Digital image processing with fourier transform

The term monochrome image or simply image,refers to a two-dimensional light intensity function f(x,y), where x and y denote spatial coordinates and the value of /at any point (x,y) is proportional to the brightness(or gray level) of the image at that point. A digital image is an image f{x,y) can be considered a matrix whose row and column indices identify a point in the image and the corresponding matrix element value identifies the gray level at that pointThe elements of such a digital image array are called image elements,picture elements or pixels. Only the brightness levels have been considered, but it is possible to represent the same picture in the frequency domain. The Fourier Transform is an important image processing tool which is used to decompose an image into its sine and cosine components. The output of the transformation represents the image in the Fourier or frequency domain, while the input image is the spatial domain equivalent.. For discrete image data, as opposed to continuous functions, there is a discrete version of the Fourier transform called the DFT. There is another more computationally efficient version of the DFT called the Fast Fourier Transform (FFT). The Fourier Transform is used in a wide range of applications, such as image analysis, image filtering, image reconstruction and image compression. In this study;section 2 introduces the Fourier transforms of one and two continuous variables,section 3 then expresses these concepts in discrete form,section 4 develops and illustrates several important properties of the 2-D Fourier transform,section 5 develops a fast Fourier transform algorithm, which can be used to reduce the number of calculations to a fraction of that required to implement the discrete Fourier transform by direct methods. Keywords: Image processing,Fourier transforms,Discrete Fourier transforms,Fast Fourier transforms,sampling XI