VU Signal and Image Processing. Image Restoration. Torsten Möller + Hrvoje Bogunović + Raphael Sahann

Transcription

1 VU Signal and Image Processing Image Restoration Torsten Möller + Hrvoje Bogunović + Raphael Sahann torsten.moeller@univie.ac.at hrvoje.bogunovic@meduniwien.ac.at raphael.sahann@univie.ac.at vda.cs.univie.ac.at/teaching/sip/17s/ Raphael 1 Sahann

2 Overview Image Restoration Noise Models Spatial Noise Only Filtering Periodic Noise Reduction by Frequency Domain Filtering Estimating Degradation and Filtering Methods 2

3 Model of the Image Degradation/ Restoration Process Gonzalez & Woods - Digital Image Processing (3rd Edition) spatial domain: g(x, y) =f(x, y)? h(x, y)+ (x, y) frequency domain: G(u, v) =F (u, v)h(u, v)+n(u, v) 3

4 Noise Probability Density Functions p(z) = 1 p 2 e (z z)2 /2 2 p(z) = 2 b (z a)e (z a)2 /b for z a 0 for z<a p(z) = ( a b z b 1 (b 1)! e az for z 0 0 for z<0 mean z z = a + p b/4 z = b a variance 2 2 = b(4 ) 4 2 = b a 2 4

5 Noise Probability Density Functions p(z) = ae az for z 0 0 for z<0 p(z) = 1 (b a) if a apple z apple b 0 otherwise p(z) = 8 < : P a P b for z = a for z = b 0 otherwise mean z = 1 a z = a + b 2 / variance 2 = 1 a 2 2 = (b a)2 12 / 5

6 Sample Image for Illustration Gonzalez & Woods - Digital Image Processing (3rd Edition) 6

7 Samples and Histograms of Noise Gonzalez & Woods - Digital Image Processing (3rd Edition) 7

8 Samples and Histograms of Noise Gonzalez & Woods - Digital Image Processing (3rd Edition) 8

9 Periodic Noise usually present due to electrical or electromechanical interference during the image acquisition process Gonzalez & Woods - Digital Image Processing (3rd Edition) 9

10 Overview Image Restoration Noise Models Spatial Noise Only Filtering Periodic Noise Reduction by Frequency Domain Filtering Estimating Degradation and Filtering Methods 10

11 Noise Only Filtering reduced model of degradation where only noise is present: g(x, y) =f(x, y)+ (x, y) Gonzalez & Woods - Digital Image Processing (3rd Edition) 11

12 Noise Only Filtering Arithmetic Mean Filter Geometric Mean Filter ˆf(x, y) = 1 mn X g(s, t) ˆf(x, y) = 2 4 Y 3 g(s, t) 5 1 mn (s,t)2sxy (s,t)2sxy Harmonic Mean Filter Contraharmonic Mean Filter P ˆf(x, y) = mn P (s,t)2sxy 1 g(s,t) ˆf(x, y) = (s,t)2sxy P (s,t)2sxy g(s, t) Q+1 g(s, t) Q 12

13 Noise Filtering Gonzalez & Woods - Digital Image Processing (3rd Edition) 13

14 Noise Filtering Gonzalez & Woods - Digital Image Processing (3rd Edition) 14

15 Wrong Order in Contraharmonic Filter Gonzalez & Woods - Digital Image Processing (3rd Edition) 15

16 Order-Statistic Filters Median Filter (50th percentile): ˆf(x, y) = median (s,t)2sxy {g(s, t)} Max Filter (100th percentile): ˆf(x, y) = max (s,t)2sxy {g(s, t)} Min Filter (1st percentile): ˆf(x, y) = min (s,t)2sxy {g(s, t)} 16

17 Order-Statistic Filters Midpoint Filter: apple ˆf(x, y) = 1 2 max (s,t)2sxy {g(s, t)} + min (s,t)2sxy {g(s, t)} computes the midpoint between minimum and maximum values Alpha-Trimmed Mean Filter: ˆf(x, y) = 1 X g r (s, t) mn d (s,t)2sxy deletes the d/2 lowest and d/2 highest pixels and computes the mean from the remaining pixels 17

18 Median Filter Application Gonzalez & Woods - Digital Image Processing (3rd Edition) 18

19 Min/Max Filter Application Gonzalez & Woods - Digital Image Processing (3rd Edition) (Input image only used pepper noise) 19

20 Mean Filter Application Gonzalez & Woods - Digital Image Processing (3rd Edition) Gonzalez & Woods - Digital Image Processing (3rd Edition) 20

21 Adaptive Filters Adaptive Local Noise Reduction Filter: ˆf(x, y) =g(x, y) 2 2 L [g(x, y) m L ] 2 - if bra is zero, the filter should return the value of g(x,y) - if the local variance is high relative to bra,an edge is found and g(x, y) a value close to g(x,y) should be returned - if the variances are equal we want the arithmetic mean of the pixels in the window to reduce noise by blurring 2 g(x, y) 21

22 Adaptive Filters Gonzalez & Woods - Digital Image Processing (3rd Edition) 22

23 Adaptive Median Filter Stage A: Stage B: A1 =z med z min B1 =z xy z min A2 =z med z max B2 =z xy z max if A1 > 0 and A2 < 0, go to stage B else increase window size if B1 > 0 and B2 < 0, output z xy else output z med if window size apple S max repeat stage A else output z med This filter aims to: remove salt-and-pepper noise, provide smoothing of other noise and reduce distortion such as thinning or thickening Stage A tries to determine whether zmed is an impulse or not. If it is no impulse stage B tries to estimate whether the center of the window zxy is an impulse. 23

24 Adaptive Median Filter Gonzalez & Woods - Digital Image Processing (3rd Edition) 24

25 Overview Image Restoration Noise Models Spatial Noise Only Filtering Periodic Noise Reduction by Frequency Domain Filtering Estimating Degradation and Filtering Methods 25

26 Frequency Domain Filtering Bandreject Filters: Gonzalez & Woods - Digital Image Processing (3rd Edition) 26

27 Bandreject Filtering Gonzalez & Woods - Digital Image Processing (3rd Edition) 27

28 Frequency Domain Filtering Bandpass Filter: Gonzalez & Woods - Digital Image Processing (3rd Edition) 28

29 Notch Filters Gonzalez & Woods - Digital Image Processing (3rd Edition) 29

30 Notch Filtering Gonzalez & Woods - Digital Image Processing (3rd Edition) Has to appear in symmetric pairs about the origin Removes periodic interference 30

31 Optimum Notch Filtering Gonzalez & Woods - Digital Image Processing (3rd Edition) 31

32 Optimum Notch Filtering Generate a notch pass filter by observing the spectrum G(u,v) N(u, v) =H NP (u, v)g(u, v) After selecting the filter obtain corresponding spatial domain representation (u, v) =F 1 {H NP (u, v)g(u, v)} To minimize impact on image information subtract noise weighted by a weighting or modulating function w(x,y): ˆf(x, y) =g(x, y) w(x, y) (x, y) 32

33 Optimum Notch Filtering Weighting function can be chosen according to need; one approach minimizes the local variance Gonzalez & Woods - Digital Image Processing (3rd Edition) 33

34 Optimum Notch Filtering Input Result Gonzalez & Woods - Digital Image Processing (3rd Edition) 34

35 Overview Image Restoration Noise Models Spatial Noise Only Filtering Periodic Noise Reduction by Frequency Domain Filtering Estimating Degradation and Filtering Methods 35

36 Estimating the Degradation Function Gonzalez & Woods - Digital Image Processing (3rd Edition) g(x, y) =h(x, y)? f(x, y)+ (x, y) G(u, v) =H(u, v)f (u, v)+n(u, v) 36

37 Estimating the Degradation Function Estimation by Image Observation - Manually looking for areas with the least amount of possible noise - process the area to obtain the closest estimate to the original image - calculate the difference between the observed and processed area to construct a degradation function H(u,v) - use resulting function in restoration process very laborious process, which is only used under specific circumstances, such as the restoration of an old photograph of historical value 37

38 Estimating the Degradation Function Estimation by Experimentation - prerequisite: similar equipment to the equipment used to acquire the degraded image is available - obtain the impulse response of the equipment by imaging a bright dot of light - Fourier Transform of an impulse is a constant, therefore: H(u, v) = G(u, v) A G(u.v) Fourier transform of observed image A constant describing strength of impulse 38

39 Estimation by Experimentation Gonzalez & Woods - Digital Image Processing (3rd Edition) 39

40 Estimating the Degradation Function Estimation by Modeling - can take physical characteristics into account (see Turbulence Model by Hufnagel and Stanley) - mathematical model can be obtained by starting from basic principles e.g. blurring by uniform linear motion between the image and the sensor during the image acquisition consistently used for many years, because of the insight it affords into the image restoration problem 40

41 Estimation by Modeling Gonzalez & Woods - Digital Image Processing (3rd Edition) 41

42 Estimation by Modeling Gonzalez & Woods - Digital Image Processing (3rd Edition) 42

43 Inverse Filtering simple estimate of the transform by dividing the transform by the degradation function ˆF (u, v) = G(u, v) H(u, v) cannot recover undegraded image fully, because N(u,v) is not known very small values of H(u,v) will dominate the estimate ˆF (u, v) =F (u, v)+ N(u, v) H(u, v) 43

44 Inverse Filtering Gonzalez & Woods - Digital Image Processing (3rd Edition) 44

45 Wiener Filtering Minimum Mean Square Error (Wiener) Filtering incorporates degradation and noise into restoration 2 3 ˆF (u, v) = 4 1 H(u, v) 2 5 G(u, v) H(u, v) H(u, v) 2 + S (u,v) S f (u,v) S... power spectrum of the noise S f... power spectrum of the undegraded image apple 1 H(u, v) 2 ˆF (u, v) = G(u, v) H(u, v) H(u, v) 2 + K 45

46 Wiener Filtering Gonzalez & Woods - Digital Image Processing (3rd Edition) 46

47 Wiener Filtering Optimal value for K needs to be guessed/ iteratively adjusted to yield optimal result Gonzalez & Woods Digital Image Processing (3rd Edition) 47