Introduction to Frequency domain

In the frequency domain, a digital image is converted from spatial domain to frequency domain. In the frequency domain, image filtering is used for image enhancement for a specific application. A Fast Fourier transformation is a tool of the frequency domain used to convert the spatial domain to the frequency domain. For smoothing an image, low filter is implemented and for sharpening an image, high pass filter is implemented. When both the filters are implemented, it is analyzed for the ideal filter, Butterworth filter and Gaussian filter.

The frequency domain is a space which is defined by Fourier transform. Fourier transform has a very wide application in image processing. Frequency domain analysis is used to indicate how signal energy can be distributed in a range of frequency.

The basic principle of frequency domain analysis in image filtering is to computer 2D discrete Fourier transform of the image.

Fourier Series and Transform

Fourier Series

Fourier series is a state in which periodic signals are represented by summing up sines and cosines and multiplied with a certain weight. The periodic signals are further broken down into more signals with some properties which are listed below:

1. Broken signals are sines and cosines.
2. New signals are harmonics of each other.

Fourier series analysis of a step edge:

Fourier decomposition

Example:

Fourier Transformation

Fourier transformation is a tool for image processing. it is used for decomposing an image into sine and cosine components. The input image is a spatial domain and the output is represented in the Fourier or frequency domain. Fourier transformation is used in a wide range of application such as image filtering, image compression. Image analysis and image reconstruction etc.

The formula for Fourier transformation:

Example: