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2014年度春学期 画像情報処理 第4回 離散フーリエ変換 (2014. 5. 7)

Jun 15, 2015

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関西大学総合情報学部 「画像情報処理」(担当:浅野晃)

  • 1. A.Asano,KansaiUniv. 2014

2. A.Asano,KansaiUniv. 3. A.Asano,KansaiUniv. 4. 2014 A.Asano,KansaiUniv. n= fT (x) = f(x)combT (x) 2(x) (Diracs delta fun (x) = 0 (x = 0), (x)dx = 1 1 0 (x) = 0 (x = 0) 1 (x = 0) 5. 2014 A.Asano,KansaiUniv. n= fT (x) = f(x)combT (x) 2(x) (Diracs delta fun (x) = 0 (x = 0), (x)dx = 1 1 0 (x) = 0 (x = 0) 1 (x = 0) f(x) x fT(x) x 1: F(x, y) = f(x, y) exp{i2(xx + yy)}dxdy (7) (x, y) sampling, 1 (sampling theorem) x f(x) f(x) T fT (x) f(x) T 6. 2014 A.Asano,KansaiUniv. n= fT (x) = f(x)combT (x) 2(x) (Diracs delta fun (x) = 0 (x = 0), (x)dx = 1 1 0 (x) = 0 (x = 0) 1 (x = 0) f(x) x fT(x) x 1: F(x, y) = f(x, y) exp{i2(xx + yy)}dxdy (7) (x, y) sampling, 1 (sampling theorem) x f(x) f(x) T fT (x) f(x) T x ...... T (x) ... (xT) (xnT) 2: (11) 0 1 fT (x) fT (x) FT[f(x)g(x)]() = FT[f(x)]() FT[g(x)]() (12 FT[f(x)] f(x) convolution, f(t) g(t) = f(y)g(t y)dy (13 7. 2014 A.Asano,KansaiUniv. n= fT (x) = f(x)combT (x) 2(x) (Diracs delta fun (x) = 0 (x = 0), (x)dx = 1 1 0 (x) = 0 (x = 0) 1 (x = 0) f(x) x fT(x) x 1: F(x, y) = f(x, y) exp{i2(xx + yy)}dxdy (7) (x, y) sampling, 1 (sampling theorem) x f(x) f(x) T fT (x) f(x) T f(x) x fT(x) x 1: F(x, y) = f(x, y) exp{i2(xx + yy)}dxdy (7) (x, y) x ...... T (x) ... (xT) (xnT) 2: (11) 0 1 fT (x) fT (x) FT[f(x)g(x)]() = FT[f(x)]() FT[g(x)]() (12 FT[f(x)] f(x) convolution, f(t) g(t) = f(y)g(t y)dy (13 8. 2014 A.Asano,KansaiUniv. n= fT (x) = f(x)combT (x) 2(x) (Diracs delta fun (x) = 0 (x = 0), (x)dx = 1 1 0 (x) = 0 (x = 0) 1 (x = 0) f(x) x fT(x) x 1: F(x, y) = f(x, y) exp{i2(xx + yy)}dxdy (7) (x, y) sampling, 1 (sampling theorem) x f(x) f(x) T fT (x) f(x) T f(x) x fT(x) x 1: F(x, y) = f(x, y) exp{i2(xx + yy)}dxdy (7) (x, y) x ...... T (x) ... (xT) (xnT) 2: (11) 0 1 fT (x) fT (x) FT[f(x)g(x)]() = FT[f(x)]() FT[g(x)]() (12 FT[f(x)] f(x) convolution, f(t) g(t) = f(y)g(t y)dy (13 9. 2014 A.Asano,KansaiUniv. n= fT (x) = f(x)combT (x) 2(x) (Diracs delta fun (x) = 0 (x = 0), (x)dx = 1 1 0 (x) = 0 (x = 0) 1 (x = 0) f(x) x fT(x) x 1: F(x, y) = f(x, y) exp{i2(xx + yy)}dxdy (7) (x, y) sampling, 1 (sampling theorem) x f(x) f(x) T fT (x) f(x) T f(x) x fT(x) x 1: F(x, y) = f(x, y) exp{i2(xx + yy)}dxdy (7) (x, y) x ...... T (x) ... (xT) (xnT) 2: (11) 0 1 fT (x) fT (x) FT[f(x)g(x)]() = FT[f(x)]() FT[g(x)]() (12 FT[f(x)] f(x) convolution, f(t) g(t) = f(y)g(t y)dy (13 f(t) T fT (t) combT (x) fT (x) = f(x)combT (x) fT (x) FT FT[fT (x)]() = FT[f(x)combT (x)]() = f(x)combT (x) exp(i2x)dx 10. 2014 A.Asano,KansaiUniv. x x f(x) fT(x) T 1 / T ... ... cc FT[f(x)]() FT[fT(x)]() 3: 11. 2014 A.Asano,KansaiUniv. x x f(x) fT(x) T 1 / T ... ... cc FT[f(x)]() FT[fT(x)]() 3: 12. 2014 A.Asano,KansaiUniv. x x f(x) fT(x) T 1 / T ... ... cc FT[f(x)]() FT[fT(x)]() 3: 13. 2014 A.Asano,KansaiUniv. x x f(x) fT(x) T 1 / T ... ... cc FT[f(x)]() FT[fT(x)]() 3: 14. 2014 A.Asano,KansaiUniv. x fT (x) T [s] [1/s] FT[f T (x)]() 1 / T [1/s] FT[fT(x)](n) 1 / NT [1/s] [1/s] 15. 2014 A.Asano,KansaiUniv. x fT (x) T [s] [1/s] FT[f T (x)]() 1 / T [1/s] FT[fT(x)](n) 1 / NT [1/s] [1/s] 16. 2014 A.Asano,KansaiUniv. x fT (x) T [s] [1/s] FT[f T (x)]() 1 / T [1/s] FT[fT(x)](n) 1 / NT [1/s] [1/s] 17. 2014 A.Asano,KansaiUniv. x fT (x) T [s] [1/s] FT[f T (x)]() 1 / T [1/s] FT[fT(x)](n) 1 / NT [1/s] [1/s] N 18. 2014 A.Asano,KansaiUniv. x fT (x) T [s] [1/s] FT[fT (x)]() 1 / T [1/s] FT[fT(x)](n) 1 / NT [1/s] [1/s] x[s] NT[s] 19. 2014 A.Asano,KansaiUniv. x fT (x) T [s] [1/s] FT[fT (x)]() 1 / T [1/s] FT[fT(x)](n) 1 / NT [1/s] [1/s] x[s] NT[s] 20. 2014 A.Asano,KansaiUniv. x fT (x) T [s] [1/s] FT[fT (x)]() 1 / T [1/s] FT[fT(x)](n) 1 / NT [1/s] [1/s] x[s] NT[s] 21. 2014 A.Asano,KansaiUniv. x fT (x) T [s] [1/s] FT[f T (x)]() 1 / T [1/s] FT[fT(x)](n) 1 / NT [1/s] [1/s] x[s] NT[s] 22. 2014 A.Asano,KansaiUniv. x fT (x) T [s] [1/s] FT[f T (x)]() 1 / T [1/s] FT[fT(x)](n) 1 / NT [1/s] [1/s] x[s] NT[s] 23. 2014 A.Asano,KansaiUniv. x fT (x) T [s] [1/s] FT[f T (x)]() 1 / T [1/s] FT[fT(x)](n) 1 / NT [1/s] [1/s] x[s] NT[s] 24. 2014 A.Asano,KansaiUniv. x fT (x) T [s] [1/s] FT[f T (x)]() 1 / T [1/s] FT[fT(x)](n) 1 / NT [1/s] [1/s] x[s] NT[s] 25. 2014 A.Asano,KansaiUniv. x fT (x) T [s] [1/s] FT[f T (x)]() 1 / T [1/s] FT[fT(x)](n) 1 / NT [1/s] [1/s] x[s] NT[s] 26. 2014 A.Asano,KansaiUniv. 27. 2014 A.Asano,KansaiUniv. u(n) = fT (nT) f(x) T NT (2) u(n) (2) 0 x[s] 28. 2014 A.Asano,KansaiUniv. u(n) = fT (nT) f(x) T NT (2) u(n) (2) 0 x[s] 29. 2014 A.Asano,KansaiUniv. u(n) = fT (nT) f(x) T NT (2) u(n) (2) 0 x[s] 30. 2014 A.Asano,KansaiUniv. u(n) = fT (nT) f(x) T NT (2) u(n) (2) 0 x[s] 31. 2014 A.Asano,KansaiUniv. u(n) = fT (nT) f(x) T NT (2) u(n) (2) 0 u(n) = fT (nT) f(x) T (2) (2) (2) U(k) = N1 n=0 u(n) exp i2 k N n (k = 0, 1, . . . , N 1) (DFT) x[s] 32. 2014 A.Asano,KansaiUniv. u(n) = fT (nT) f(x) T NT (2) u(n) (2) 0 u(n) = fT (nT) f(x) T (2) (2) (2) U(k) = N1 n=0 u(n) exp i2 k N n (k = 0, 1, . . . , N 1) (DFT) x[s] 33. 2014 A.Asano,KansaiUniv. u(n) = fT (nT) f(x) T NT (2) u(n) (2) 0 u(n) = fT (nT) f(x) T (2) (2) (2) U(k) = N1 n=0 u(n) exp i2 k N n (k = 0, 1, . . . , N 1) (DFT) x[s] 34. 2014 A.Asano,KansaiUniv. x fT (x) T [s] [1/s] FT[fT (x)]() 1 / T [1/s] u(n) 1[] = 1[Ts] n k[] U(k) 1[] x[s] n[] N[] = N[Ts] = NT[s] FT[fT(x)](n) 1 / NT [1/s] [1/s] 1: 35. 2014 A.Asano,KansaiUniv. 36. 2014 A.Asano,KansaiUniv. 37. 2014 A.Asano,KansaiUniv. 38. 2014 A.Asano,KansaiUniv.