Image Transformations

Introduction:

Two dimensional unitary transforms play an important role in image processing. The term image transform refers to a class of unitary matrices used for representation of images.

In analogy with I-D signals that can be represented by an orthogonal series of basis functions , we can similarly represent an image in terms of a discrete set of basis arrays called " basis images ". These are generated by unitary matrices.

Alternatively an image can be represented as vector. An image transform provides a set of coordinates or basis vectors for the vector space.

I - D-Transforms:

For a one dimensional sequence representing a vector of size N , a unitary transform is :=

	for	(4.1.1)

where = (unitary)

This implies , =

or ,	for	(4.1.2)

Equation (4.1.2) can be viewed as a series representation of sequence u(n) . The columns of i.e the vectors

are called the "basis vectors" of .

The series coefficients v(k) give a representation of original sequence u(n) and are useful in compression , filtering , feature extraction and other analysis.