IMAGE TRANSFORM

IMAGE TRANSFORM

Depending on the transform chosen an operator or function selects an image as input and produces an image as output. KL analysis, fourier transforms and various spatial filters are image transformation procedure.

HAAR TRANSFORM

One of the simplest wave transform is Haar transform. Fourier transform cross multiplies a function against with two phase and many stretches against sine wave same as Haar transform cross multiplies a function with various shift and stretches against the Haar wavelet.

It is a transform which is derived from the Harr matrix. It is a kind of sampling process in ehich transformation matrix rows behaves as a sample of finer resolution.

Example of Haar transform matrix

SLANT TRANSFORM

Slant transform is developed by Pratt et al and introduced by Enomoto and Shibata in 1971.

[V] = [Sn][U][Sn]r

It is a 2D slant transform. Where, U is the original image of size N X N. and Sn is the unitary slant matrix.

This transform is a member of orthogonal transform. And for the first row it has a constant function and for the second row it has a linear function of the column index.

Properties of slant transform:

Real and orthogonal

Fast transform

Good energy compaction

KL TRANSFORM

KL transform is introduced by the Karhunen and Louve for continuous random process as a series expansion. It is an adaptive technique of signal processing and it is a tool used for data compression and pattern recognition.

Properties of KL transform are

KL transform decorrelates the process

Total mean square is minimized by the KL expansion

Explained variance

Representation entropy property is minimum

2-D Discrete Fourier Transform(DFT)

2-D DFT of an NxN image {u(m, n)} Is given as

N −1 N −1 v ( k , l ) = ∑∑ u ( m, n )WN WN n , km l 0 ≤ k, l ≤ N −1 m =0 n =0

and it is a seperable transform.

And inverse transform of 2D DFT is given as

N −1 N −1 v ( k , l ) = ∑∑ u ( m, n ) WN WN n , km l 0 ≤ k, l ≤ N −1 m=0 n =0

PROPERTIES OF 2D DFT:

Conjugate symmetry

Symmetric and unitary

It is a fast transform so 2-D DFT is separable.

SAMPLING THEOREM

Sampling theorem is also known as Nyquist-Shanon sampling theorem. This theorem is given by Harry Nyquist and Claude Shanon.

Sampling is a process of converting continuous time or space function into discrete time or space functionor converting a signal into a numeric sequence.

and for all

Where, X(f) is the fourier transform of function x(t) which is band limited. And at the intervals of T seconds when x(t) is sampled uniformly then result is denoted by x(nT) and the sample rate is given by

fs = 1/T

fs > 2B and B < fs / 2 is the sufficient condition to reconstruct the x(t) where, 2B is the Nyquist rate and fs / 2 is the Nyquist frequency. The condition described by these inequalities are called as Nyquist criterion.

QUESTION AND ANSWER

1. What is an image transform??

ANSWER

Depending on the transform chosen an operator or function selects an image as input and produces an image as output

2. What are the image transformation procedure??

ANSWER

Image transformation procedure are

KL analysis, fourier transform and various spatial filters.

3. What are the properties of slant transform??

ANSWER

Properties of slant transform are

Good energy compaction

Fast transform

Real and orthogonal

4. What is sampling??

ANSWER

Sampling is a process of converting continuous time and space function into discrete time and space function. Sampling theorem is also known as Nyquist-Shannon sampling theorem and it is named after Harry Nyquist and Claude Shanon.