2. Image Transformation
Concept of Image Transformation
Unitary Matrix
Orthogonal & Orthonormal Basis Vector
Arbitrary 1D Signal Representation as Series Summation of O
An Arbitrary Image Representation as Series
Summation of Orthonormal Basis Vector
Computational Complexity of Image Transformation
Operation
Problem
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6. Unitary Matrix
Unitary Matrix: A matrix A is a unitary matrix if
A-1
= A* T
Where A*
is conjugate of A.
A unitary matrix is also a Basis Image
8. Orthogonal & Orthonormal Basis
Vector
A set of real valued continuous function
{an(t)} = {a0(t), a1(t), ……….)}/(t0,t0+T)
Is said to be Orthogonal over interval (t0, t0+T)
iff
Where K is some constant
If K = 1 Then set is said to Orthonormal basis
function
10. Arbitrary 1D Signal Representation as
Series Summation of Orthogonal Basis
Vector
Suppose we have arbitrary signal
X(t) and t0 ≤ t ≤ t0+T
Representation as Series Summation
cn is nth
coefficient of expansion
How to find nth
coefficient ?
11. Cont..
By using Orthogonalilty definition we get
0 0 0
By Orthogonalilty definition we get constant K only where M = N and for all others
values we get 0
12. Cont..
The orthogonal basis function is complete or close if at least one of
the following condition hold
1 . There is no signal x(t) with
Such that
2. For any piecewise continuous signal
x(t) with
If their exist N and є < 0
Such that
14. An Arbitrary Image Representation as
Series Summation of Orthonormal Basis
Vector
Let {u(n); 0 ≤ n ≤ N -1 }
Is 1-D discrete set of sample size N we can
simply represent by vector u of dimension N
Au where A is a Unitary matrix of size NxN
Hence a Transformed vector v can be
represented as v = Au ( A is Transformation
matrix)
So a series summation form we can write
16. Cont..
If the basis vector has to be orthogonal or
orthonormal
And A*T
is set of basis vector then
Dot product of any two distinct column should
be zero and dot product of column with itself
should not be zero i.e
17. Image Transformation
Let u(m,n) is a Image where 0 ≤ m,n ≤ N-1
Hence a transformed Image v(k,l) is
Inverse Transformation is
Unitary matrix
Input Image
Transformed Image
Conjugate of Unitary matrix
Transformed Image
Input Image
19. Computational Complexity
To compute v (k,l) (Transformed Image)
The no. of complex multiplication and
complex addition needed is performed in the
order of O(N2
). For every value k & l and for
computing each coefficient we need O(N4
)
computation
If computational complexity is order of
O(N4
) then it is difficult for transforming images
of size 256 x 256, 512 x 512 and so on
To reduce computational complexity we need
Separable Unitary Transformation
20. Separable Unitary
Transformation
akl(m,n) is seperable iff it can be represented like
ak(m).bl(n) ≈ a(k,m).b(l,n)
Here
{ ak(m) k = 0,1, … , N-1}
{ bl(n) k = 0,1, … , N-1}
They are 1D complete orthogonal basis vector and
A ≈ {ak(k,m)} B ≈ {bl(l,n)}
Both A and B are unitary matrix
Hence
AA*T
= AT
A* = I
21. Separable Unitary
Transformation
To reduce complexity we assume A and B are same
So
In matrix form It can be written as
V = A U AT
where V coefficient matrix and U is input image
Inverse Transformation is written as
In matrix form It can be written as
U = A*T
V A*
22. Separable Unitary
Transformation
No V = A U AT
can also be represented as
VT
= A [ A U ]T
It transform each column of U with matrix and
then transform each row of result with matrix A
Here A and U are NxN matrices
Hence for multiplication we know the complexity
is O(N3
) i.e. So total no. of multiplication in
separable unitary transformation is O(2N3
)
Hence the complexity is reduced to
O(2N3
) from O(N4
)
24. Example
Find Transformed image and Basis Images where Input image (U)
and unitary matrix (A) is
Transformed Image (V)
V = AUAT
25. Basis Image
Basis Image: Let a*
k → kth
column of A*T
Basis Image is computed as
A*
kl = a*
k. a*T
l (product of kth
column & lth
row)
Given
Basis Images are
26. Inverse Transformation
After Inverse Transformation we should get
original Image
U = A*T V A* where V is Transformed image
U is the original image