Image Restoration - univie.ac.atvda.univie.ac.at/.../LectureNotes/16_Image_Restoration.pdf052600 VU Signal and Image Processing Image Restoration Torsten Möller + Hrvoje Bogunović

052600 VU Signal and Image Processing

Image Restoration

Torsten Möller + Hrvoje Bogunović + Raphael Sahann

[email protected] [email protected]

[email protected]

vda.cs.univie.ac.at/Teaching/SIP/17s/

1© Raphael Sahann

mailto:[email protected]



Overview

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

2© Raphael Sahann

Model of the Image Degradation/Restoration Process

3

g(x, y) = f(x, y) ? h(x, y) + ⌘(x, y)

Gonzalez & Woods - Digital Image Processing

(3rd Edition)

spatial domain:

frequency domain: G(u, v) = F (u, v)H(u, v) +N(u, v)

© Raphael Sahann

Noise Probability Density Functions

4

p(z) =1p2⇡�

e�(z�z̄)2/2�2

p(z) =

⇢2b (z � a)e�(z�a)2/b

for z � a0 for z < a

p(z) =

(abzb�1

(b�1)! e�az

for z � 0

0 for z < 0

z̄mean

variance

z̄ =b

az̄ = a+

p⇡b/4

�2 �2 =b(4� ⇡)

4�2 =

b

a2

© Raphael Sahann

Noise Probability Density Functions

5

p(z) =

⇢ 1(b�a) if a z b

0 otherwise

p(z) =

⇢ae�az

for z � 0

0 for z < 0

p(z) =

8<

:

Pa for z = aPb for z = b0 otherwise

mean

variance

z̄ =1

az̄ =

a+ b

2/

/�2 =1

a2�2 =

(b� a)2

12© Raphael Sahann

Sample Image for Illustration

6


(3rd Edition)

© Raphael Sahann

Samples and Histograms of Noise

7


(3rd Edition)

© Raphael Sahann

Samples and Histograms of Noise

8


(3rd Edition)

© Raphael Sahann

Periodic Noise

• usually present due to electrical or electromechanical interference during the image acquisition process

9


(3rd Edition)

© Raphael Sahann

Overview


10© Raphael Sahann

Noise Only Filtering

11

g(x, y) = f(x, y) + ⌘(x, y)

reduced model of degradation where only noise is present:


(3rd Edition)

© Raphael Sahann

Noise Only Filtering

12

Arithmetic Mean Filter Geometric Mean Filter

f̂(x, y) =

P(s,t)2S

xy

g(s, t)Q+1

P(s,t)2S

xy

g(s, t)Qf̂(x, y) =

mnP(s,t)2S

xy

1g(s,t)

f̂(x, y) =

2

4Y

(s,t)2S

xy

g(s, t)

3

5

1mn

f̂(x, y) =1

mn

X

(s,t)2S

xy

g(s, t)

Harmonic Mean Filter Contraharmonic Mean Filter

© Raphael Sahann

13


(3rd Edition)

© Raphael Sahann

Noise Filtering

14© Raphael Sahann


(3rd Edition)

Noise Filtering

Wrong Order in Contraharmonic Filter

15© Raphael Sahann

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

Order-Statistic Filters• Median Filter (50th percentile):

• Max Filter (100th percentile):

• Min Filter (1st percentile):

16© Raphael Sahann

f̂(x, y) = median(s,t)2S

xy

{g(s, t)}

ˆ

f(x, y) = max

(s,t)2S

xy

{g(s, t)}

f̂(x, y) = min(s,t)2S

xy

{g(s, t)}

Order-Statistic Filters• Midpoint Filter:computes the midpoint between minimum and maximum values

• Alpha-Trimmed Mean Filter:deletes the d/2 lowest and d/2 highest pixels and computes the mean from the remaining pixels

17© Raphael Sahann

ˆ

f(x, y) =

1

2

max

(s,t)2S

xy

{g(s, t)}+ min

(s,t)2S

xy

{g(s, t)}�

f̂(x, y) =1

mn� d

X

(s,t)2S

xy

gr(s, t)

18

Median Filter Application


(3rd Edition)

© Raphael Sahann

19

Min/Max Filter Application


(3rd Edition)

(Input image only used pepper noise)

© Raphael Sahann

20

Mean Filter Application


(3rd Edition)

© Raphael SahannGonzalez & Woods - Digital Image Processing (3rd Edition)

• Adaptive Local Noise Reduction Filter:- 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 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

21

Adaptive Filters

© Raphael Sahann

g(x, y)

g(x, y)

�2⌘

�2⌘

f̂(x, y) = g(x, y)��

2⌘

�

2L

[g(x, y)�mL]

Adaptive Filters

22© Raphael Sahann


(3rd Edition)

Adaptive Median Filter

23© Raphael Sahann

A1 = zmed

� zmin

A2 = zmed

� zmax

if A1 > 0 and A2 < 0, go to stage B

else increase window size

if window size Smax

repeat stage A

else output zmed

B1 = zxy

� zmin

B2 = zxy

� zmax

if B1 > 0 and B2 < 0, output zxy

else output zmed

Stage A: Stage B:

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 impulsestage B tries to estimate whether the center of the window zxy is an impulse.

Adaptive Median Filter

24


© Raphael Sahann

Overview


25© Raphael Sahann

Frequency Domain Filtering• Bandreject Filters:

26© Raphael Sahann


Bandreject Filtering

27© Raphael Sahann


(3rd Edition)

Frequency Domain Filtering

• Bandpass Filter:

28


(3rd Edition)

© Raphael Sahann

Notch Filters

29


(3rd Edition)

© Raphael Sahann

Notch Filtering

• Has to appear in symmetric pairs about the origin

• Removes periodic interference

30© Raphael Sahann


(3rd Edition)

Optimum Notch Filtering

31


© Raphael Sahann

Optimum Notch Filtering• Generate a notch pass filter by observing the spectrum G(u,v)

• After selecting the filter obtain corresponding spatial domain representation

• To minimize impact on image information subtract noise weighted by a weighting or modulating function w(x,y):

32

N(u, v) = HNP(u, v)G(u, v)

⌘(u, v) = F�1 {HNP(u, v)G(u, v)}

f̂(x, y) = g(x, y)� w(x, y)⌘(x, y)

© Raphael Sahann

Optimum Notch Filtering• Weighting function can be chosen according to need; one

approach minimizes the local variance

33


© Raphael Sahann

Optimum Notch Filtering

34

Input ResultGonzalez & Woods - Digital Image Processing (3rd Edition)

© Raphael Sahann

Overview


35© Raphael Sahann

Estimating the Degradation Function

36© Raphael Sahann


g(x, y) = h(x, y) ? f(x, y) + ⌘(x, y)

G(u, v) = H(u, v)F (u, v) +N(u, v)

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© Raphael Sahann

• 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:G(u.v) … Fourier transform of observed imageA … constant describing strength of impulse

38


H(u, v) =G(u, v)

A

© Raphael Sahann

Estimation by Experimentation

39© Raphael Sahann


• 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


© Raphael Sahann

Estimation by Modeling

41


(3rd Edition)

© Raphael Sahann

Estimation by Modeling

42


(3rd Edition)

© Raphael Sahann

Inverse Filtering• simple estimate of the transform by dividing the transform by the

degradation function

43

F̂ (u, v) =G(u, v)

H(u, v)

F̂ (u, v) = F (u, v) +N(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

© Raphael Sahann

Inverse Filtering

44© Raphael Sahann


(3rd Edition)

© Raphael Sahann

Wiener Filtering• Minimum Mean Square Error (Wiener) Filtering

• incorporates degradation and noise into restoration

45

S⌘ ... power spectrum of the noise

Sf ... power spectrum of the undegraded image

F̂ (u, v) =

2

4 1

H(u, v)

|H(u, v)|2

|H(u, v)|2 + S⌘(u,v)Sf (u,v)

3

5G(u, v)

F̂ (u, v) =

1

H(u, v)

|H(u, v)|2

|H(u, v)|2 +K

�G(u, v)

© Raphael Sahann

Wiener Filtering

46© Raphael Sahann


Wiener Filtering

47

• Optimal value for K needs to be guessed/iteratively adjusted to yield optimal result

© Raphael Sahann


(3rd Edition)

Documents

Image Restoration - univie.ac.atvda.univie.ac.at/.../LectureNotes/16_Image_Restoration.pdf052600 VU Signal and Image Processing Image Restoration Torsten Möller + Hrvoje Bogunović