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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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DFT and Inverse DFT

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Fast Fourier Transform (FFT)


DFT and Inverse DFT

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DFT and Inverse DFT

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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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