Chapter 4 Image Enhancement in the Frequency Domain

Transcription

1 Chapter 4 Image Enhancement in the Frequency Domain

2 3. Fourier transorm -D Let be a unction o real variable,the ourier transorm o is F { } F u ep jπu d j F { F u } F u ep[ jπ u ] du F u R u + ji u or F u F U e [ ] I u F u R u + I u φ u tan R u jφ u

3 3. Fourier transorm -D cont. The magnitude unction Fu is called the Fourier spectrum o φu is the phase angle P u F u R u + I u Power spectrum o spectral density u:requency variable

4 Eample A Fu A 3 3

5 u ep j π u d F A ep j π u d A j π u e j λ u A j π u [ e j π u e j π u ] e j π u A π u sin π u e jπ u e j cos + j sin Eulor

6 Fourier spectrum is F u A πu sin πu e jπu A sin π u λu

7 D-F.T. F {, y } F u, v, y ep[ jπ u + vy ] ddy F { F u, v }, y F u, v ep[ jπ u + vy ] dudv u,v : requency variables F + u, v [ R u, v I u, v ] P u, v F u, v

8 Eample F u, v, yep[ jπ u + vy] ddy A X A ep jπu d ep e j πu j λ u y e j πu j λvy Y jπvy dy

9 Cont. AXY sin π ux jπu sin πuy e λu πuy e jπvy F u, v AXY sin u ux sin vy vy

10 Chapter 4 Image Enhancement in the Frequency Domain

11 3. DFT Δ + Δ + Δ Δ X {, +, +[N-] +,, N- denotes any N uniormly spaced samples. N DFT Fu j u N N ep / or u,,,..,n- π

12 N F uep jπu/ N or,,,.,n- u Δu nδ D-DFT N F u, v, y ep jπ u + vy N y N or u,,,.,- v,,, N- N, y F u, v ep jπ u + vy u u N or,,,- y,,..,n- Δu Δ Δv NΔy

13 EX: F F 4 u N N ep [ j π u N ] 3 N F ep 4 e j3 + 4 π + 4 F 4 + j jπ 4 j e 4 + 3e jπ + 4e jπ

14 F F F 3 + j F F

15 Chapter 4 Image Enhancement in the Frequency Domain

16 Chapter 4 Image Enhancement in the Frequency Domain

17 Chapter 4 Image Enhancement in the Frequency Domain

18 Impulse unction condition δ d δ d Eg A a α

19 g δ + T + δ + δ T gα α * g

20 Convolution Theorem * g F u G u g F u* G u D-Discrete :,,... A g, g, g... g B I and g are with same period, then condition is period with How to select >A+B- Otherwise the individual periods i convolution with overlap wraparound error

21 Etended sequence e ge g e * g e /,.- <<A- A<<- <<B- B<<- e m g e m or

22 Digital Image Processing, nd ed. D-continuous β α β α β α d d y g y g y,,, *,, *,,,,,, *, v u G v u F y g y v u G v u F y g y

23 D-discrete, y A* B g, y C * D array array Let >A+C- N>B+D-

24 Digital Image Processing, nd ed. <<A-, <y<b- A<<-, B<y<N- <<C-, <y<d- c<<-, D<y<N- For, -, y,..n-,, / *,,,, m N n e e e e e n y m g n m N g y g y g y y The etended sequence

25 Chapter 4 Image Enhancement in the Frequency Domain

26 Chapter 4 Image Enhancement in the Frequency Domain

27 Chapter 4 Image Enhancement in the Frequency Domain

28 Chapter 4 Image Enhancement in the Frequency Domain

29 Chapter 4 Image Enhancement in the Frequency Domain

30 Chapter 4 Image Enhancement in the Frequency Domain

31 Chapter 4 Image Enhancement in the Frequency Domain

32 Chapter 4 Image Enhancement in the Frequency Domain

33 Chapter 4 Image Enhancement in the Frequency Domain

34 Chapter 4 Image Enhancement in the Frequency Domain

35 Chapter 4 Image Enhancement in the Frequency Domain

36 Chapter 4 Image Enhancement in the Frequency Domain

37 Chapter 4 Image Enhancement in the Frequency Domain

38 Chapter 4 Image Enhancement in the Frequency Domain

39 Chapter 4 Image Enhancement in the Frequency Domain

40 Chapter 4 Image Enhancement in the Frequency Domain

41 Chapter 4 Image Enhancement in the Frequency Domain

42 Chapter 4 Image Enhancement in the Frequency Domain

43 Chapter 4 Image Enhancement in the Frequency Domain

44 Chapter 4 Image Enhancement in the Frequency Domain

45 Chapter 4 Image Enhancement in the Frequency Domain

46 Chapter 4 Image Enhancement in the Frequency Domain

47 Chapter 4 Image Enhancement in the Frequency Domain

48 Chapter 4 Image Enhancement in the Frequency Domain

49 Chapter 4 Image Enhancement in the Frequency Domain

50 Chapter 4 Image Enhancement in the Frequency Domain

51 Chapter 4 Image Enhancement in the Frequency Domain

52 Chapter 4 Image Enhancement in the Frequency Domain

53 Chapter 4 Image Enhancement in the Frequency Domain

54 Chapter 4 Image Enhancement in the Frequency Domain

55 Chapter 4 Image Enhancement in the Frequency Domain

56 Chapter 4 Image Enhancement in the Frequency Domain

57 Chapter 4 Image Enhancement in the Frequency Domain

58 Digital Image Processing, nd ed. Separability or u,v,,,,n- or,y,,,,n- Where [ ] [ ] ep, ep, N y N vyn j y un j N v u F π π [ ] [ ] ep, ep, N v N u N vy j v u F N u j N y π π D D [ ] N u j v F N v u F N π ep,, [ ] N vy j y N N v F N y π ep,,

59 For each, with value v,,,n- -DFT is computing one row or -DFT F,v is obtained by taking a transorm

60 < correlation > match ilter *:comple conjugate Convolution and correlation ormula is similar the only dierence is that the unction g is not olded about the origin.

61 discrete: or,,,.., - D-continuous

62 D-discrete Digital Image Processing, nd ed. or,,, m-, y,,,,n- correlation theorem Application : template or prototype matching bind maimum

63 <Sampling> recovering -D eg. band-limit unction sampling un s S ^^^ sample date convolution in the ug domain is Fig 3.7 with period may have overlap region center at i w overlap A

64 To avoid overlap, we must choose Shannon theorem : Complete recovery o a band-limited unction rom sampling whose spacing satisies A To recover Isolate Fu

65 Practical case : inite sample window un distortion impossible to recover completely The FT can be isolated only when is band limited and periodic, with a period equal to alowing complete recovery ater revering, the unction is etended rom - to

66 Conclusion :. no unction o inite duration can be band-limited.a unction o band-limited must etend rom - to, in domain

67 Along each row o,y and multiplying the result by N. Fu,v is taking a rom transorm along each column o F,v N- y N-,y Row tran. N F,v column Fu,v

68 <translation>, yep[ jπ u + v y/ N] <> F u u, v v, y y <> F u, vep[ jπ u + vy / N] let u v N jπ + y + y ep[ jπ u + v y N] e cosπ + y + y N N, y <> F u, v FT o,y can be moved to the center o its corresponding N * N requency square. While the magintude remains the same. F u, vep[ jπ u + vy / N] F u, v

69 < Periodicity and conjugate symmetry> period N. Fu,v Fu+N,v Fu,v+N Fu+N,v+N conjugate Fu,v * -u,-v Fu,v F-u,-v N/, N/ Fig 3-9 < Rotation > let r cosθ y r sinθ uωcosφ vωsinφ F u, v F ω, φ, y r, θ r, θ + θ F ω, φ + θ same rotation angle Fig 3-

70 Digital Image Processing, nd ed. < Distributivity & Scaling > < Average Value > < Laplacian > outlining edge,, F y N, y y +, }, { v u F v u y + <> π ϑ }, { }, { },, { y y y y ϑ ϑ ϑ + +,,,, b v a u ab F ay a v u af y a

71 < Convolution > -D continuous The convolution o and g is deined as * g g α dα eg. α g α u v g N α g α α g g α α α,,

72 D Function Sampling process, y δ, y y ddy y o, δ, y is a -D impulse unction

73 -D train o impulse S,y δ, y Δy Δ A sampled un. is obtained by orming the product δ, y y,

74 Δ w u Δ y w v For N N image Δ u in req domain δ, v is a train o impulse with separation Δ, Δ y in u and v direction δ u,v N Δ, Δ v N Δ y u F u, v

75 Chapter 4 Image Enhancement in the Frequency Domain

76 Chapter 4 Image Enhancement in the Frequency Domain

77 Chapter 4 Image Enhancement in the Frequency Domain

78 Chapter 4 Image Enhancement in the Frequency Domain

79 Chapter 4 Image Enhancement in the Frequency Domain

80 Fast Fourier Transorm FFT F N u ep[ jπu / N ] N D-DFT The number o comple multiplication and addition is N FFT N log N

81 FFT Algorithm π Let W N ep[ j N ] A Fi table o W u N can be computed and build or ep[ jπu / N ] F N u W N u N assume N n

82 Digital Image Processing, nd ed. W u F u W u u W [ W W u u W W W u u u u F [ + +

83 u W u Deine even u, - F F u odd u + W F u [ Feven u + Fodd u ] Q W and W u u, - A u+ W u W u + u W

84 F u + [ F W u even u Fodd u ] B <observation> An N-point transorm can be computed by dividing the original epression into two parts in AB The irst part A requires evaluation N N o two -point o Feven and Fodd,

85 Computation mn-multiplication, an addition or n N n,n need F+F F even point itsel F odd itsel F one multiplication,one addition F one addition m,a

86 n,n4 A our point trans. can be divided into two parts.the irst hal evaluates two point m a 7, mm+ aa+4 mnmn-+ n- anan-+ n Where m, a

87 Number o operation mn/ n log n /Nn an n log n Nn ONn

88 Inverse FFT N u F uep-jπu/n Take comple conjugate and divide by N /N * /N N u F u*ep-jπu/n Taking FFT o Fu * * N

89 Digital Image Processing, nd ed. Separable reduce computation compleity and invese transorm g: orword transormation kernal h: invese transormation kernal g,y,u,vg,ug y,v separable g,y,u,vg,yg y,v separable,,,,, N N y v u y g y v u T,,,,, N N y v u y h v u T y

90 g,y,u,v Digital Image Processing, nd ed. / N ep[ jπ u + vy / N] g u, g y, v / N ep[ jπu / N ]/ N ep[ jπvy / N ] D D First: -D transorm along each now o,y T,v Net: -D transorm along each column o T,v Tu,v N N, y g, y g TAFA BTBBAFAB y, v, u FBTA BA -

91 +p7

92 Digital Image Processing, nd ed. -D Fourier Transorm ] / ep[ N N u j N u F π Let ] / ep[ ] / ep[ N u j N j u N N π ω π ω N u N N u F ω N n Let u u F ω

93 where and [ + + ω ω + ω u u ω u + + ω u ω u ω u ω ω u + u 3 ω u 3 ] + +

94 Digital Image Processing, nd ed. ] [ ] [ ] / ep[ u u u u u u u u u F u j ω ω ω ω ω ω ω π ω Q ] [ u u Fodd u Feven ω +...

95 where Feven u Fodd u or u,,,..., ω u + ω u Two urther multiplication and additions are necessarry to obtain F and F m +, a +

96 To obtain m n n n By indiction, m n log n n a n log Nn It is m n On F and + F3 n,, two more additions m a n +, a + a n + Nn n +

97 A are orthonormal vectors A A T T A y + m Use k longest eigenvalues and eigenvectors A k k n y : k-dimension. T ˆ A y + k m

98 mean squar even e ms m j λ j k j λ j n j k+ λ j e ms when λ j+

99 A are orthonormal vectors A A T T A y + Use k longest eigenvalues and eigenvectors A k k n y : k-dimension. T ˆ A y + k m m

100 Chapter 4 Image Enhancement in the Frequency Domain