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Image Enhancement in Frequency Domain - Fourier Transform

Learn about Fourier Transform and its application in image enhancement. Explore filters, convolution theorem, smoothing filters, high-pass filters, and more.

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Image Enhancement in Frequency Domain - Fourier Transform

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  1. Chapter 4 Image Enhancement in the Frequency Domain

  2. Fourier Transform • 1-D Fourier Transform • 1-D Discrete Fourier Transform (DFT) • Magnitude • Phase • Power spectrum

  3. 2D DFT • Definition: if f(x,y) is real

  4. Centered Fourier Spectrum • It can be shown that:

  5. Example • SEM Image

  6. Filtering in the Frequency Domain • Multiply the input image by (-1)^x+y to center the transform • Compute F(u,v), the DFT of input • Multiply F(u,v) by a filter H(u,v) • Computer the inverse DFT of 3 • Obtain the real part of 4 • Multiply the result in 5 by (-1)^(x+y)

  7. Fourier Domain Filtering

  8. Some Basic Filters • Notch filter:

  9. Lowpass and Highpass Filters

  10. Convolution Theorem • Definition • Theorem • Need to define the discrete version of impulse function to prove these results.

  11. Gaussian Filters • Difference of Gaussians (DoG)

  12. Illustration

  13. Smoothing Filters • Ideal lowpass filters • Butterworth lowpass filters • Gaussian lowpass filters

  14. Ideal Lowpass Filters

  15. Example

  16. Ringing Effect

  17. Butterworth Lowpass Filters • Definition:

  18. Example

  19. Ringing Effect

  20. Gaussian Lowpass Filters • Definition:

  21. Example

  22. More example

  23. Sharpening Filters • High-pass filters • In general, • Ideal highpass filter • Butterworth highpass filter: • Gaussian highpass filters

  24. Relationship between Lowpass and Highpass Filters

  25. Spatial Domain Representation

  26. Ideal Highpass Example

  27. Butterworth Highpass Example

  28. Gaussian Highpass Example

  29. Laplacian in the Frequency Domain • It can be shown that: • Therefore,

  30. Illustration

  31. Other Filters • Unsharp masking: • High-boost filtering: • High-frequency emphasis filtering:

  32. Homomorphic Filtering

  33. Example

  34. DFT: Implementation Issues • Rotation • Periodicity and conjugate symmetry • Separability • Need for padding • Circular convolution • FFT

  35. Properties of 2D FT (1)

  36. Properties of 2D FT (2)

  37. FT Pairs

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