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Transform ácie obrazu

Transform ácie obrazu. Gonzales, Woods: Digital Image Processing k apitola : Image transforms. Fourierova transformácia. Jean Baptiste Joseph Fourier (1768-1830) Akákoľvek funkcia f(x) môže byť vyjadrená ako vážený súčet sínusov a kosínusov. Suma s í nusov a kos ínusov.

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Transform ácie obrazu

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  1. Transformácieobrazu Gonzales, Woods: Digital Image Processing kapitola: Image transforms

  2. Fourierova transformácia Jean BaptisteJosephFourier (1768-1830) Akákoľvek funkcia f(x) môže byť vyjadrená ako vážený súčet sínusov a kosínusov

  3. Suma sínusov a kosínusov • A sin(ωx+φ)

  4. Time and Frequency • example : g(t) = sin(2pf t) + (1/3)sin(2p(3f) t)

  5. Time and Frequency • example : g(t) = sin(2pf t) + (1/3)sin(2p(3f) t) = +

  6. Frequency Spectra • example : g(t) = sin(2pf t) + (1/3)sin(2p(3f) t) = +

  7. Frequency Spectra

  8. Frequency Spectra = + =

  9. Frequency Spectra = + =

  10. Frequency Spectra = + =

  11. Frequency Spectra = + =

  12. Frequency Spectra = + =

  13. Frequency Spectra =

  14. Frequency Spectra

  15. FT: Just a change of basis M * f(x) = F(w) = * . . .

  16. IFT: Just a change of basis M-1 * F(w)= f(x) = * . . .

  17. Fourierova transformácia Ak f(x) je spojitá funkcia s reálnou premenou x, potom Fourierovou transformáciou F(x) je Inverznou Fourierovou transformáciou nazývame pár Fourierovej transformácie

  18. Inverse Fourier Transform Fourier Transform F(u) f(x) F(u) f(x) Fourier Transform Pre každé u od0 poinf, F(u)obsahujeamplituduA and fázuf odpovedajúceho sinusu

  19. Definitions • F(u)sú komplexne čísla: • Magnitúda FT (spektrum): • Fázový uhol FT: • Reprezentácia pomocou magnitúdy a fázy: • Power of f(x): P(u)=|F(u)|2=

  20. FT – je periodická s periódou N, to znamená, ženajejurčeniestačíjednaperiódavofrekvenčnejoblasti.

  21. 2D Fourierova transformácia Fourierova transformáci je ľahko rozšíriteľná do 2D

  22. Príklad 2D funkcie

  23. Sinusoidne vzory sa zobrazia vo frekvenčnom spektre ako body.

  24. Nízke frekvencie sú pri strede a vysoké na okrajoch

  25. Sampling Theorem Continuous signal: Shah function (Impulse train): Sampled function:

  26. Sampling

  27. Sampling and the Nyquist rate • Pri vzorkovaní spojitej funkcie môže vzniknút aliasingak vzorkovacia frekvencia nie je dostatočne vyskoká • Vzorkovacia frekvencia musí byť taky vyskoká aby zachytila aj tie najvyžšie frekvencie obrazu

  28. Sampling and the Nyquist rate • Predísť aliasingu: • Vzorkovacia frekvencia > 2 * max frekvencia v obraze • Treba viac ako 2 vzorky na periodu • Minimálna vzorkovacia frekvencia sa nazýva Nyquist rate

  29. Diskrétna Fourierova transformácia

  30. RozšírenieDFT do2D • Predpokladajme že f(x,y) jeM x N. • DFT • InverznáDFT:

  31. Filtre

  32. Visualizing DFT • Typically, we visualize |F(u,v)| • The dynamic range of |F(u,v)| is typically very large • Apply streching:(c is const) original image before scaling after scaling

  33. DFT Properties: (1) Separability • The 2D DFT can be computed using 1D transforms only: Forward DFT: Inverse DFT: kernel is separable:

  34. DFT Properties: (1) Separability • Rewrite F(u,v) as follows: • Let’s set: • Then:

  35. ) DFT Properties: (1) Separability • How can we compute F(x,v)? • How can we compute F(u,v)? N x DFT of rows of f(x,y) DFT of cols of F(x,v)

  36. DFT Properties: (1) Separability

  37. DFT Properties: (2) Periodicity • The DFT and its inverse are periodic with period N

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