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

 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

Two variables Fourier transform Inverse Fourier transform

 Discrete Fourier transform (DFT) Original variable Transformed variable

 DFT The discrete Fourier transform and its inverse always exist f(x) is finite in the book

 Sines and cosines

 Time domain  Time components  Frequency domain  Frequency components

 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

Magnitude or spectrum Phase angle or phase spectrum Power spectrum or spectral density

 Samples

 Some references e/other/dft/ e/other/dft/ R2/fourier.htm R2/fourier.htm

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