Image Transformation

Image Transformation
Two-dimensional image transforms are extremely important areas of study in
image processing. The image output in the transformed space may be
analyzed, interpreted, and further processed for implementing diverse image
processing tasks. These transformations are widely used, since by using these
transformations, it is possible to express an image as a combination of a set
of basic signals, known as the basis functions. In case of Fourier transform
of an image these basis signals are sinusoidal signals with different periods
which describe the spatial frequencies in an image. Thus such transforms, such as the Fourier transform, reveal spectral structures embedded in the image that may be used to characterize the image.
The term Image transform are usually refers to a class of unitary matrices used for representing images.
large class of image processing transformations is linear in nature; an output image is formed from linear combinations of pixels of an input image. Such transforms include convolutions, correlations, and unitary transforms. Linear transforms have been utilized to enhance images and to extract various features from images. For example, the Fourier transform is used in highpass and lowpass filtering as well as in texture analysis. Another application is image coding in which bandwidth reduction is achieved by deleting low-magnitude transform coefficients.

Fourier Transform
Fourier transform is one of the most important tools which have been extensively used not only for understanding the nature of an image and its formation but also for processing the image.
Using Fourier transform, it has been possible to analyze an image as a set of spatial sinusoids in various directions, each sinusoid having a precise frequency.
Discrete Fourier Transform (DFT)
Since the signal is discretized, the operation of integration in continuous Fourier trunsfomn (CFT) is replaced by summation operations in DFT. We present the one-dimensional DFT and also the two-dimensional DFT in the following subsections.
One-Dimensional DFT
The one-dimensional discrete Fourier transform of a function f(x) of size N with integer index x running from 0 to N - 1,
is represented by