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Image Enhancement in the Frequency Domain

Image Enhancement in the Frequency Domain. Spring 2006, Jen-Chang Liu. Outline. Introduction to the Fourier Transform and Frequency Domain Magnitude of frequencies Phase of frequencies Fourier transform and DFT Filtering in the frequency domain Smoothing Frequency Domain Filters

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Image Enhancement in the Frequency Domain

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  1. Image Enhancement in the Frequency Domain Spring 2006, Jen-Chang Liu

  2. Outline • Introduction to the Fourier Transform and Frequency Domain • Magnitude of frequencies • Phase of frequencies • Fourier transform and DFT • Filtering in the frequency domain • Smoothing Frequency Domain Filters • Sharpening Frequency Domain Filters • Homomorphic Filtering • Implementation of Fourier transform

  3. Background • 1807, French math. Fourier • Any function that periodically repeats itself can be expressed as the sum of of sines and/or cosines of different frequencies, each multiplied by a different coefficient (Fourier series)

  4. Periodic function f(t) = f(t+T), T: period (sec.) 1/T: frequency (cycles/sec.)

  5. Frequency Weight f1 w1 f2 w2 f3 w3 f4 w4 Periodic function f

  6. min Minimize squared error How to measure weights? • Assume f1 , f2 ,f3 ,f4 are known • How to measure w1 , w2 ,w3 ,w4 ?

  7. Minimize MSE calculation min

  8. Orthogonal condition 正交 • f1 and f2 are orthogonal if • f1 , f2 ,f3 ,f4 are orthogonal to each other

  9. Recall in linear algebra: projection Minimization calculation • To satisfy min We have =>

  10. Weight = Projection magnitude • Represent input f(x) with another basis functions Functional space Vector space f v projection f1 (1,0)

  11. Summary 1 • A function f can be written as sum of f1 , f2 ,f3 , … If f1 , f2 , f3 , … are orthogonal to each other Weight (magnitude)

  12. Summary 1: sine, cosine bases • Let f1 , f2 ,f3 , … carry frequency information • Let them be sines and cosines n, k:integers => They all satisfy orthogonal conditions

  13. Summary 1: orthogonal

  14. DC 頻率=3 頻率=1 頻率=2 Fourier series (Assume periodic outside) • For

  15. Outline • Introduction to the Fourier Transform and Frequency Domain • Magnitude of frequencies • Phase of frequencies • Fourier transform and DFT • Filtering in the frequency domain • Smoothing Frequency Domain Filters • Sharpening Frequency Domain Filters • Homomorphic Filtering • Implementation of Fourier transform

  16. 相位 Correlation with different phase • Weight calculation f 相關係數 f1

  17. Correlation with different phase (cont.) • Weight calculation 相關係數? f f1

  18. Corr(q) q 2p 0 q0 Deal with phase: method 1 • For example, expand f(t) over the cos(wt) basis function • Consider different phases Problem: weight(w, q)

  19. With frequency w: Deal with phase: method 2 • Complex exponential as basis j 1 real Advantage: Derive magnitude and phase q simultaneously

  20. Deal with phase 2: example • Input phase magnitude

  21. DC Fourier series with phase (Assume periodic outside) • For Complex weight 頻率k=1 k=2 k=3

  22. Outline • Introduction to the Fourier Transform and Frequency Domain • Magnitude of frequencies • Phase of frequencies • Fourier transform and DFT • Filtering in the frequency domain • Smoothing Frequency Domain Filters • Sharpening Frequency Domain Filters • Homomorphic Filtering • Implementation of Fourier transform

  23. Fourier transform • Functions that are not periodic can be expressed as the integral of sines and/or cosines multiplied by a weighting functions • Frequency up to infinity • Perfect reconstruction Functions -- Fourier transform Operation in frequency domain without loss of information

  24. 1-D Fourier Transform • Fourier transform F(u) of a continuous function f(x) is: Inverse transform: Forward Fourier transform:

  25. 2-D Fourier Transform • Fourier transform F(u,v) of a continuous function f(x,y) is: Inverse transform: x u F y v

  26. Future development • 1950, fast Fourier transform (FFT) • Revolution in the signal processing • Discrete Fourier transform (DFT) • For digital computation

  27. 1-D Discrete Fourier Transform • f(x),x=0,1,…,M-1 . discrete function • F(u),u=0,1,…,M-1. DFT of f(x) Inverse transform: Forward discrete Fourier transform:

  28. Frequency Domain 頻率域 • Where is the frequency domain? j Euler’s formula: 1 F(u) frequency u

  29. Fourier transform

  30. Physical analogy • Mathematical frequency splitting • Fourier transform • Physical device • Galss prism 三稜鏡 • Split light into frequency components

  31. F(u) Complex quantity? imaginary • Polar coordinate m real magnitude phase Power spectrum

  32. Some notes about sampling in time and frequency axis • Time index • Frequency index • Also follow reciprocal property

  33. Extend to 2-D DFT from 1-D • 2-D: x-axis then y-axis

  34. Complex Quantities to Real Quantities • Useful representation magnitude phase Power spectrum

  35. DFT: example log(F)

  36. Properties in the frequency domain • Fourier transform works globally • No direct relationship between a specific components in an image and frequencies • Intuition about frequency • Frequency content • Rate of change of gray levels in an image

  37. +45,-45 degree artifacts

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