Noise models presented by Nisha Menon K

1. 1
NISHA MENON K
MTECH
ROLL NO: 16
15-Nov-13
NOISE MODELS & RESTORATION
OF NOISY IMAGES

2. NOISE AND IMAGES
15-Nov-13
The sources of noise in digital images
arise during image acquisition
(digitization) and transmission
Imaging sensors can be affected by
ambient conditions
 Interference can be added
to an image during transmission

2

3. NOISE MODEL
15-Nov-13
Consider a noisy image is modelled as follows:
g ( x, y)  f ( x, y)   ( x, y)
f(x, y)
η(x, y)
g(x, y)
original image pixel
noise term
resulting noisy pixel
3

4. NOISE MODELS
Spatially independent noise models







15-Nov-13

Gaussian noise
Rayleigh noise
Erlang (Gamma) noise
Uniform noise
Exponential noise
Impulse (salt-and-pepper) noise
Spatially dependent noise model
4

Periodic noise

5. Gaussian
NOISE MODELS
Rayleigh

Gaussian





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15-Nov-13
There are many different
models for the image
noise term η(x, y):
Most common model
Rayleigh
Erlang
Exponential
Uniform
Impulse

Erlang
Exponential
Uniform
Salt and pepper noise
Impulse
5

6. Noise cannot be predicted but can be approximately described in
statistical way using the probability density function (PDF)
15-Nov-13
Gaussian noise


Electronic circuit noise, sensor noise due
to poor illumination and/or high
temperature
The PDF of a Gaussian random
1
variable, z,
 ( z   ) 2 / 2 2
p( z ) 
e
2 
μ

Mean:

Standard deviation:

Variance:
σ
2
σ
6

7. RAYLEIGH NOISE
Range imaging



The PDF of Rayleigh noise,
15-Nov-13

2

ab
( )
z 2
za  / for
( )

e
za

p
( 
z b
)

0
for
za


Mean:
 a  /4
 b
b4 
( )
Variance:
2

7
4

8. ERLANG (GAMMA) NOISE
The PDF of Erlang noise,
integer,
a  0, b is a positive
15-Nov-13

bb z
a a
z1

e for
z0

p)  
(  b1
z ( )!
 0
for
z0

b


Mean:

Variance:
a
b
σ = 2
a
2
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9. Exponential noise
15-Nov-13

p z) aeaz
(
Uniform noise
1


p( a
z b
)

0

for b
az


otherwise
Impulse (salt & pepper) noise
P
a

p ) b
( P
z

0

for a
z

for b
z

otherwise
9

10. NOISE EXAMPLE
This is a suitable pattern to use because
it is composed of simp1e, constant areas
that span the gray scale from black to
near white in only three increments
15-Nov-13
The test pattern to the right is ideal for
demonstrating the addition of noise
Image
Histogram to go here
10
Histogram

11. NOISE EXAMPLE (CONT…)
15-Nov-13
11
Gaussian
Rayleigh
Erlang

12. NOISE EXAMPLE (CONT…)
15-Nov-13
Histogram to go here
12
Exponential
Uniform
Impulse

13. PERIODIC NOISE
15-Nov-13
Typically arises due to
electrical or electromagnetic
interference
Gives rise to regular noise
patterns in an image
Frequency domain techniques in the
Fourier domain are most effective at
removing periodic noise
Periodic noise can be reduced by setting
frequency components corresponding
to noise to zero
13

14. HOW TO ESTIMATE NOISE
PARAMETERS?


noise
Analyze the FT
 Noise

15-Nov-13
 Periodic
PDFs
Sensor specification
Crop the vertical strips and obtain the histogram.
μ=
 z p(z )
S
i
i
zi
σ =
2
 (z
S
zi
i
 μ) p(zi )
2
14

15. 15-Nov-13
We cannot use the image
histogram to estimate
noise PDF.
It is better to use the
histogram of one area
of an image that has
constant intensity to
estimate noise PDF. 15

16. RESTORATION IN PRESENCE OF NOISE BY
SPATIAL FILTERING





15-Nov-13
Spatial filters that are based on ordering the pixel
values that make up the neighbourhood operated on
by the filter
Useful spatial filters include
Mean filter
Median filter
Max and min filter
Midpoint filter
Alpha trimmed mean filter
16

17. MEAN FILTERS
15-Nov-13
Arithmetic mean filter
(AMF, average)
Geometric mean filter
Harmonic mean filter
Contraharmonic mean filter
17

18. Original image
subimage
15-Nov-13
Statistical parameters
Mean, Median, Mode,
Min, Max, Etc.
Moving
window
18
Output image

19. MEAN FILTERS
15-Nov-13

Arithmetic mean filter
ˆx ) 1 g,)
f ,  
(y
(t
s
mn
(,)S
s
t xy

Geometric mean filter
1
mn


ˆx )  g , )
f(,y t
(
s
(
s
,) S

xy
 t

19

20. Harmonic mean filter

15-Nov-13

Works well for salt noise, but fails for pepper noise
mn
1
 ( ,t)
s
( ,) S g
st 
xy
ˆx )
f( ,y 
20

21. 
Contraharmonic mean filter
: eliminates pepper noise
Q0
15-Nov-13
Q0
: eliminates salt noise
gst Q1
(, ) 

( xy
ˆ xy s,t) S
f( , )
Q
g,
(
 st)
( ,) S
st xy
For Q = 0, the filter reduces to an arithmetic mean filter.
For Q = -1, the filter reduces to a harmonic mean filter.
21

22. Image
Corrupted by
guassian noise
15-Nov-13
Filtering
with
Arithmetic
mean filter
Filtering with
geometric
mean filter
22

23. 15-Nov-13
Corrupted by
pepper noise
Corrupted by
salt
noise
Contraharmonic
filter Q= -1.5
Contraharmonic
filter Q=1.5
23

24. 
Usage

Image
obtained
using a contraharmonic
mean filter
With Q=-1.5
15-Nov-13

Arithmetic and geometric mean filters: suited for
Gaussian or uniform noise
Contraharmonic filters: suited for impulse noise but
necessary to know whether noise is dark or light for
selecting Q value
Image
obtained
using a contraharmonic
mean filter
With Q=1.5
24

25. ORDER-STATISTICS FILTERS
15-Nov-13
Median filter
Max and min filters
Midpoint filter
Alpha-trimmed mean filter
25

26. MEDIAN FILTER
15-Nov-13
ˆ
f ( x, y )  median{g ( s, t )}
( s ,t )S xy
Excellent at noise removal, without the smoothing
effects that can occur with other smoothing filters
Particularly good when salt and pepper noise is
present.
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27. MIDPOINT FILTER
15-Nov-13
Midpoint Filter:
ˆ ( x, y )  1  max {g ( s, t )}  min {g ( s, t )}
f

( s ,t )S xy
2 ( s ,t )S xy


Good for random Gaussian and uniform noise
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28. MAX AND MIN FILTER
ˆ
f ( x, y )  max {g ( s, t )}
15-Nov-13
Max Filter:
( s ,t )S xy
Min Filter:
ˆ
f ( x, y )  min {g ( s, t )}
( s ,t )S xy
Max filter is good for pepper noise and min is good for
salt noise
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29. ALPHA-TRIMMED MEAN FILTER
ˆ
f ( x, y) 
1
S gr (s, t )
mn  d ( s ,t ) xy
15-Nov-13
Alpha-Trimmed Mean Filter:
We can delete the d/2 lowest and d/2 highest grey
levels
So gr(s, t) represents the remaining mn – d pixels
Useful in situations involving multiple types of
noise, such as a combination of salt-and-pepper and
Gaussian noise
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30. 15-Nov-13
Corrupted by
salt and
pepper noise
After
passing
through
Median
filter
30

31. 15-Nov-13
max filter
min filter
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32. Image
corrupted
by additive
uniform
noise
Arithmetic mean
filter
Geometric
mean filter
15-Nov-13
Median filter
Image
additionally
corrupted
by additive
salt-andpepper
noise
Alpha trimme
mean filter
32

33. ADAPTIVE FILTERS
15-Nov-13
The filters discussed so far are applied to an entire
image without any regard for how image
characteristics vary from one point to another
The behaviour of adaptive filters changes depending
on the characteristics of the image inside the filter
region
33

34. General concept:
-Filter behavior depends on statistical characteristics of local areas
inside mxn moving window
Statistical characteristics:
Local mean:
1
mL =
 g(s,t)
mn (s,t)S xy
Noise variance:

2
Local variance:
1
σ =
(g(s,t)  mL )2

mn (s,t)S xy
2
L
34
15-Nov-13
- More complex but superior performance compared with “fixed”
filters

35. Adaptive, Local Noise Reduction Filter
Purpose: want to preserve edges
15-Nov-13
Concept:
If 2 is zero,  No noise
the filter should return g(x,y) because g(x,y) = f(x,y)
If L2 is high relative to 2,  Edges (should be preserved),
the filter should return the value close to g(x,y)
If L2 = 2,  Local area has same properties as overall image.
The filter should return the arithmetic mean value mL
Formula:
x
ˆ,)g ) (y 
f y (y  ,)
(  g m
x
x
,

2
2
L
L
35

36. Adaptive Median Filtering
15-Nov-13
zmin = minimum gray level value in Sxy
zmax = maximum gray level value in Sxy
zmedian = median of gray levels in Sxy
zxy = gray level value at pixel (x,y)
where
Smax = maximum allowed size of Sxy
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37. Algorithm
Return zxy  to preserve original details
Else return zmedian  to remove impulse
15-Nov-13
Level A:
Determine
A1= zmedian – zmin
whether zmedian
A2= zmedian – zmax
is an impulse or not
If A1 > 0 and A 2 < 0,  zmedian is not an impulse
goto level B
Else  Window is not big enough
increase window size
If window size <= Smax repeat level A
Else return zxy
Level B:
Determine
B1= zxy – zmin
whether zxy
B2= zxy – zmax
is an impulse or not
If B1 > 0 and B2 < 0  zxy is not an impulse
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38. Purposes of the algorithm
Remove salt-and-pepper (impulse) noise

Provide smoothing of other noise

15-Nov-13

Reduce distortion, such as excessive thinning or
thickening of object boundaries
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39. ADAPTIVE FILTERING EXAMPLE
15-Nov-13
Image corrupted by salt
and pepper noise with
probabilities Pa = Pb=0.25
Result of filtering with a 7
* 7 median filter
Result of adaptive
median filtering with
Smax = 7
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40. CONCLUSION


Different types of noise pdfs have been discussed
Estimation of noise parameters
Restoration of images using
Order- Statistic filters
Adaptive filters
15-Nov-13

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41. 15-Nov-13
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