Chapter 4: Filtering in the Frequency Domain. Fourier Analysis R. C. Gonzalez & R. E. Woods

Transcription

1 Fourier Analysis R. C. Gonzalez & R. E. Woods

2 Properties of δ (t) and (x) δ : f t) δ ( t t ) dt = f ( ) f x) δ ( x x ) = f ( ) ( 0 t0 x= ( 0 x R. C. Gonzalez & R. E. Woods

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5 Sampling R. C. Gonzalez & R. E. Woods

6 Sampling R. C. Gonzalez & R. E. Woods

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10 Reconstruction of analog signal with aliasing R. C. Gonzalez & R. E. Woods

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12 2-D discrete impulse R. C. Gonzalez & R. E. Woods

13 2-D Fourier transform R. C. Gonzalez & R. E. Woods

14 R. C. Gonzalez & R. E. Woods 2-D impulse train and sampling 2-D Fourier transform pair: dtdz e z t f F z t j + = ) ( 2 ), ( ), ( ν µ π µ ν ν µ µ ν ν µ π d d e F z t f z t j + = ) ( 2 ), ( ), (

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36 Lowpass and Highpass Filtering R. C. Gonzalez & R. E. Woods

37 Gaussian lowpass filtering with/without padding R. C. Gonzalez & R. E. Woods

38 Padding with full size R. C. Gonzalez & R. E. Woods

39 Frequency response and IDFT with/without padding R. C. Gonzalez & R. E. Woods

40 IDFT based on phase only R. C. Gonzalez & R. E. Woods

41 Low-pass filtering using DFT and full-size padding R. C. Gonzalez & R. E. Woods

42 Guassian Filters R. C. Gonzalez & R. E. Woods

43 Image and its spectrum with interest on line edges R. C. Gonzalez & R. E. Woods

44 High-pass filtering with spatial direction R. C. Gonzalez & R. E. Woods

45 Ideal low-pass filter R. C. Gonzalez & R. E. Woods

46 DFT of padded image with shrinking different powers enclosed by circles R. C. Gonzalez & R. E. Woods

47 Ideal LPF with different cut-off frequencies Blurring and Ringing happened! R. C. Gonzalez & R. E. Woods

48 Ideal LPF R. C. Gonzalez & R. E. Woods

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50 Butterworth low-pass filtered images Blurring happened but less ringing R. C. Gonzalez & R. E. Woods

51 Ringing of BLPF increases with order of filter R. C. Gonzalez & R. E. Woods

52 Guassian LPF without ringing R. C. Gonzalez & R. E. Woods

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54 Guassian LP filtered image without ringing R. C. Gonzalez & R. E. Woods

55 Guassian LP filtering R. C. Gonzalez & R. E. Woods

56 Gaussian LP Filtering Example R. C. Gonzalez & R. E. Woods

57 Gaussian LP Filtering Example R. C. Gonzalez & R. E. Woods

58 Ideal, Butterworth and Gaussian Highpass Filters R. C. Gonzalez & R. E. Woods

59 Spatial-domain HPFs R. C. Gonzalez & R. E. Woods

60 Highpass filtered images with ideal filters R. C. Gonzalez & R. E. Woods

61 Highpass filtered images with BLPF R. C. Gonzalez & R. E. Woods

62 Highpass filtered images with GLPF R. C. Gonzalez & R. E. Woods

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64 Highpass filtering and thresholding R. C. Gonzalez & R. E. Woods

65 R. C. Gonzalez & R. E. Woods Laplacian operator, y f x f y x f ), ( + = In frequency domain, ) ( 4 ), ( v u v u H + = π Centered spectrum, ), ( 4 ] 2) / ( 2) / [( 4 ), ( v u D Q v P u v u H π π = + = Spatial-domain enhancement, ), ( ), ( ), ( 2 y x f c y x f y x g + = DFT implementation with c=1, )}, ( )], ( 4 {[1 ), ( 2 2 v u F v u D IDFT y x g π + = Frequency domain Laplacian

66 Frequency domain Laplacian processed image R. C. Gonzalez & R. E. Woods

67 Highpass, Highpass with high-frequency emphasis and Histogram processing R. C. Gonzalez & R. E. Woods

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