1 / 23

The Fourier Transform

The Fourier Transform. Jean Baptiste Joseph Fourier. A. A. sin(x). 3 sin(x). B. + 1 sin(3x). A+B. + 0.8 sin(5x). C. A+B+C. + 0.4 sin(7x). D. A+B+C+D. A sum of sines and cosines. =. …. Higher frequencies due to sharp image variations (e.g., edges, noise, etc.).

batey
Download Presentation

The Fourier Transform

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. The Fourier Transform Jean Baptiste Joseph Fourier

  2. A A sin(x) 3 sin(x) B + 1 sin(3x) A+B + 0.8 sin(5x) C A+B+C + 0.4 sin(7x) D A+B+C+D A sum of sines and cosines = …

  3. Higher frequencies dueto sharp image variations (e.g., edges, noise, etc.)

  4. The Continuous Fourier Transform Basis functions:

  5. Complex Numbers Imaginary Z=(a,b) b |Z|  Real a

  6. The Continuous Fourier Transform Basis functions:

  7. The 1D Basis Functions 1 x 1/u • The wavelength is 1/u . • The frequency is u .

  8. The Continuous Fourier Transform The InverseFourier Transform The Fourier Transform An orthonormal basis  1D Continuous Fourier Transform: Basis functions:

  9. Some Fourier Transforms Fourier Transform Function

  10. The Continuous Fourier Transform The InverseFourier Transform The Fourier Transform 2D Continuous Fourier Transform: The Inverse Transform The Transform 1D Continuous Fourier Transform:

  11. The 2D Basis Functions V u=-2, v=2 u=-1, v=2 u=0, v=2 u=1, v=2 u=2, v=2 u=-2, v=1 u=-1, v=1 u=0, v=1 u=1, v=1 u=2, v=1 U u=0, v=0 u=-2, v=0 u=-1, v=0 u=1, v=0 u=2, v=0 u=-2, v=-1 u=-1, v=-1 u=0, v=-1 u=1, v=-1 u=2, v=-1 u=-2, v=-2 u=-1, v=-2 u=0, v=-2 u=1, v=-2 u=2, v=-2 The wavelength is . The direction is u/v .

  12. Discrete Functions f(x) f(n) = f(x0 + nDx) f(x0+2Dx) f(x0+3Dx) f(x0+Dx) f(x0) 0 1 2 3 ... N-1 x0+2Dx x0+3Dx x0 x0+Dx The discrete function f: { f(0), f(1), f(2), … , f(N-1) }

  13. The Finite Discrete Fourier Transform 2D Finite Discrete Fourier Transform: (u = 0,..., N-1; v = 0,…,M-1) (x = 0,..., N-1; y = 0,…,M-1) 1D Finite Discrete Fourier Transform: (u = 0,..., N-1) (x = 0,..., N-1)

  14. Fourier spectrum |F(u,v)| The Fourier Image Fourier spectrum log(1 + |F(u,v)|) Image f

  15. Frequency Bands Image Fourier Spectrum Percentage of image power enclosed in circles (small to large) : 90%, 95%, 98%, 99%, 99.5%, 99.9%

  16. Low pass Filtering 90% 95% 98% 99% 99.5% 99.9%

  17. Noise-cleaned image Fourier Spectrum Noise Removal Noisy image

  18. High Pass Filtering Original High Pass Filtered

  19. High Frequency Emphasis + Original High Pass Filtered

  20. High Frequency Emphasis Original High Frequency Emphasis

  21. High Frequency Emphasis Original High Frequency Emphasis

  22. High pass Filter High Frequency Emphasis High Frequency Emphasis Original

  23. Properties of the Fourier Transform – Developed on the board…(e.g., separability of the 2D transform, linearity, scaling/shrinking, derivative, shift  phase-change, rotation, periodicity of the discrete transform.)We also developed the Fourier Transform of various commonly used functions, and discussed applications which are not contained in the slides (motion, etc.)

More Related