Discrete Fourier transforms

Department of Data Communication Networks and Systems

Lecturer Shukhrat Palvanov

What is Fourier Transform (FT)?�

A powerful mathematical tool that converts a signal or image from the spatial (time/space) domain into the frequency domain.

Instead of analyzing the values of pixels directly, FT focuses on how often intensity values change.

Basic Idea

• Any complex signal or image can be expressed as a sum of simple sinusoidal components (sines and cosines).
• Low frequencies correspond to smooth variations (background, illumination).
• High frequencies correspond to rapid changes (edges, textures, noise).

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Importance of FT in Image Processing�

• Provides an alternative view of the image for analysis.
• Makes filtering (blurring, sharpening, noise removal) easier and more effective.
• Used in compression (JPEG), image enhancement, pattern recognition, and medical imaging.
• Helps separate important information (edges, structures) from redundant details or noise.

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Mathematical Background�Fourier Series vs Fourier Transform�

Fourier Series: Represents periodic signals as a sum of sinusoids.

Fourier Transform (FT): Extends the concept to non-periodic signals.

In image processing, signals (images) are generally non-periodic → FT is used.

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2D Fourier Transform�

• Images are 2D signals, so 2D Fourier Transform is used to analyze them.
• Converts an image f(x,y) in spatial domain into its frequency representation F(u,v).

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Discrete Fourier Transform (DFT)

Why DFT?

• Images are digital and consist of discrete pixels.
• Continuous Fourier Transform cannot be applied directly.
• Therefore, Discrete Fourier Transform (DFT) is used.

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