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Unit - IIImage Transforms
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The Discrete Fourier Transform
Continuous function f(x) is discretized into a sequence
N samples x units apart
Where x = discrete values = 0,1,2, .N-1
The sequence any N uniformly spaced samples from a correspondingcontinuous function
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2-D functions and their Fourier spectra
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Sampling continuous function
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1D-Discrete Fourier Transform (DFT) pair :
DFT and Inverse DFT
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2D-Discrete Fourier Transform pair:
DFT and Inverse DFT
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2D-Discrete Fourier Transform pair:
DFT and Inverse DFT
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Fast Fourier Transform (FFT)
1D-Discrete Fourier Transform pair:
DFT and Inverse DFT
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2D-Discrete Fourier Transform pair:
DFT and Inverse DFT
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2D-Discrete Fourier Transform pair:
DFT and Inverse DFT
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The number of complex multiplications and additions in DFT
N2
The number of complex multiplications and additions in FFT N log2N.
Decomposition procedure FFT algorithm
The reduction in proportionality from N2to N log2N operations.
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FFT Algorithm
Discrete Fourier Transform
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Number of operations
The number of complex multiplications and additions required toimplement FFT Algorithm:
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Inverse FFT
1-D DFT and Inverse DFT
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Taking Complex conjugate and dividing both sides by N:
Similarly for 2-D
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Implementation
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FORTAN implementation of successive doubling algorithm
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Other Separable Transforms
1-D Discrete Fourier Transform (DFT)
T(u) Forward Transformion of f(x)g(x,u)Forward Transform Kernelu = 0,1,.,N-1
1-D Inverse Discrete Fourier Transform (DFT)
f(x) Inverse Transformh(x,u)Inverse Transformation Kernel
x = 0,1,.,N-1
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2-D Discrete Fourier Transform (DFT)
T(u,v) Forward Transformion of f(x)
g(x,y,u,v)Forward Transform Kernelu = 0,1,.,N-1
v = 0,1,.,N-1
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2-D Inverse Discrete Fourier Transform (DFT)
f(x,y) Inverse Transform
h(x,y,u,v)Inverse Transformation Kernel
x = 0,1,.,N-1
y = 0,1,.,N-1
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Forward kernel is separable
Forward kernel is symmetric
g1 = g2
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Example
Forward kernel
Forward kernel is separable and symmetric
Inverse Fourier kernel is also separable and symmetric
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Separable kernel computed in two steps
1.1-D Transform along each row of f(x,y):
x,v = 0,1,2,,N-1
2. 1-D Transform along each column of T(x,v):
u,v = 0,1,2,,N-1
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Walsh Transform
N = 2n
1-D Forward Walsh Transform Kernel:
1-D Forward Discrete Walsh Transform of f(x):
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Values of 1-D Walsh Transformation Kernel for N=8
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1-D Inverse Walsh Transform Kernel:
1-D Inverse Walsh Transform:
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2-D Forward Walsh Transform Kernel:
2-D Forward Discrete Walsh Transform of f(x):
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2-D Inverse Walsh Transform Kernel:
2-D Inverse Walsh Transform:
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Walsh Transform kernel are separable and symmetric:
M difi ti f th i d bli FFT l ith f
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Modification of the successive doubling FFT algorithm forcomputing the fast Walsh transform
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Hadamard Transform
1-D Forward Hadamard Transform Kernel:
1-D Forward Hadamard Transform:
where N = 2n
foru= 0,1,2,.,N-1
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1-D Inverse Hadamard Transform Kernel:
1-D Inverse Hadamard Transform
forx= 0,1,2,.,N-1
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2-D Forward Hadamard Transform Kernel:
2-D Forward Hadamard Transform:
foru= 0,1,2,.,N-1v= 0,1,2,.,N-1
2 D I H d d T f K l
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2-D Inverse Hadamard Transform Kernel:
2-D Inverse Hadamard Transform:
forx= 0,1,2,.,N-1
y= 0,1,2,.,N-1
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Hadamard Transform kernel are separable and symmetric:
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Discrete Cosine Transform (DCT)
1-D Forward Discrete Cosine Transform (DCT) Kernel:
1-D Forward DCT :
foru= 0,1,2,.,N-1
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1-D Inverse DCT Kernel:
1-D Inverse DCT
forx= 0,1,2,.,N-1
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2-D Forward Discrete Cosine Transform (DCT) Kernel:
2-D Forward DCT :
where N = 2n
foru= 0,1,2,.,N-1
v= 0,1,2,.,N-1
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2-D Inverse DCT Kernel:
2-D Inverse DCT
forx= 0,1,2,.,N-1
y= 0,1,2,.,N-1
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Haar Transform
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Haar Transform
Haar Transform based on Haar functions hk(z)
Continuous and closed interval z[0, 1]
for k = 0,1,2,,N-1 where N = 2n
First step in generating Haar Transform, integer k decomposed
uniquelyk = 2p+ q 1 where N = 2n
Where 0 p n-1
q= 0or 1 for p = 01 q 2
p for p 0
if N=4,
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k, p and q values:
Haar Functions:
and
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Slant Transform
Slant Transform matrix of order N x N is the recursive expression
SN:
IM Identity matrix of order M x M
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The coefficients are
and
For N>1
Example
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Slant matrix S4:
Hotelling Transform
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Developed based on stastical properties of vector representations
Tool for Image processing (H.T. has several useful properties)
Population of random vectors of the form:
The mean vector of the population:
E{arg} expected value of the argument
Subscript m associated with the populationof x vectors
The Covariance matrix of the vector population:
Wh T V t T iti
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Where T Vector Transposition
x is n dimensional
Cx & (x-mx) (x-mx)T matrices of order n x n
Element cii of Cx variance of xi
ith componeny of the x vectors in the population
Element cij of Cx covariance between elements xi and xj
Matrix Cx real & symmetric
If elements xi and xj are correlated,
the covariance = 0 & cij = cji = 0
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For M vector samples from the random population,
The mean vector & covariance from the samples:
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