Digital Image Processing

Chapter 4: Image Enhancement in the Frequency Domain

Background

The French mathematiian Jean Baptiste Joseph Fourier Born in 1768 Published Fourier series in 1822 Fourier’s ideas were met with

skepticism Fourier series

Any periodical function can be expressed as the sum of sines and/or cosines of different frequencies, each multiplied by a different coefficient

Fourier transform Functions can be expressed as the

integral of sines and/or cosines multiplied by a weighting function

Functions expressed in either a Fourier series or transform can be reconstructed completely via an inverse process with no loss of information

1),( yx2222222 2/)()(222 vuyx AeeA

Applications Heat diffusion Fast Fourier transform (FFT) developed

in the late 1950s

Introduction to the Fourier Transform and the Frequency Domain

The one-dimensional Fourier transform and its inverse Fourier transform

Inverse Fourier transform

dxexfuF uxj 2)()(

dueuFxf uxj 2)()(

Two variables

dxdyeyxfvuF vyuxj )( 2),(),(

dudvevuFyxf vyuxj )( 2),(),(

Fourier transform

Inverse Fourier transform

Discrete Fourier transform (DFT) Original variable

Transformed variable

1,...,2,1,0),( Mxxf

1,...,2,1,0),( MuuF

1,...,2,1,0

,)(1

)(1

0

/ 2

Mu

exfM

uFM

x

Muxj

1,...,2,1,0

,)()(1

0

/ 2

Mx

euFxfM

u

Muxj

DFT The discrete Fourier transform and its

inverse always exist f(x) is finite in the book

Sines and cosines

sincos je j

1

0

]/ 2sin/ 2)[cos(1

)(M

x

MuxjMuxxfM

uF

Time domain

Time components

Frequency domain

Frequency components

x

)(xf

u

)(uF

Fourier transform and a glass prism Prism

Separates light into various color components, each depending on its wavelength (or frequency) content

Fourier transform Separates a function into various

components, also based on frequency content

Mathematical prism

Polar coordinates

Real part

Imaginary part

)()()( ujeuFuF

)(uR

)(uI

Magnitude or spectrum

Phase angle or phase spectrum

Power spectrum or spectral density

2

122 )()()( uIuRuF

)(

)(tan)( 1

uR

uIu

)()()()( 222uIuRuFuP

Samples

)()( 0 xxxfxf

)()( uuFuF

xMu

1

Some references http://local.wasp.uwa.edu.au/~pbourke/

other/dft/ http://homepages.inf.ed.ac.uk/rbf/HIPR2

/fourier.htm

Examples test_fft.c fft.h fft.c Fig4.03(a).bmp test_fig2.bmp