms
 For a 1-D sequence represented
as a vector u of size N, a unitary transformation
is written as
{ ( ),0 1}u n n N  
1
0
( ) ( , ) ( ) , 0 1
N
n
v k a k n u n k N


    v = Au

1
*
0
( ) ( ) ( , ) , 0 1
N
k
u n v k a k n n N


   
*T
u = A v
v(k) is the series representation of the sequence u(n).
The columns of A*T, that is, the vectors
are called the basis vectors of A.
*
{ ( , ), 0 1}T
a k n n N   *
ka
One dimensional orthogonal and unitary
transforms

Two-dimensional orthogonal and
unitary transforms
 A general orthogonal series expansion for an N x
N image u(m,n) is a pair of transformations of
the form
1 1
,
0 0
( , ) ( , ) ( , )
N N
k l
m n
y k l x m n a m n
 
 
 
1 1
*
,
0 0
( , ) ( , ) ( , )
N N
k l
k l
x m n y k l a m n
 
 
 
 ,where ( , ) ,k la m n called an image transform, is a set of complete
orthonormal discrete basis functions.

Separable unitary transforms
 Complexity : O(N4)
 Reduced to O(N3) when transform is separable i.e.
ak,l(m,n) = ak(m) bl(n) =a(k,m)b(l,n) where
{a(k,m), k=0,…,N-1},{b(l,n), l=0,…,N-1}
are 1-D complete orthonormal sets of basis vectors.

Separable unitary transforms
 A={a(k,m)} and B={b(l,n)} are unitary
matrices i.e. AA*T = ATA* = I.
 If B is same as A
1 1
0 0
( , ) ( , ) ( , ) ( , )
N N
m n
y k l a k m x m n a l n
 
 
   T
Y AXA
1 1
* * *
0 0
( , ) ( , ) ( , ) ( , )
N N
T
k l
x m n a k m y k l a l n
 
 
  *
X = A YA

Basis Images
 Let denote the kth column of . Define the matrices
then
*
ka *T
A
* * *T
k,l k lA = a a
1 1
0 0
( , )
( , ) ,
N N
k l
y k l
y k l
 
 


 *
k,l
*
k,l
X A
X A
, 0,..., 1k l N *
k,lA
The above equation expresses image X as a linear
combination of the N2 matrices , called
the basis images.

8x8 Basis images for discrete cosine transform.

Example
 Consider an orthogonal matrix A and image X
1 11
1 12
A
 
   







43
21
X
1 1 1 2 1 1 5 11
1 1 3 4 1 1 2 02
       
                 
T
Y = AXA
To obtain basis images, we find the outer product of the
columns of A*T
* *
0,1 1,0
1 11
1 12
T
A A
 
   
*
1,1
1 11
1 12
A
 
   
 *
0,0
1 1 11 1
1 1
1 1 12 2
A
   
    
   
The inverse transformation gives
1 1 5 1 1 1 1 21
1 1 2 0 1 1 3 42
     
             
*T *
X = A YA

Properties of Unitary Transforms
Energy Conservation
In unitary transformation, y = Ax and ||y||2 = ||x||2
1 1
2 2
0 0
( ) ( )
N N
k n
y k x n
 
 
    
2 2*T *T *T *T
y y y = x A Ax = x x x
Proof:
This means every unitary transformation is simply a rotation
of the vector x in the N-dimensional vector space.
Alternatively, a unitary transformation is rotation of the basis
coordinates and the components of y are the projections of x
on the new basis.

Properties of Unitary Transforms
 Energy compaction
 Unitary transforms pack a large fraction of the
average energy of the image into a relatively
few components of the transform coefficients.
i.e. many of the transform coefficients contain
very little energy.
 Decorrelation
 When the input vector elements are highly
correlated, the transform coefficients tend to be
uncorrelated.
 Covariance matrix E[ ( y – E(y) ) ( y – E(y) )*T ].
 small correlation implies small off-diagonal terms.

1-D Discrete Fourier Transform
1
0
1
( ) ( ) , 0,..., -1
N
nk
N
n
y k x n W k N
N


 
2
expN
j
W
N
 
  
 
The discrete Fourier transform (DFT) of a sequence {u(n), n=0,…,N-1} is
defined as
where
The inverse transform is given by
1
0
1
( ) ( ) , 0,..., -1
N
nk
N
k
x n y k W n N
N



 
The NxN unitary DFT matrix F is given by
1
, 0 , 1nk
NF W k n N
N
 
    
 

DFT Properties
 Circular shift u(n-l)c = x[(n-l)mod N]
 The DFT and unitary DFT matrices are
symmetric i.e. F-1 = F*
 DFT of length N can be implemented by a fast
algorithm in O(N log2N) operations.
 DFT of a real sequence {x(n), n=0,…,N-1} is
conjugate symmetric about N/2.
i.e. y*(N-k) = y(k)

The Two dimensional DFT
1 1
0 0
1
( , ) ( , ) , 0 , -1
N N
km ln
N N
m n
y k l x m n W W k l N
N
 
 
  
1 1
0 0
1
( , ) ( , ) , 0 , -1
N N
km ln
N N
k l
x m n y k l W W m n N
N
 
 
 
  
The 2-D DFT of an N x N image {x(m,n)} is a separable
transform defined as
Y = FXF * *
X = F YF
The inverse transform is
In matrix notation &

Properties of the 2-D DFT
 Symmetric,
unitary.
 Periodic
 Conjugate
Symmetry
 Fast transform
 Basis Images
T -1
  * *
F F*
F F, F F =
( , ) ( , ), ,
( , ) ( , ), ,
y k N l N y k l k l
x m N n N x m n m n
   
   
*
( , ) ( , ), 0 , 1y k l y N k N l k l N     
 * ( )
,
1
, 0 , 1 , 0 , 1T km ln
k l k l NA W m n N k l N
N
 
         
O(N2log2N)

2-D pulse DFT
Square Pulse
2D sinc function

FT is Shift Invariant
After shifting:
• Magnitude stay constant
• Phase changes

Rotation
• FT of a rotated image also
rotates

The Cosine Transform (DCT)
1
0
(2 1)
( ) ( ) ( )cos , 0 1
2
N
n
n k
y k k x n k N
N





   
1
, 0,0 1
( , )
2 (2 1)
cos , 1 1,0 1
2
k n N
N
C k n
n k
k N n N
N N


   

 
      

1
0
(2 1)
( ) ( ) ( )cos , 0 1
2
N
k
n k
x n k y k n N
N





   
The N x N cosine transform matrix C={c(k,n)}, also known
as discrete cosine transform (DCT), is defined as
1 2
(0) , ( ) = for 1 1
N
k k N
N
    
The 1-D DCT of a sequence {x(n), 0 ≤ n ≤ N-1} is defined as
The inverse transformation is given by
where

Properties of DCT
 The DCT is real and orthogonal
i.e. C=C*C-1=CT
 DCT is not symmetric
 The DCT is a fast transform : O(N log2N)
 Excellent energy compaction for highly
correlated data.
 Useful in designing transform coders and
Wiener filters for images.

2-D DCT
(2 1) (2 1)
( , , , ) ( ) ( )cos cos
2 2
m k n l
C m n k l k l
N N
 
 
 

1
0
( )
2
1 1
k
N
k
k N
N




 
   
The 2-D DCT Kernel is given by
where
Similarly for ( )l

DCT example
a) Original image b) DCT image

The Sine Transform
 ( , )k nΨ
2 ( 1)( 1)
( , ) sin , 0 , 1
1 1
n k
k n k n N
N N


 
   
 
The N x N DST matrix is defined as
The sine transform pair of 1-D sequence is defined as
1
0
1
0
( ) ( ) ( , ), 0 1
( ) ( , ) ( ), 0 1
N
n
N
k
y k x n k n k N
x n k n y k n N






   
   



The properties of Sine
transform
 The Sine transform is real, symmetric, and
orthogonal
 The sine transform is a fast transform
 It has very good energy compaction property
for images
* T -1
Ψ = Ψ = Ψ = Ψ

The Hadamard transform
 The elements of the basis vectors of the
Hadamard transform take only the binary
values ±1.
 Well suited for digital signal processing.
 The transform matrices Hn are N x N matrices,
where N=2n, n=1,2,3.
 Core matrix is given by
1 11
1 12
 
   
1H

The Hadamard transform
1 1
1 1
1 1
1
2
 

 
 
     
n n
n n
n n
H H
H H H
H H
The matrix Hn can be obtained by kroneker product recursion
3 2 1 2 1 1
3
&
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 11
1 1 1 1 1 1 1 18
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
   
 
     
    
 
    
    
 
    
    
 
     
H H H H H H
H
Example

The Hadamard transform properties
 The number of sine changes in a row is called
sequency. The sequency for H3 is 0,7,3,4,
1,6,2,5.
 The transform is real, symmetric and
orthogonal.
 The transform is fast
 Good energy compaction for highly correlated
data.
* T -1
H = H = H = H

The Haar transform
The Haar functions hk(x) are defined on a continuous
interval, x [0,1], and for k = 0,…,N-1, where N = 2n.
The integer k can be uniquely decomposed as k = 2p + q -1
where 0 ≤ p ≤ n-1; q=0,1 for p=0 and 1 ≤ q ≤ 2p for p≠0. For
example, when, N=4
k 0 1 2 3
p 0 0 1 1
q 0 1 1 2

The Haar transform
•The Haar functions are defined as
0 0,0
/ 2
/ 2
,
1
( ) ( ) , [0,1]
1
1 22 ,
2 2
1
1 2( ) 2 ,
2 2
0, [0,1]
p
p p
p
k p q p p
h x h x x
N
q
q
x
q
q
h x h x
N
otherwise for x
  

 
 



    







Haar transform example
The Haar transform is obtained by letting x take discrete
values at m/ N, m=0,…,N-1. For N = 4, the transform is
1 1 1 1
1 1 1 11
2 2 0 04
0 0 2 2
 
   
 
 
  
Hr

Properties of Haar transform
 The Haar transform is real and
orthogonal
Hr = Hr* and Hr-1 = HrT
 Haar transform is very fast: O(N)
 The basis vectors are sequency
ordered.
 It has poor energy compaction for
images.

KL transform
Hotelling transform
 Originally introduced as a series
expansion for continuous random
process by Karhunen and Loeve.
 The discrete equivalent of KL series
expansion – studied by Hotelling.
 KL transform is also called the
Hotelling transform or the method of
principal components.

KL transform
 Let x = {x1, x2,…, xn}T be the n x 1 random
vector.
 For K vector samples from a random
population, the mean vector is given by
 The covariance matrix of the population is
given by
1
1 K
k
kK 
 xm x
1
1 K
T T
k k
kK 
 x x xC x x m m

KL Transform
 Cx is n x n real and symmetric matrix.
 Therefore a set on n orthonormal eigenvectors
is possible.
 Let ei and i, i=1, 2, …, n, be the eigenvectors
and corresponding eigenvalues of Cx, arranged
in descending order so that j ≥ i+1 for j = 1, 2,
…, n.
 Let A be a matrix whose rows are formed from
the eigenvectors of Cx, ordered so that first row
of A is eigenvector corresponding to the largest
eigenvalue, and the last row is the eigenvector
corresponding to the smallest eigenvalue.

KL Transform
 Suppose we use A as a transformation
matrix to map the vectors x’s into the
vectors y’s as follows:
y = A(x – mx)
This expression is called the Hotelling
transform.
 The mean of the y vectors resulting from
this transformation is zero; that is my =
E{y} =0.

KL Transform
 The covarianve matrix of the y’s is given in
terms of A and Cx by the expression
Cy = ACxAT
 Cy is a diagonal matrix whose elements along
the main diagonal are the eigenvalues of Cx
1
2
0
0 n



 
 
 
 
 
 
yC

KL Transform
 The off-diagonal elements of this
covariance matrix are 0, so that the
elements of the y vectors are
uncorrelated.
 Cx and Cy have the same eigenvalues
and eigenvectors.
 The inverse transformation is given by
x = ATy + mx

KL transform
 Suppose, instead of using all the eigenvectors
of Cx we form a k x n transformation matrix
Ak from k eigenvectors corresponding to k
largest eigenvalues, the vector reconstructed
by using Ak is
 The mean square error between x and is
T
k xx = A y + m
x
1 1 1
n k n
ms j j j
j j j K
e   
   
    

KL Transform
 As j’s decrease monotonically, the error can be
minimised by selecting the k eigenvectors
associated with the largest eigenvalues.
 Thus Hotelling transform is optimal i.e. it
minimises the min square error between x and
 Due to the idea of using the eigenvectors
corresponding to the largest eigenvalues, the
Hotelling transform is also known as the
principal components transform.
x

KL transform example
2 3 4
0 1 1 1
0 0 1 0
0 0 0 1
       
                 
              
1x x x x
3 1 1
1
1 3 1
16
1 1 3
 
  
  
xC =
3
1
1
4
1
 
   
  
xm
0.5774 0.5774 0.5774
-0.1543 -0.7715 0.6172
0.8018 0.2673 0.5345
A
 
 
 
  
=
1 2 30.0625 0.2500 0.2500    
0.1443 -0.4330 0.1443 0.1443
0.1543 -0.0000 -0.7715 0.6172
-0.8018 0.0000 0.2673 0.5345
 
 
 
  
y =
0
0
0
 
   
  
ym
0.0833 0.0000 0.0000
0.0000 0.3333 0.0000
0.0000 0.0000 0.3333
 
 
 
  
yC =

a) Original Image,
b) Reconstructed using all the three principal components,
c) Reconstructed image using two largest principal components,
d) Reconstructed image using only the largest principal component
KL Transform Example
a b
c d

Unit ii

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Unit ii