Image Transformations
Introduction:
Two dimensional unitary transforms play an important role in
image processing. The term image transform refers to a class of
unitary matrices used for representation of images.
In anal
ogy with I

D signals that can be represented by a
n
orthogonal series of basis functions , we can similarly represent an
image in terms of a discrete set of basis arrays called “
basis
images
”. These are generated by unitary matrices.
Alternatively
a
n
( N N )
image can be represented as
2
1
( N )
vector. An image transform provides a set of coordinates or basis
vectors for the vector space.
I

D

Transforms
:
For a one dimensional sequence
01 1
u( n),n,......N
representing a vector
u
of size
N
, a unitary transform is :
v
=
A
u
v
(
k
)
=
1
0
N
n
a( k,n)u( n)
,
for
0 1
K N
(1)
(1)
where
1
A
=
T
A
*
T
*
(unitary)
This implies ,
u
=
T
A
*
v
or
,
u(n)
=
1
0
N
k
v( k )a* ( k,n)
,
for
0
n
1
N
(2)
Equation (2) can be viewed as
a series representation of sequence
u(n)
. The columns
of
T
A
*
i.e the vectors
*
k
a
0 1
T
*
a ( k,n),n N
are called the “basis
vectors” of
A
.
The series coeffi
cients
v(k)
give a representation of origin
al
sequence u(n) and are useful in compression , filtering , feature
extraction and other analysis.
Two dimensional Orthogonal and Unitary transforms:
As applied to image processing, a general orthogonal series
expansion for an
N N
image is a pair of transformations of the
form :
v(k,l
) =
1
0
N
k,l
m,n
u( m,n)a ( m,n)
,
0 1
k,l N
(3)
u(m,n)
=
1
0
N
*
k,l
k,l
v( k,l )a ( m,n)
,
0 1
m,n N
(4)
where
k,l
a ( m,n)
is called an ” image transform.”
It is a set of complete orthogonal discrete basis
functions
satisfying the properties
:

1)
Orthonormality
:
1
0
//
N
*
k,l
k,l
m,n
a ( m,n)a ( m,n)
=
δ
( k k,l l )
2) Completeness
:
1
0
N
k
k,l
k,l
k,l
a ( m,n)a ( m,n )
=
δ
( m m,n n )
The elements
v
( k,l )
are transform coefficients and
V
v( k,l )
is
the transformed image.
The orthonomality property assures that any truncated series
expansion of the form
P,Q
U ( m,n)
1
0
P
k
1
0
Q
*
k,l
l
v( k,l )a ( m,n)
, for
P N
,
Q N
w
ill minimize the sum of squares error
1
2
2
0
σ
N
e P,Q
m,n
u( m,n) U ( m,n)
where coefficients
v( k,l )
are given by (3).
The completeness property assures that this error
will be zero for
P Q N
Separable Unitary Transforms:
The number of multiplications and additions required to compute
transform coefficients
v( k,l )
in equation(3) is
4
O( N )
. This is too
large for practica
l size images.
If the transform is restricted to be separable,
i.e
k,l k l
a ( m,n ) a ( m)b ( n )
a( k,m)b( l,n)
where
0 1 1
k
a ( m),k ( )n
,
and
0 1 1
l
b ( n),l ( N
are 1D complete orthogonal sets of
basis vectors.
On imposition of completeness and orthonormality properties we
can show that
A
a( k,m)
,
and
B
b( l,n)
are unitary matrices.
i.e
T
*
AA
=
I
=
T *
A A
and
T
*
BB
=
I
=
T *
B B
O
ften one chooses
B
same as
A
v( k,l )
=
1
0
N
m,n
a( k,m)u( m,n)a( l,n)
V
=
A
U
T
A
(5)
And
u( m,n)
=
1
0
N
* *
k,l
a ( k,m)v( k,l )a ( l,n)
U
=
T
*
A
V
*
A
(6)
Eqn (5) can be written as
T
V
=
T
A( AU )
Eqn (5) can be performed by first transforming each column of
U
and then tr
ansforming each row of the result to obtain rows of
V
.
Basi
s
Images
:
Let
*
k
a
denote
th
k
column of
T
*
A
.
Let us define the matrices
T
*
* *
k l
k,l
A a a
and matrix inn
er
product of two
N N
matrices
F
and
G
as
F,G
=
1
0
N
*
m,n
f ( m,n)g ( m,n)
Then equ (6) and (5) give a series representation
.
U
=
1
0
N
k,l
v( k,l )
*
k,l
A
and
v( k,l )
=
*
k,l
u,A
A
ny image
U
can be expressed as linear combination of
2
N
matrices.
*
k,l
A
called “basi
s
images”.
Therefore any
N N
image can be expanded in a series using a
complete set of
2
N
basis images.
Example:
Let
A
=
1 1
1
1 1
2
;
U
=
1 2
3 4
Transformed image
V
=
A
U
T
A
=
5 1
2 0
And Basis images are found as outer product of columns of
*
T
A
i.e
0 0
*
,
A
=
1
1
(1 1)
1
2
0 1
*
,
A
=
1
1
(1 1)
1
2
=
1 1
1
1 1
2
=
1 0
*
,
T
A
11
*
,
A
=
1 1
1
1 1
2
1
(1 1)
1
The inverse transformation
* *
T
A V A
=
1 1
1
1 1
2
5 1
2 0
1 1
1 1
=
1 2
3 4
=
U
Dimensionality of Image transforms
The
3
2
N
computations for
V
can also be reduced by
r
estricting the
choice of
A
to
fast transforms
.
This implies that
A
has a
structure that allows factorization of the type,
A
=
1 2
( ) ( ) ( )
.........
p
A A A
where
( )
i
A
,
1
i
,
p( p N )
are matrices with just a few non zero
entries say
r
where
r N
Therefore
a
multiplication of the type
:
y
=
Ax
is accomplished
in
rpN
o
perations.
For several transforms like Fourier, Sine, Cosine, Hadamard etc,
2
p log N
,
and operations reduce to the
order of
2
Nlog N
or
2
2
N log N
(for
N N
images).
Depending on the transform
,
an operation is defined as
1
multiplication + 1 addition.
Or, 1 addition or subtraction as in Hadamard Transform
.
Kronecker products
:
If
A
and
B
are
1 2
M M
and
1 2
N N
matrices w
e define
Kronecker
product as:
A
B
a( m,n)B
Consider the transform,
V
=
A
U
T
A
or,
v( k,l )
=
1
0
N
m,n
a( k,m)u( m,n)a( l,n)
(
7)
If
k
v
and
k
u
denote
th
k
and
th
m
row vecto
rs of
V
and
U
then (1)
becomes
,
T
k
v
=
m
a( k,m)
T
m
AU
=
T
m
m
k,m
A A u
where
k,m
is the
th
( k,m)
block of
A A
If
U
and
V
are row ordered into vector
s
v
and
u
respectively,
then
V
=
T
AUA
v
=
(
A A
)
u
The
number of operations required for implementing equation(
7
)
reduces from
4
O( N )
to
3
2
O( N )
.
Properties of Unitary transforms
:

1)
Energy conservatio
n
:
In the unitary transformation,
v
=
Au
,
2
v
=
2
u
Proof
2
v
2
1
0
N
k
v( k )
=
T
*
v v
=
T
*
u
T
A
Au
=
2
u
.
unitary transformation preserves signal
energy or equivalently the length of v
ector
u
in
N
dimensional vector space. That is , every unitary
transformation is simply a rotation of
u
in
N
dimensional
vector space.
Alternatively , a unitary transfor
m is a
rotation of basis coordinates and components of
v
are
projections of
u
on the new basis.
Similarly , for 2D
unitary transformations, it can be proved that
2
1
0
,
)
,
(
N
n
m
n
m
u
=
2
1
0
,
)
,
(
N
l
k
l
k
v
Example: Consider the vector
x
=
1
0
x
x
and
A
=
cos
sin
sin
cos
(
diagram
)
This
,
0
y
=
x
a
T
0
;
1
y
=
x
a
T
1
Transformation
y
=
x
A
can be written as
y
=
1
0
y
y
=
cos
sin
sin
cos
1
0
x
x
=
cos
sin
sin
cos
1
0
1
0
x
x
x
x
with new basis as
0
a
,
1
a
.
For 2D unitary transforms we have
2
)
,
(
m
n
n
m
u
=
2
)
,
(
k
l
l
k
v
.
2)
Energy Compaction Property
:
Most unitary transforms have a tenden
cy to pack a large
fraction of average energy of an image into relatively few
transform coefficients. Since total energy is preserved this
implies that many transform coefficients will contain very
little energy. If
u
and
u
R
denote the mean and covariance
of vector
u
then corresponding quantities for
v
are
v
v
E
=
]
[
u
A
E
=
u
A
And
v
R
=
T
u
v
v
v
E
*
)
(
)
(
=
]
)
(
)
(
[
*
T
u
u
A
u
A
A
u
A
E
=
A
]
)
(
)
(
[
*
T
u
u
A
u
A
A
u
A
E
T
A
*
=
T
A
R
A
u
*
Varia
nces of the transform coefficients are given by the
diagonal elements of
v
R
i.e
k
k
v
R
,
=
)
(
2
k
v
)
(
2
k
v
=
k
k
kT
u
A
R
A
,
Since
A
is unitary , it implies:
2
1
0
)
(
N
k
v
k
=
v
v
T
*
=
u
A
A
T
T
*
*
=
2
1
0
)
(
N
n
u
u
and
)
(
1
0
2
k
N
k
v
=
T
A
R
A
T
u
r
*
=
u
r
R
T
=
)
(
1
0
2
n
N
n
n
2
1
)
(
k
v
E
N
k
=
1
0
2
)
(
N
n
n
u
E
The average energy
2
)
(
k
v
E
of transform coefficients
)
(
k
v
tends to be unevenly distributed, although it may be evenly
distributed for input sequence
)
(
n
u
.
For a 2D random field
)
,
(
n
m
u
,
with mean
n
m
u
,
and
covariance
n
m
n
m
r
,
;
,
,
its transform coefficients
l
k
v
,
satisfy the properties
,
)
,
(
l
k
v
=
)
,
(
)
,
(
)
,
(
1
0
,
n
m
n
l
a
m
k
a
N
n
m
u
)
,
(
l
k
v
=
1
0
,
,
)
,
(
)
,
(
N
n
m
n
m
u
E
n
l
a
m
k
a
and
)
,
(
2
l
k
v
=
2
,
,
l
k
l
k
v
E
u
=
n
l
a
m
k
a
m
n
m
n
,
,
n
l
a
m
k
a
n
m
n
m
r
,
,
,
;
,
*
*
If covariance of
n
m
u
,
is separable i.e
n
m
n
m
r
,
;
,
=
n
n
r
m
m
r
,
,
2
1
Then variances of transform coefficients
can be written as a
sepa
rable product
l
k
v
,
2
=
k
k
2
2
2
1
k
k
k
k
T
T
A
R
A
A
R
A
,
*
2
,
*
1
where
1
R
=
m
m
r
,
1
;
n
n
r
R
,
2
2
3) Decorrelation : When input
vector elements are highly
correlated , the transform coefficients tend to be uncorrelated. That
is , the off

diagonal terms of covariance matrix
v
R
tend to be small
compared to diagonal elements.
4)
Other properties
:
(a) The determin
ant and eigenvalues of
a unitary matrix have unity magnitute.
(b) Entropy of a random vector is observed under unitary
transformation average information of the random vector
is preserved.
Example
:
Given the entropy of an
1
N
Gaussian
random vector
u
with mean
and covariance
u
R
, as :
u
H
=
2
1
2
2
log
2
u
R
e
N
To show
u
H
is invariant under any unitary
transformation.
Let
v
=
u
A
u
=
v
A
1
=
v
A
T
*
u
A
H
=
N
u
eR
N
1
2
2
log
2
Use the definition of
u
R
we have
N
u
u
T
A
u
A
A
u
A
E
1
*
=
N
T
u
A
R
A
1
*
=
N
u
R
1
Now
N
v
R
1
=
N
A
1
N
u
R
1
N
T
A
1
*
=
4
1
u
R
v
R
=
T
A
R
A
u
*
Also
T
A
A
*
=
*
A
A
T
=
I
A
R
A
v
=
A
A
R
A
A
T
u
*
=
T
T
u
A
A
R
*
=
u
R
u
A
H
=
N
v
A
R
A
e
N
T
1
*
2
log
2
=
)
2
log(
2
1
1
1
N
N
v
N
A
R
A
ex
N
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