Digital Image Processing Chapter 5: Image Restoration

Transcription

1 Digital Image Processing Chapter 5: Image Restoration

2 Concept of Image Restoration Image restoration is to restore a degraded image back to the original image while image enhancement is to maniplate the image so that it is sitable for a specific application. Degradation model: g x y f x y h x y + η x y where hxy is a system that cases image distortion and ηxy is noise. Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition.

3 Noise Models Noise Models Noise cannot be predicted bt can be approximately described in statistical way sing the probability density fnction PDF Gassian noise: / 1 σ μ πσ z e z p Rayleigh noise < a for z 0 for / a z e a z b z p b a z Erlang Gamma noise < 0 for z 0 0 for 1! 1 z e a z b z a z p az b b

4 Noise Models cont. Exponential noise p z ae az Uniform noise p z 1 b - a 0 for a z otherwise b Implse salt & pepper noise Pa for z a p z Pb for z b 0 otherwise

5 PDF: Statistical Way to Describe Noise PDF tells how mch each z ale occrs. Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition.

6 Image Degradation with Additie Noise g x y f x y + η x y Degraded images Original image Histogram Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition.

7 Image Degradation with Additie Noise cont. g x y f x y + η x y Degraded images Original image Histogram Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition.

8 Chapter 4 -D DFT Properties Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition.

9 Chapter 4 -D DFT Properties cont. Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition.

10 Chapter 4 -D DFT Properties cont. Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition.

11 Chapter 4 -D DFT Properties cont. Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition.

12 Estimation of Noise We cannot se the image histogram to estimate noise PDF. It is better to se the histogram of one area of an image that has constant intensity to estimate noise PDF. Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition.

13 Periodic Noise Periodic noise looks like dots In the freqency domain Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition.

14 Periodic Noise Redction by Freq. Domain Filtering Degraded image DFT Periodic noise can be redced by setting freqency components corresponding to noise to zero. Band reject filter Restored image Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition.

15 Band Reject Filters Use to eliminate freqency components in some bands Periodic noise from the preios slide that is Filtered ot. Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition.

16 Notch Reject Filters A notch reject filter is sed to eliminate some freqency components. Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition.

17 Notch Reject Filter: Degraded image DFT Notch filter freq. Domain Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition. Noise Restored image

18 Example: Image Degraded by Periodic Noise Degraded image DFT no shift DFT of noise Noise Restored image Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition.

19 Mean Filters Mean Filters Arithmetic mean filter or moing aerage filter from Chapter 3 S xy t s t s g mn y x f 1 ˆ Geometric mean filter mn S t s xy t s g y x f 1 ˆ mn size of moing window Degradation model: y x y x h y x f y x g η + To remoe this part

20 Geometric Mean Filter: Example Original image Image corrpted by AWGN Image obtained sing a 3x3 arithmetic mean filter Image obtained sing a 3x3 geometric mean filter AWGN: Additie White Gassian Noise Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition.

21 Harmonic and Contraharmonic Filters Harmonic mean filter fˆ x y s t g s t S xy mn 1 Works well for salt noise bt fails for pepper noise Contraharmonic mean filter mn size of moing window fˆ x y s t S xy s t S g s t g s t xy Q+ 1 Q Positie Q is sitable for eliminating pepper noise. Negatie Q is sitable for eliminating salt noise. Q the filter order For Q 0 the filter redces to an arithmetic mean filter. For Q -1 the filter redces to a harmonic mean filter.

22 Contraharmonic Filters: Example Image corrpted by pepper noise with prob. 0.1 Image corrpted by salt noise with prob. 0.1 Image obtained sing a 3x3 contraharmonic mean filter With Q 1.5 Image obtained sing a 3x3 contraharmonic mean filter With Q-1.5 Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition.

23 Contraharmonic Filters: Incorrect Use Example Image corrpted by pepper noise with prob. 0.1 Image corrpted by salt noise with prob. 0.1 Image obtained sing a 3x3 contraharmonic mean filter With Q-1.5 Image obtained sing a 3x3 contraharmonic mean filter With Q1.5 Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition.

24 Order-Statistic Filters: Reisit Original image sbimage Statistic parameters Mean Median Mode Min Max Etc. Moing window Otpt image

25 Order-Statistics Filters Median filter fˆ x y median s t S xy { g s t } Max filter Min filter fˆ x y max s t S xy { g s t } Redce dark noise pepper noise fˆ x y min s t S xy { g s t } Redce bright noise salt noise Midpoint filter fˆ x y 1 max s t S { g s t } + { g s t } min s t xy S xy

26 Median Filter : How it works A median filter is good for remoing implse isolated noise Pepper noise Median Salt noise Degraded image Salt noise Pepper noise Moing window Sorted array Filter otpt Normally implse noise has high magnitde and is isolated. When we sort pixels in the moing window noise pixels are sally at the ends of the array. Therefore it s rare that the noise pixel will be a median ale.

27 Median Filter : Example Image corrpted by saltand-pepper noise with p a p b Images obtained sing a 3x3 median filter Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition.

28 Max and Min Filters: Example Image corrpted by pepper noise with prob. 0.1 Image corrpted by salt noise with prob. 0.1 Image obtained sing a 3x3 max filter Image obtained sing a 3x3 min filter Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition.

29 Alpha-trimmed Mean Filter Formla: fˆ x y 1 mn d s t g S xy r s t where g r st represent the remaining mn-d pixels after remoing the d/ highest and d/ lowest ales of gst. Lowest Gray Leel Total mn pixels d/ d/ Highest Gray Leel This filter is sefl in sitations inoling mltiple types of noise sch as a combination of salt-and-pepper and Gassian noise.

30 Alpha-trimmed Mean Filter: Example Image corrpted by additie niform noise 1 Image additionally corrpted by additie salt-andpepper noise Image Image obtained obtained sing a 5x5 sing a 5x5 arithmetic geometric mean filter mean filter Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition.

31 Alpha-trimmed Mean Filter: Example cont. Image corrpted by additie niform noise Image obtained sing a 5x5 median filter 1 Image additionally corrpted by additie salt-andpepper noise Image obtained sing a 5x5 alphatrimmed mean filter with d 5

32 Alpha-trimmed Mean Filter: Example cont. Image obtained sing a 5x5 arithmetic mean filter Image obtained sing a 5x5 geometric mean filter Image obtained sing a 5x5 median filter Image obtained sing a 5x5 alphatrimmed mean filter with d 5

33 Adaptie Filter General concept: -Filter behaior depends on statistical characteristics of local areas inside mxn moing window - More complex bt sperior performance compared with fixed filters Statistical characteristics: Local mean: m L Local ariance: σ 1 mn 1 s t g s t S xy g s t L m L mn s t S xy Noise ariance: σ η

34 Adaptie Local Noise Redction Filter Prpose: want to presere edges Concept: Formla: 1. If σ η is zero No noise the filter shold retrn gxy becase gxy fxy. If σ L is high relatie to σ η Edges shold be presered the filter shold retrn the ale close to gxy 3. If σ L σ η Areas inside objects the filter shold retrn the arithmetic mean ale m L fˆ x y σ η σ L g x y m g x y L

35 Adaptie Noise Redction Filter: Example Image corrpted by additie Gassian noise with zero mean and σ 1000 Image obtained sing a 7x7 geometric mean filter Image obtained sing a 7x7 arithmetic mean filter Image obtained sing a 7x7 adaptie noise redction filter Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition.

36 Adaptie Median Filter Prpose: want to remoe implse noise while presering edges Algorithm:Leel A: A1 z median z min A z median z max If A1 > 0 and A < 0 goto leel B Else increase window size If window size < S max repeat leel A where Leel B: Else retrn z xy B1 z xy z min B z xy z max If B1 > 0 and B < 0 retrn z xy Else retrn z median z min minimm gray leel ale in S xy z max maximm gray leel ale in S xy z median median of gray leels in S xy z xy gray leel ale at pixel xy S max maximm allowed size of S xy

37 Adaptie Median Filter: How it works Leel A: A1 z median z min A z median z max Determine whether z median is an implse or not If A1 > 0 and A < 0 goto leel B Else Window is not big enogh increase window size If window size < S max repeat leel A Else retrn z xy Leel B: z median is not an implse B1 z xy z min B z xy z max Determine whether z xy is an implse or not If B1 > 0 and B < 0 z xy is not an implse retrn z xy to presere original details Else retrn z median to remoe implse

38 Adaptie Median Filter: Example Image corrpted by salt-and-pepper noise with p a p b 0.5 Image obtained sing a 7x7 median filter Image obtained sing an adaptie median filter with S max 7 More small details are presered Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition.

39 Estimation of Degradation Model Degradation model: or g x y f x y h x y + η x y G F H + N Prpose: to estimate hxy or H Why? If we know exactly hxy regardless of noise we can do deconoltion to get fxy back from gxy. Methods: 1. Estimation by Image Obseration. Estimation by Experiment 3. Estimation by Modeling Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition.

40 Estimation by Image Obseration Original image nknown Degraded image fxy fxy*hxy gxy Obseration Estimated Transfer fnction H H s G Fˆ This case is sed when we know only gxy and cannot repeat the experiment! s s DFT Sbimage G s x y ˆ F s DFT ˆ f s g s Restoration process by estimation Reconstrcted Sbimage x y

41 Estimation by Experiment Used when we hae the same eqipment set p and can repeat the experiment. Response image from Inpt implse image the system System H Aδ x y DFT { Aδ x y } A DFT H G A g x y G DFT Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition.

42 Estimation by Modeling Used when we know physical mechanism nderlying the image formation process that can be expressed mathematically. Original image Seere trblence Example: Atmospheric Trblence model Mild trblence k Low trblence H e k + 5/ 6 k k Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition.

43 Estimation by Modeling: Motion Blrring Assme that camera elocity is x0 t y0 t The blrred image is obtained by where T exposre time. T g x y f x + x0 t y + y0 0 t dt G T g x T 0 0 f f y e x + x + jπ x+ y x x 0 0 t t dxdy y + y + y y 0 0 t dt e t e jπ x+ y jπ x+ y dxdy dxdy dt

44 Estimation by Modeling: Motion Blrring cont. G T 0 T 0 [ jπ x t + y t F e ] 0 0 F f x + T 0 e x 0 jπ x t 0 y t + y 0 + y t 0 dt t e dt jπ x+ y Then we get the motion blrring transfer fnction: H T 0 e jπ x0 t + y0 t For constant motion x 0 t y0 t at bt H T e 0 jπ a+ b dt dt T π a + sin π a b dxdy dt + b e jπ a+ b

45 Motion Blrring Example For constant motion H T π a + b sin π a + b e jπ a+ b Original image Motion blrred image a b 0.1 T 1 Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition.

46 Inerse Filter From degradation model: G F H + N after we obtain H we can estimate F by the inerse filter: ˆ G F F + H N H Noise is enhanced when H is small. In practical the inerse filter is not Poplarly sed. To aoid the side effect of enhancing noise we can apply this formlation to freq. component with in a radis D 0 from the center of H. Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition.

47 Inerse Filter: Example Original image Reslt of applying the fll filter Reslt of applying the filter with D 0 40 Blrred image De to Trblence H e / 6 Reslt of applying the filter with D 0 70 Reslt of applying the filter with D 0 85

48 Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition. Wiener Filter: Minimm Mean Sqare Error Filter Wiener Filter: Minimm Mean Sqare Error Filter Objectie: optimize mean sqare error: { } ˆ f f E e / 1 / ˆ * * G S S H H H G S S H H G S H S S H F f f f f η η η Wiener Filter Formla: where H Degradation fnction S η Power spectrm of noise S f Power spectrm of the ndegraded image

49 Approximation of Wiener Filter Approximation of Wiener Filter / 1 ˆ G S S H H H F f + η Wiener Filter Formla: Approximated Formla: 1 ˆ G K H H H F + Difficlt to estimate Practically K is chosen manally to obtained the best isal reslt!

50 Wiener Filter: Example Original image Reslt of the fll inerse filter Reslt of the inerse filter with D 0 70 Blrred image De to Trblence Reslt of the fll Wiener filter

51 Wiener Filter: Example cont. Original image Reslt of the inerse filter with D 0 70 Blrred image De to Trblence Reslt of the Wiener filter

52 Example: Wiener Filter and Motion Blrring Image degraded by motion blr + AWGN σ η 650 Reslt of the inerse filter Reslt of the Wiener filter σ η 35 Note: K is chosen manally σ η 130 Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition.

53 Degradation model: y x y x h y x f y x g η + Written in a matrix form Constrained Least Sqares Filter Constrained Least Sqares Filter η Hf g + Objectie: to find the minimm of a criterion fnction [ ] M x N y y x f C Sbject to the constraint ˆ η Hf g ˆ * G P H H F + γ We get a constrained least sqare filter where P Forier transform of pxy where w w w T

54 Constrained Least Sqares Filter: Example Constrained Least Sqares Filter: Example ˆ * G P H H F + γ Constrained least sqare filter γ is adaptiely adjsted to achiee the best reslt. Reslts from the preios slide obtained from the constrained least sqare filter

55 Constrained Least Sqares Filter: Example cont. Image degraded by motion blr + AWGN σ η 650 Reslt of the Constrained Least sqare filter Reslt of the Wiener filter σ η 35 σ η 130

56 Constrained Least Sqares Filter:Adjsting Constrained Least Sqares Filter:Adjsting γ Define f H g r ˆ It can be shown that r r r T γ φ We want to adjst gamma so that a ± η r where a accracy factor 1. Specify an initial ale of γ. Compte 3. Stop if is satisfied Otherwise retrn step after increasing γ if or decreasing γ if Use the new ale of γ to recompte 1 1 r a < η r + a > η r ˆ * G P H H F + γ

57 Constrained Least Sqares Filter:Adjsting Constrained Least Sqares Filter:Adjsting γ cont. cont. ˆ * G P H H F + γ ˆ F H G R M x N y y x r MN r [ ] M x N y m y x MN η σ η η M x N y y x MN m η η [ ] η η σ m MN η η r For compting For compting

58 Constrained Least Sqares Filter: Example Original image Use correct noise parameters Correct parameters: Initial γ 10-5 Correction factor 10-6 a 0.5 σ η 10-5 Blrred image De to Trblence Use wrong noise parameters Wrong noise parameter σ η 10 - Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition. Reslts obtained from constrained least sqare filters

59 Geometric Mean filter Geometric Mean filter ˆ 1 * * G S S H H H H F f α η α β + This filter represents a family of filters combined into a single expression α 1 the inerse filter α 0 the Parametric Wiener filter α 0 β 1 the standard Wiener filter β 1 α < 0.5 More like the inerse filter β 1 α > 0.5 More like the Wiener filter Another name: the spectrm eqalization filter

60 Geometric Transformation These transformations are often called rbber-sheet transformations: Printing an image on a rbber sheet and then stretch this sheet according to some predefine set of rles. A geometric transformation consists of basic operations: 1. A spatial transformation : Define how pixels are to be rearranged in the spatially transformed image.. Gray leel interpolation : Assign gray leel ales to pixels in the spatially transformed image.

61 Geometric Transformation : Algorithm x y x y Image f to be restored 1 3 Distorted image g 1. Select coordinate xy in f to be restored. Compte x r x y 4. get pixel ale at g x y By gray leel interpolation y s x y 5. store that ale in pixel fxy 3. Go to pixel x y in a distorted image g 5

62 Spatial Transformation To map between pixel coordinate xy of f and pixel coordinate x y of g x r x y y s x y For a bilinear transformation mapping between a pair of Qadrilateral regions x r x y c x + c y + c xy + c y s x y c x + c y + c xy + c x y x y To obtain rxy and sxy we need to know 4 pairs of coordinates x y and its corresponding x y which are called tiepoints. Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition.

63 Gray Leel Interpolation: Nearest Neighbor Since x y may not be at an integer coordinate we need to Interpolate the ale of g x y Example interpolation methods that can be sed: 1. Nearest neighbor selection. Bilinear interpolation 3. Bicbic interpolation Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition.

64 Gray Leel Interpolation: Bilinear interpolation

65 Geometric Distortion and Restoration Example Original image and tiepoints Tiepoints of distorted image Distorted image Restored image Use nearest neighbor intepolation Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition.

66 Geometric Distortion and Restoration Example cont. Original image and tiepoints Tiepoints of distorted image Distorted image Restored image Use bilinear intepolation Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition.

67 Example: Geometric Restoration Original image Geometrically distorted Image Use the same Spatial Trans Bilinear. as in the preios example Restored image Difference between aboe images Images from Rafael C. Gonzalez and Richard E. Wood Digital Image Processing nd Edition.