Lecture - 10 Image Enhancement in the Frequency Domain

Transcription

1 Lectre - Image Enhancement in the Freqenc Domain Cosimo Distante

2 Backgrond An fnction that periodicall repeats itself can be epressed as the sm of sines and/or cosines of different freqencies each mltiplied b a different coefficient Forier series. Even fnctions that are not periodic bt whose area nder the crve is finite can be epressed as the integral of sines and/or cosines mltiplied b a weighting fnction Forier transform.

3 Backgrond The freqenc domain refers to the plane of the two dimensional discrete Forier transform of an image. The prpose of the Forier transform is to represent a signal as a linear combination of sinsoidal signals of varios freqencies.

4 Introdction to the Forier Transform and the Freqenc Domain The one-dimensional Forier transform and its inverse Forier transform continos case j π F f e d Inverse Forier transform: f j π F e d The two-dimensional Forier transform and its inverse Forier transform continos case where j π + v F v f e dd Inverse Forier transform: f j π + v F v e ddv j θ e j cosθ + j sinθ

5 Introdction to the Forier Transform and the Freqenc Domain The one-dimensional Forier transform and its inverse Forier transform discrete case DCT M / j π F f e M Inverse Forier transform: M f F e jπ/ M M for for... M... M

6 Since Introdction to the Forier Transform and the Freqenc Domain and the fact then discrete Forier transform can be redefined F θ e j cosθ + j sinθ cos θ cosθ M M f [cosπ / M for... M Freqenc time domain: the domain vales of over which the vales of F range; becase determines the freqenc of the components of the transform. Freqenc time component: each of the M terms of F. j sin π / M ]

7 Introdction to the Forier Transform and the Freqenc Domain F can be epressed in polar coordinates: R: the real part of F I: the imaginar part of F Power spectrm: I R F P + [ ] phase angle or phase spectrm tan magnitdeor spectrm where + R I I R F e F F j φ φ Also referred to spectral densit

8 The One-Dimensional Forier Transform Eample

9 The One-Dimensional Forier Transform Some Eamples The transform of a constant fnction is a DC vale onl. The transform of a delta fnction is a constant.

10 The One-Dimensional Forier Transform Some Eamples The transform of an infinite train of delta fnctions spaced b T is an infinite train of delta fnctions spaced b /T. The transform of a cosine fnction is a positive delta at the appropriate positive and negative freqenc.

11 The One-Dimensional Forier Transform Some Eamples The transform of a sin fnction is a negative comple delta fnction at the appropriate positive freqenc and a negative comple delta at the appropriate negative freqenc. The transform of a sqare plse is a sinc fnction.

12 Introdction to the Forier Transform and the Freqenc Domain The two-dimensional Forier transform and its inverse Forier transform discrete case DFT M N / j π M F v f e MN for... M v Inverse Forier transform: f for M N v F v e... M... N jπ / M + v / N... N + v / N v : the transform or freqenc variables : the spatial or image variables

13 Introdction to the Forier Transform and the Freqenc Domain We define the Forier spectrm phase angle and power spectrm as follows: Rv: the real part of Fv Iv: the imaginar part of Fv [ ] power spectrm phase angle tan spectrm v I v R v F v P v R v I v v I v R v F + + φ

14 Introdction to the Forier Transform and the Freqenc Domain Some properties of Forier transform: [ ] smmetric conjgate smmetric * average shift v F v F v F v F f MN F N v M F f M N I +

15 The Two-Dimensional DFT and Its Inverse The D DFT Fv can be obtained b. taking the D DFT of ever row of image f F. taking the D DFT of ever colmn of F af bf cfv

16 The Two-Dimensional DFT and Its Inverse shift

17 The Two-Dimensional DFT and Its Inverse

18 The Propert of Two-Dimensional DFT Rotation DFT DFT

19 The Propert of Two-Dimensional DFT Linear Combination A DFT B DFT.5 * A +.75 * B DFT

20 The Propert of Two-Dimensional DFT Epansion A DFT B DFT Epanding the original image A b a factor of n n filling the empt new vales with zeros B reslts in the same DFT.

21 Two-Dimensional DFT with Different Fnctions Sine wave Its DFT Rectangle Its DFT

22 Two-Dimensional DFT with Different Fnctions D Gassian fnction Its DFT Implses Its DFT

23 Filtering in the Freqenc Domain

24 Basics of Filtering in the Freqenc Domain

25 Basics of Filtering in the Freqenc Domain

26 Some Basic Filters and Their Fnctions Mltipl all vales of Fv b the filter fnction notch filter: if v M / N / H v otherwise. All this filter wold do is set F to zero force the average vale of an image to zero and leave all freqenc components of the Forier transform ntoched.

27 Some Basic Filters and Their Fnctions Lowpass filter Highpass filter

28 Some Basic Filters and Their Fnctions

29 Correspondence between Filtering in the Spatial and Freqenc Domain Convoltion theorem: The discrete convoltion of two fnctions f and h of size M N is defined as Let Fv and Hv denote the Forier transforms of f and h then Eq Eq v H v F h f M m N n n m h m n f MN h f v H v F h f

30 Correspondence between Filtering in the Spatial and Freqenc Domain :an implse fnction of strength A located at coordinates where : a nit implse located at the origin The Forier transform of a nit implse at the origin Eq : M N As A s δ A δ M N s s δ + / / M N N v M j MN e MN v F π δ δ

31 Correspondence between Filtering in the Spatial and Freqenc Domain Let then the convoltion Eq Combine Eqs with Eq we obtain h MN n m h m n MN h f M m N n δ f δ [ ] v H h v H MN h MN v H h v H v F h f I δ δ

32 Correspondence between Filtering in the Spatial and Freqenc Domain Let H denote a freqenc domain Gassian filter fnction given the eqation where σ : the standard deviation of the Gassian crve. The corresponding filter in the spatial domain is h H πσae Note: Both the forward and inverse Forier transforms of a Gassian fnction are real Gassian fnctions. Ae π σ / σ

33 Correspondence between Filtering in the Spatial and Freqenc Domain

34 Correspondence between Filtering in the Spatial and Freqenc Domain One ver sefl propert of the Gassian fnction is that both it and its Forier transform are real valed; there are no comple vales associated with them. In addition the vales are alwas positive. So if we convolve an image with a Gassian fnction there will never be an negative otpt vales to deal with. There is also an important relationship between the widths of a Gassian fnction and its Forier transform. If we make the width of the fnction smaller the width of the Forier transform gets larger. This is controlled b the variance parameter σ in the eqations. These properties make the Gassian filter ver sefl for lowpass filtering an image. The amont of blr is controlled b σ. It can be implemented in either the spatial or freqenc domain. Other filters besides lowpass can also be implemented b sing two different sized Gassian fnctions.

35 Smoothing Freqenc-Domain Filters The basic model for filtering in the freqenc domain G v H v F v where Fv: the Forier transform of the image to be smoothed Hv: a filter transfer fnction Smoothing is fndamentall a lowpass operation in the freqenc domain. There are several standard forms of lowpass filters LPF. Ideal lowpass filter Btterworth lowpass filter Gassian lowpass filter

36 Ideal Lowpass Filters ILPFs The simplest lowpass filter is a filter that cts off all highfreqenc components of the Forier transform that are at a distance greater than a specified distance D from the origin of the transform. The transfer fnction of an ideal lowpass filter if D v D H v if D v > D where Dv : the distance from point v to the center of ther freqenc rectangle [ ] M / + v / D v N

37 Ideal Lowpass Filters ILPFs

38 Ideal Lowpass Filters ILPFs

39 Ideal Lowpass Filters

40 Btterworth Lowpass Filters BLPFs With order n H v + [ D v / D ] n

41 Btterworth Lowpass Filters BLPFs n D 5538and 3

42 Btterworth Lowpass Filters BLPFs Spatial Representation n n n5 n

43 Gassian Lowpass Filters FLPFs H v e D v/ D

44 Gassian Lowpass Filters FLPFs D 5538and 3

45 Additional Eamples of Lowpass Filtering

46 Additional Eamples of Lowpass Filtering

47 Sharpening Freqenc Domain Filter H v H v hp lp Ideal highpass filter if H v if D v D D v > D Btterworth highpass filter H v + [ D / D v ] n Gassian highpass filter H v e D v/ D

48 Highpass Filters Spatial Representations

49 Ideal Highpass Filters H v if if D v D v > D D

50 Btterworth Highpass Filters H v + [ D / D v ] n

51 Gassian Highpass Filters H v e D v/ D

52 The Laplacian in the Freqenc Domain The Laplacian filter H v + v Shift the center: M H v + v N Freqenc domain Spatial domain

53 g where f f f : the Laplacian- filtered image in the spatial domain For displa prposes onl

54 Implementation Some Additional Properties of the D Forier Transform Periodicit smmetr and back-to-back properties shift

55 Implementation Some Additional Properties of the D Forier Transform Separabilit

56 Smmar of Some Important Properties of the -D Forier Transform

57 Smmar of Some Important Properties of the -D Forier Transform

58 Smmar of Some Important Properties of the -D Forier Transform

59 Smmar of Some Important Properties of the -D Forier Transform