Chapter 4 Image Processing: Image Transformation

CHAPTER4
IMAGE TRANSFORMATION
Dr. Varun Kumar Ojha
and
Prof. (Dr.) Paramartha Dutta
Visva Bharati University
Santiniketan, West Bengal, India

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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Concept of Image
Transformation
Image
NxN
Another
Image
NxN
Original
Matrix
Coefficient
Matrix
Inverse Image
Transformation
Image
Transformation
“Image Transformation represents a given image as a series
summation of a set of unitary matrices.”

Application of Image
Transformation
 Preprocessing
 Image Filtering (By modifying o-efficient matrix by
suppressing high frequency)
 Image Enhancement
 Data Compression
 Feature Extraction
 Edge Detection
 Corner Detection

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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

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

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 ?

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

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

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


Cont..
 Representation in the series summation of set
of basis vector
 Where it is basis of
A

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

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

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

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

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*

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
)

Example
 Find Transformed image and Basis Images where Input image (U)
and unitary matrix (A) is
 Transformed Image (V)
V = AUAT


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

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

Chapter 4 Image Processing: Image Transformation

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