Image and Video Compression

23

3

Image Transforms

3.1 Introduction

According to the Merriam-Webster dictionary, transform in its broadest sense

means “change in the external form or in the inner nature.” Continuing,

mathematical transform means: “to change to a different form having the

same value.” In this chapter, we are focusing on the image transforms.

Usually, an image is represented as a function of two spatial variables

xy(,)

and represented as

fxy(,)

. The intensity at a particular point in an image is

the value taken by the function

fxy(,)

at that spatial location. This domain

of representation is most common for image storage and display. Since an

image is a representation in space through spatial coordinates, domain is

termed as the spatial domain. The term image transform refers to the math-

ematical process of converting and representing an image into its alternative

form. For example, an image can be represented as a series summation of

sinusoids with varying degree of magnitude and frequencies by the cosine

transform. This alternative representation of an image is known as frequency

domain. A typical transformation process between spatial and frequency

domain is shown in Figure3.1.

From Figure3.1, it should be noted that the conversion from the spatial to

the frequency domain is known as a forward transform and the other way

around is known as an inverse transform. As noted in the denition earlier,

these transforms should be lossless, meaning that the information content of

the signal should not be altered or lost due to the process of transformation.

An important question that must be pondered upon is: Why do we need

these transforms when the information content should not change? These

transforms help in visualizing the same information content but from a dif-

ferent perspective. It is useful to visualize certain feature(s) of a signal in

transform domain when they are not visible in the parent domain. For exam-

ple, information about the frequency content of the signal is available only in

the frequency domain and not in its time domain visualization. In another

reasoning, certain image processing operations can be best realized by trans-

forming the signal, carrying out the desired operation, and returning to the

parent domain. The general principle in applying the transform is depicted

24 Image and Video Compression

in Figure3.2. Thus, input image is transformed from spatial domain to the

frequency domain. In that domain the desired operation is carried out and

then inverse transform is applied to go back to the spatial domain.

Image transform is useful in many applications and can be used for a

range of purposes. Some of the important applications are as follows:

Image enhancement—Transform can be used to exploit limitations of the

human visual system and improve perceptual quality and computa-

tional efciency of the image.

Filtering—Transform can help in isolating certain components of inter-

est in an image or can help to remove spurious noise components

from the band of interest.

Image compression—Image transform helps in de-correlating the data,

preserving the principal components of a signal, and quantizing

other components to obtain the signal compression.

Pattern recognition—Image transform captures many features like

edges, corners, and statistical moments to identify objects and/or to

classify them.

Convolution—It is a computationally expensive operation in time or

spatial domain, which in the frequency (transform) domain reduces

to a multiplication operation by using an image transform.

Forward

Transform

Desired

Operation

Inverse

Transform

Spatial

Domain

Spatial

Domain

Freq.Freq.

Domain Domain

FIGURE 3.2

The general principle in carrying out a specic operation in the transform domain.

Image in

spatial domain

Image in

frequency

domain

Forward

Transform

Inverse

Transform

FIGURE 3.1

A transformation in between the spatial and frequency domain.