1. Digital Image Processing Chapter 4: Image Enhancement in the Frequency Domain

2. 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

4.  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

5.  Applications Heat diffusion Fast Fourier transform (FFT) developed in the late 1950s

6. Introduction to the Fourier Transform and the Frequency Domain  The one-dimensional Fourier transform and its inverse Fourier transform Inverse Fourier transform

7. Two variables Fourier transform Inverse Fourier transform

8.  Discrete Fourier transform (DFT) Original variable Transformed variable

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

11.  Sines and cosines

12.  Time domain  Time components  Frequency domain  Frequency components

13.  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

14.  Polar coordinates Real part Imaginary part

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

17.  Samples

18.  Some references http://local.wasp.uwa.edu.au/~pbourk e/other/dft/ http://local.wasp.uwa.edu.au/~pbourk e/other/dft/ http://homepages.inf.ed.ac.uk/rbf/HIP R2/fourier.htm http://homepages.inf.ed.ac.uk/rbf/HIP R2/fourier.htm

19.  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