Filtering and Frequency

In: Computers and Technology

Submitted By loveneesh
Words 3417
Pages 14
Content
This chapter is concerned primarily with helping the reader develop a basic understanding of the Fourier transform and the frequency domain, and how they apply to image enhancement.

Background Introduction to the Fourier Transform and the Frequency Domain DFT Smoothing Frequency-Domain Filters Sharpening Frequency-Domain Filters

4.1 Background
• Any function that periodically repeats itself can be expressed as the sum of sines and/or cosines of different frequencies, each multiplied by a different coefficient (Fourier series). • Even functions that are not periodic (but whose area under the curve is finite) can be expressed as the integral of sines and/or cosines multiplied by a weighting function (Fourier transform). • The advent of digital computation and the “discovery” of fast Fourier Transform (FFT) algorithm in the late 1950s revolutionized the field of signal processing, and allowed for the first time practical processing and meaningful interpretation of a host of signals of exceptional human and industrial importance.

• The frequency domain refers to the plane of the two dimensional discrete Fourier transform of an image. • The purpose of the Fourier transform is to represent a signal as a linear combination of sinusoidal signals of various frequencies.

=
Any function that periodically repeats itself can be expressed as a sum of sines and cosines of different frequencies each multiplied by a different coefficient – a Fourier series

4.2 Introduction to the Fourier Transform and the Frequency Domain
• The one-dimensional Fourier transform and its inverse – Fourier transform (continuous case)
F (u ) = ∫ f ( x)e − j 2πux dx where j = − 1
−∞ ∞

– Inverse Fourier transform: f ( x) = ∫ F (u )e j 2πux du
−∞ ∞

e jθ = cos θ + j sin θ

• The two-dimensional Fourier transform and its inverse – Fourier transform (continuous case)…...

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