Digital Image Processing. Filtering in the Frequency Domain

Transcription

1 2D Linear Systems 2D Fourier Transform and its Properties The Basics of Filtering in Frequency Domain Image Smoothing Image Sharpening Selective Filtering Implementation Tips 1

2 General Definition: System f x, y H g x, y,, g x y H f x y 2

3 Linearity,,,, H af x y bf x y ah f x y bh f x y Shift Invariant or Spatially Invariant:,,,, g x y H f x y g x x y y H f x x y y Causality: Same as before Don t worry about it! Stability: Same as before 3

4 Unit Impulse Function (Pinhole): xy, xy, 0,0 xy 0, 0,0 x, y dxdy 1 4

5 Point Spread Function (Impulse Response):, ;,, H x y x y H x x y y PSD of Hubble Telescope Linear Shift Invariant Systems:, ; 0, 0 0, 0 0, 0,, H x y x y H x x y y H x x y y H x y H x y 5

6 Convolution/Correlation Integral:,,,, f x y h x y f s t h x s y t dsdt,,,, f x y h x y f s t h x s y t dsdt Convolution/Correlation Summation: M1N1,,,, f m n h m n f p q h m p n q p0 q0 M1N1,,,, f m n h m n f p q h m p n q p0 q0 6

7 The 2-D Continuous Fourier Transform j 2 ux vy F u v f x y e dxdy,, j 2 ux vy f x y F u v e dudv,, x y A rect rect A TZsinc ut sinc vz Z T 7

8 2-D Sampling and Sampling Theorem 2-D impulse train: x mx, y ny m m Error Free Reconstruction: 1 1 x, y 2u 2v max max 8

9 Aliasing in Images: See Figures for practical examples. 9

10 2-D Discrete Fourier Transform M 1N 1 1 um vn F u, v f m, n exp 2 1 j m 0 n 0 M N M 1N 1 1 um vn f m, n Fu, v exp j 2 MN u0 v 0 M N Spatial and Frequency Intervals 1 1 u, v M x N y 10

11 Phase Significance 11

12 Phase Significance 12

13 Phase Significance 13

14 Symmetry Properties 14

15 Definition and Properties (1) 15

16 Definition and Properties (2) 16

17 DFT Pairs (1) 17

18 DFT Pairs (2) 18

19 FFT Shift, Centering and Symmetry 19

20 Fourier Transform Centering fftshift in Matlab 20

21 Example Without Shift With Shift 21

22 Translation Rotation 22

23 Phase Changes: Original Translated Rotated 23

24 Line Pattern in Spatial and Frequency Domain Strong ±45 edge in Spatial Strong ±45 edge in Frequency 24

25 Frequency Domain Filtering Fundamentals: Convolution Theorem: Zero Padding: f h g Zero-Padding is necessary to avoid Wraparound error. Circular vs. Linear Convolution,,,, f m n h m n F u v H u v 1 1 M N PQ M P N Q 25

26 Frequency Domain Manipulation Set F(0,0) to zero Clip negative value 26

27 Lowpass, Highpass, Highboost No shift 27

28 Zero-Padding Effect: Blurring With Gaussian Original No Zero-Padding Zero-Padding 28

29 Inherent Periodicity of DFT and Zero-Padding With (Right) and Without (Left) Padding 29

30 Zero-Padding Side effect Read Pg ! Ideal Filter (TL) Time Domain (BL) Zero Padding (TR) Ringing Effect (BR) 30

31 Effect of small changes in phase Why we prefer zero-phase filters? j 0.5 F IDFT F e j 0.25 F IDFT F e 31

32 Steps for Frequency Domain Filtering a) Original b) Padding c) Multiply by (-1) x+y d) FFT e) GLP (Centered) f) Multiply and (e) g) Multiply IFFT Real by (-1) x+y h) Cropping 32

33 Spatial-Frequency Correspondences h x, y H u, v The most used Filter (Gaussian) e 2 2 x y u v e

34 Example (1) 34

35 Example (2) An image and its spectrum 35

36 Filtering in Spatial and Frequency Domain 36

37 Image Smoothing in Frequency Domain Ideal Lowpass Filter Butterworth Lowpass Filter Gaussian Lowpass Filter More Examples 2 2, D u v u v 37

38 Ideal Lowpass Filter 38

39 Test Pattern and Energy Circles 460 (99.2%) 160 (97.8%) 60 (95.7%) 30 (93.1%) 10 (87.0%) 39

40 Ideal Lowpass Filter 10, 30, 60, 160, and 460 (Radius) Smoothing Blurring Ringing Effect 40

41 Origin of Ringing Effect 41

42 Butterworth Lowpass Filter: 42

43 Butterworth Lowpass Filter: Order (2) Same radius Smoothing Blurring Less Ringing Effect 43

44 Less Ringing Effect of Butterworh Filter: Order 1,2,5, and 20 44

45 Gaussian Lowpass Filter: 45

46 Gaussian Lowpass Filter: Smoothing Blurring No Ringing Effect! 46

47 Low Resolution Images Repairing 47

48 Pre-print Processing (Smooth and soft-looking) Original, D 0 =100, and D 0 =80 48

49 Remove Unwanted Pattern: Remove Horizontal Lines (Imaging System Deficiency) Large Recognizable Features Original, D 0 =50, and D 0 =20 49

50 Image Sharpening in Frequency Domain Ideal Highpass Filter Butterworth Highpass Filter Gaussian Highpass Filter More Examples, 1, H u v H u v HP LP 50

51 Highpass Filters 51

52 Spatial Representation of Highpass Filters Ringing (Ideal, Butterworth, and Gaussian) 52

53 Ideal Highpass Filter D 0 =30 D 0 =60 D 0 =160 53

54 Butterworth (n=2) Highpass Filters D 0 =30 D 0 =60 D 0 =160 54

55 Gaussian Highpass Filter D 0 =30 D 0 =60 D 0 =160 55

56 Thumb Print Processing: Original (Left) Butterworth Highpass Filter with n=4, D 0 =50 (Middle) Thresholding (Right) Setting Negative Value to Black and Positive value to White 56

57 Laplacian in Frequency Domain: , 4 4, H u v u v D u v Image Laplacian: 2 f x, y 1 H u, v F u, v Enhanced Image: 2,,, 1 F u, v H u, v F u, v g x y f x y c f x y u v F u v 1 4 D,, 57

58 Example (Laplacian): Similar But Not identical to Spatial Domain 58

59 Unsharp Masking, Highboost, High-Frequency Emphasis: gmask x, y f x, y flp x, y 1 f x, y H u, v F u, v LP LP Unsharp Masking (K=1) and Highboost Filters(K>1): g x, y f x, y kg x, y High Frequency Emphasing: 1 g x, y 1 kh u, vf u, v mask 1 LP, 1 1,, g x y k H u v F u v 1, y k k H u, vf u, v g x 1 2 HP HP 59

60 Example: GHPF, D 0 =40 HFE, k 1 =0.5, k 2 =0.25 Histogram EQ. 60

61 Homomorphic Filtering: Linear Process is not Possible: Summary of Steps,,, f x y i x y r x y ln f x, y ln i x, y ln r x, y 61

62 Illumination-Reflection Control Filter cd 2 u, v D, 1 2 H u v 0 H L e L 62

63 Example: Whole Body PET Scan Enhancement L H 2 c 1 D 80 63

64 Selective Filtering: Bandpass Band Reject Notch 64

65 Bandreject and Bandpass Filter H BP (u, v)= 1- H BR (u, v) 65

66 Notch Filters: Pass/Reject predefined, both (u 0, v 0 ) and (-u 0, -v 0 ) H NP (u, v)= 1- H NR (u, v) 66

67 Example (1) Spot in Frequency Domain Spectrum Multiplied Spectrum 67

68 Example (2) Vertical sin Pattern 68

69 Example (2) Cont. Extract Vertical sin Pattern 69

70 Matlab Command fft2, ifft2, fftshift, ifftshift freqz2, fspecial 70