Digital Image Processing. Lecture 8 (Enhancement in the Frequency domain) Bu-Ali Sina University Computer Engineering Dep.

Transcription

1 Digital Image Processing Lectre 8 Enhancement in the Freqenc domain B-Ali Sina Uniersit Compter Engineering Dep. Fall 009

2 Image Enhancement In The Freqenc Domain

3 Otline Jean Baptiste Joseph Forier The Forier series & the Forier transform Image Processing in the freqenc domain Image smoothing Image sharpening Fast Forier Transform

4 Jean Baptiste Joseph Forier Forier was born in Aerre France in 768 Most famos for his work La Théorie Analitiqe de la Chaler pblished in 8 Translated into English in 878: The Analtic Theor of Heat Nobod paid mch attention when the work was first pblished One of the most important mathematical theories in modern engineering

5 The Big Idea An fnction that periodicall repeats itself can be epressed as a sm of sins and cosines of different freqencies each mltiplied b a different coefficient a Forier series

6 The Big Idea Freqenc domain signal processing eample

7 Forier Series Forier series: a periodic fnction can be represented b the sm of sines/cosines of different freqencies mltiplied b a different coefficient Forier series T w nwt B nwt A f f T f n n sin cos + + π

8 One dimensional Forier Transform Non-periodic fnctions can also be represented as the integral of sines/cosines mltiplied b weighting fnction Forier transform f: continos fnction of a real ariable Forier transform of f: I { } F f f ep[ jπ] d where j

9 One dimensional Forier Transform Gien F f can be obtained b the inerse Forier transform: I { F } f F ep[ jπ] d The aboe two eqations are the Forier transform pair.

10 Two dimensional Forier Transform Forier transform pair for a fnction f of two ariables: I{ f } F I { F } f f and ep[ F ep[ jπ + ] dd jπ + ] dd where are the freqenc ariables.

11 Discrete Forier Transform A continos fnction f is discretized into a seqence: { f 0 0 f 0 + f f + [ ] } b taking N or M samples nits apart.

12 Discrete Forier Transform Where assmes the discrete ales 03M- then f f 0 + The seqence {f0ff fm-} denotes an M niforml spaced samples from a corresponding continos fnction.

13 Discrete Forier Transform F M M 0 f ep[ jπ / M] For 0 M- and M 0 f f ep[ jπ / M] For 0 M-

14 Discrete Forier Transform The ales 0 M- correspond to samples of the continos transform at ales 0 M-. i.e. F represents F where: M

15 Discrete Forier Transform e jθ cosθ + j sinθ cos θ cosθ F M M f [cosπ / M j sinπ / M] 0

16 Discrete Forier Transform The Forier transform of a real fnction is generall comple and we se polar coordinates: ] [ tan ] [ / I R F P R I I R F e F F ji R F j φ φ Magnitde or Spectrm Phase or Angle Power Spectrm

17 Forier Transform F magnitde fnction is the Forier spectrm of f and φ its phase angle. The sqare of the spectrm P F R + I is referred to as the power spectrm of f spectral densit.

18

19 Two Dimensional Discrete Forier Transform ] / / ep[ M M j f M F π ] / / ep[ M M j F f π For 0 M- and 0 N- For 0 M- and 0 N- AND: M

20 Two Dimensional Discrete Forier Transform second Matlab notation ] / / ep[ M M j f F π ] / / ep[ M M j F M f π For 0 M- and 0 N- For 0 M- and 0 N- AND: M

21 Introdction to the Forier Transform Forier spectrm: [ ] / I R F + Phase: tan R I φ Power spectrm: I R F P +

22 Discrete Forier Transform When images are sampled in a sqare arra MN and the FT pair becomes: For 0 N- For 0 N- AND: ] / ep[ j f F π ] / ep[ j F f π

23 Basic Properties I [ + ] f F M / / F00 is at M/ and N/ Shifts the origin of F to M/ N/ i.e. the center of MN of the -D DFT freqenc rectangle Freqenc rectangle: from 0 to M- and 0 to N- integers MN een nmbers In compter: smmations are from to M and to N center of transform: M/ + and N/ +

24 Basic Properties Vale of transform at 00: F00 M M 0 0 f which means that the ale of FT at the origin the aerage gra leel of the image FT is also conjgate smmetric: FF*-- so F F-- which means that the FT spectrm is smmetric.

25 Basic Properties

26 Basic Properties

27 Filtering in the Freqenc Domain Basic Steps for Filtering in the Freqenc Domain. Mltipl inpt Image b - +. Compte F the DFT of image 3. Mltipl b a filter fnction H 4. Compte inerse DFT of the reslt in 3 5. Obtain the real part of reslt in 4 6. Mltipl the reslt in 5 b - + Smmar: G H F Filtered Image I [ G ]

28 Filtering in the freqenc domain

29 Freqenc Domain Filtering Edges and sharp transitions e.g. noise in an image contribte significantl to highfreqenc content of FT. Low freqenc contents in the FT are responsible to the general appearance of the image oer smooth areas. Blrring smoothing is achieed b attenating range of high freqenc components of FT.

30 Basic Filters To force the aerage ale of an image to 0: F00 gies the aerage ale of an image then since F000 take the inerse H Notch filter 0 if otherwise M/ /

31 Basic Filters

32 Correspondence between Filtering in the Spatial & freqenc Domain Book notation Discrete conoltion of two fnctions MN 0 0 * M m n n m h n m f M h f f*h FH fh F*H

33 Correspondence between Filtering in the Spatial & freqenc Domain Matlab Notation Discrete conoltion of two fnctions MN 0 0 * M m n n m h n m f h f H F M fh H F f*h *

34 Conoltion Theorem GF H gh*f Mltiplication in Freqenc Domain Conoltion in Time Domain f is the inpt image g is the filtered h: implse response Filtering in Freqenc Domain with H is eqialent to filtering in Spatial Domain with h. For eample Gassian filter

35 Eample Of Filters

36 Image Enhancement in the Freqenc Domain Tpes of enhancement that can be done: Lowpass filtering: redce the high-freqenc content -- blrring or smoothing Highpass filtering: increase the magnitde of high-freqenc components relatie to lowfreqenc components -- sharpening.

37 Image Enhancement in the Freqenc Domain

38 Low Pass filtering or Smoothing in the Freqenc Domain G H F Ideal Btterworth parameter: filter order Gassian

39 Ideal low-pass filter ILPF H 0 D D D > D 0 0 / D [ M / + / ] M/N/: center in freqenc domain D 0 is called the ctoff freqenc.

40 Shape of ILPF Freqenc domain Spatial domain

41 Calclation of Ctoff freqenc Calclate P T the total Power of image α P T 00 M 0 0 P A circle with radis r origin at the center of the freqenc rectangle encloses a percentage of the power which is gien b the epression P / The smmation is taken within the circle D 0 P T

42 Image Enhancement in the Freqenc Domain

43 ringing and blrring

44 Btterworth Lowpass Filters BLPF Smooth transfer fnction no sharp discontinit no clear ctoff freqenc. H n + D D0

45 Btterworth Lowpass Filters BLPF

46 o serios ringing artifacts

47 Smooth transfer fnction Smooth implse response No ringing Its inerse is also Gassian 0 D D e H Gassian Lowpass Filters GLPF D D Ae D h Ae H π π

48 Gassian Lowpass Filters GLPF

49 o serios ringing artifacts

50 Image Enhancement in the Freqenc Domain

51 Sharpening High-pass Filters Hhp-Hlp Ideal: Btterworth: Gassian: n D D H H 0 D D > 0 D D 0 / D D e H

52

53 High-pass Filters

54 Ideal High-pass Filtering ringing artifacts

55 Btterworth High-pass Filtering

56 Gassian High-pass Filtering

57 Laplacian f f + f f f + + f f f f + + f f f [ f + + f + f + + f ] 4 f

58 Laplacian in the Freqenc Domain F j f F j f n n n n n n π π I I It can be shown [ ] 4 F f + I π

59 The Laplacian can be implemented in the FD b sing the filter FT pair: Laplacian in the Freqenc Domain 4 H + π ] / / [ 4 F M f + π

60 Laplacian in the Freqenc Domain

61 Sbtract Laplacian from the Original Image to Enhance It enhanced image Original image Laplacian otpt Spatial domain g f f Freqenc domain new operator G F + 4π + F H + 4π + H Laplacian

62 Image Enhancement in the Freqenc Domain

63 Unsharp Masking High-boost Filtering Unsharp masking: f hp f-f lp H hp -H lp High-boost filtering: f hb Af-f lp f hb A-f+f hp H hb A-+H hp One more parameter to adjst the enhancement H hfe a+bh hp

64 An image formation model We can iew an image f as a prodct of two components: i: illmination. It is determined b the illmination sorce. r: reflectance. It is determined b the characteristics of imaged objects. 0 0 < < < < r i r i f

65 Homomorphic Filtering In some images the qalit of the image has redced becase of non-niform illmination. Homomorphic filtering can be sed to perform illmination correction. i r f The aboe eqation cannot be sed directl in order to operate separatel on the freqenc components of illmination and reflectance.

66 F F Z r i + r i f z ln ln ln ' ' r i e g r i s s + Homomorphic Filtering Z H S ln : DFT : H : DFT - : ep :

67 Homomorphic Filtering B separating the illmination and reflectance components homomorphic filter can then operate on them separatel. Illmination component of an image generall has slow ariations while the reflectance component ar abrptl. B remoing the low freqencies highpass filtering the effects of illmination can be remoed.

68 Homomorphic Filtering

69 Homomorphic Filtering