1. 5. 1 Model of Image degradation and restoration
g(x,y)=h(x,y)f(x,y)+(x,y)
Note: H is a linear, position-invariant process.

2. Spatial and frequency property of noise
5.2 Noise Model
Spatial and frequency property of noise
White noise (random noise)
A sequence of random positive/negative numbers whose mean is zero.
Independent of spatial coordinates and the image itself.
In the frequency domain, all the frequencies are the same.
All the frequencies are corrupted by an additional constant frequency.
Periodic noise

3. Noise Probability Density Functions (1)
Gaussian noise
 : mean;  : variance
70% [(-), (+)]
95 % [(-2), (+2)]
Because of its tractability, Gaussian (normal) noise model is often applicable at best.

4. Implementation
The problem of adding noise to an image is identical to that of adding a random number to the gray level of each pixel.
Noise models describe the distribution (probability density function, PDF) of these random numbers.
How to match the PDF of a group of random numbers to a specific noise model?
Histogram matching.

5. Noise Probability Density Functions (2)
Rayleigh noise
 = a+(b/4)1/2
 = b(4-)/4

6. Noise Probability Density Functions (3)
Erlang (gamma) noise
=b/a;  =b/a2

7. Noise Probability Density Functions (4)
Exponential noise
=1/a;  =1/a2
A special case Erlang noise model when b = 1.

8. Noise Probability Density Functions (5)
Uniform noise
=(a+b)/2;  =(b-a)2/12

9. Noise Probability Density Functions (6)
Impulse noise (salt and pepper noise)

10. Example

11. Results of adding noise

12. Results of adding noise

13. The Principle Use of Noise Models
Gaussian noise:
electronic circuit noise and sensor noise due to poor illumination or high temperature.
Rayleigh noise:
Noise in range imaging.
Erlang noise:
Noise in laser imaging.
Impulse noise:
Quick transients take place during imaging.
Uniform noise:
Used in simulations.

15. Impulse noise is caused by
Malfunctioning pixels in camera sensors
Fault memory locations in hardware
Transmisison in a noisy channel
Two types:
Salt-and-pepper Noise
Uniformly-Distrubuted Random Noise

23. 5.2.4 Estimation of Noise Parameters
How do we know which noise model adaptive to the currently available imaging tool?
Image a solid gray board that is illuminated uniformly.
Crop a small patch of constant grey level and analyze its histogram to see which model matches.

24. 5.2.4 Estimation of Noise Parameters
Gaussian n:
Find the mean  and standard deviation  of the histogram (Gaussian noise).
Rayleigh, Erlang, and uniform noise:
Calculate the a and b from  and .
Impulse noise:
Compute the height of peaks at gray levels 0 and 255 to find Pa and Pb.

25. Estimation of Noise Parameters

35. Non-Local Means Algorithm
(Buades 2005)
The recently proposed Nonlocal Means Algorithm (NLmeans)
has offered remarkably promising results.
Unlike previous denoising methods that rely on the local
regularity assumption, the NL-means exploits spatial
correlation in the entire image for noise removal.
It adjusts each pixel value with a weighted average of other pixels
whose neighborhood has a similar geometrical configuration.
Since image pixels are highly correlated while noise is
typically independently and identically distributed (i.i.d.),
averaging of these pixels results in noise cancellation and
yields a pixel that is similar to its original value.

36. New Idea: NL-Means Filter (Buades 2005)
Same goals: ‘Smooth within Similar Regions’
KEY INSIGHT: Generalize, extend ‘Similarity’
Bilateral: Averages neighbors with similar intensities;
NL-Means: Averages neighbors with similar neighborhoods!

37. NL-Means Method: Buades (2005)
For each and
every pixel p:

38. NL-Means Method: Buades (2005)
For each and
every pixel p:
Define a small, simple fixed size neighborhood;

39. NL-Means Method: Buades (2005)
0.74
0.32
0.41
0.55 …
For each and
every pixel p:
Define a small, simple fixed size neighborhood;
Define vector Vp: a list of neighboring pixel values.
Vp =

40. NL-Means Method: Buades (2005)q
‘Similar’ pixels p, q
 SMALL vector distance; p
|| Vp – Vq ||2

41. NL-Means Method: Buades (2005)q
‘Dissimilar’ pixels p, q
 LARGE vector distance; q p
|| Vp – Vq ||2

42. NL-Means Method: Buades (2005)
‘Dissimilar’ pixels p, q
 LARGE vector distance;
Filter with this! q p
|| Vp – Vq ||2

43. NL-Means Method: Buades (2005)
p, q neighbors define
a vector distance;
Filter with this: No spatial term! q
|| Vp – Vq ||2 p

44. NL-Means Method: Buades (2005)
pixels p, q neighbors Set a vector distance;
Vector Distance to p sets weight for each pixel q q
|| Vp – Vq ||2 p

45. NL-Means Filter (Buades 2005)
Noisy source image:

46. NL-Means Filter (Buades 2005)
Gaussian Filter
Low noise,
Low detail

47. NL-Means Filter (Buades 2005)
Anisotropic Diffusion
(Note ‘stairsteps’: ~ piecewise constant)

48. NL-Means Filter (Buades 2005)
Bilateral Filter
(better, but similar ‘stairsteps’:

49. NL-Means Filter (Buades 2005)
Sharp,
Low noise,
Few artifacts.

50. Order-Statistics filters
Median filter
Max filter: find the brightest points to reduce the pepper noise
Min filter: find the darkest point to reduce the salt noise
Midpoint filter: combining statistics and averaging.

51. Median filter

52. The median filter is used for removing noise.
It can remove isolated impulsive noise and at the same time it preserves the edges and other structures in the image.
Contrary to average filtering it does not smooth the edges.

53. Unlike the mean filter, the median filter is non- linear
Unlike the mean filter, the median filter is non- linear. This means that for two images A(x) and B(x):

58. Removal of Line Artifacts by Median Filtering

59. The original image. b) Original image corrupted by salt and pepper noise (p=5 % that a bit is flipped.
c) After smoothing with a 3 x 3 filter most of the noise has been eliminated.

60. d) If we smooth the image with a larger median filter, e. g
d) If we smooth the image with a larger median filter, e.g. 7 x 7, all the noise pixels disappear.
e) Alternatively, we can pass a 3 x 3 filter over the image 3 times in order to remove the noise with less loss of detail.

61. Salt and pepper noise (5 %)
Smoothed by 3 x 3 window
Smoothed by 3 x 3 window

62. Salt and pepper noise (5 %)
Median filtered by 3 x 3 window
Median filtered by 3 x 3 window

63. Example 5.3
Iteratively applying median filter to an image corrupted by impulse noise.

64. Alpha-trimmed mean filter
is windowed filter of nonlinear class, by its nature is hybrid of the mean and median filters. The basic idea behind filter is for any element of the signal (image) look at its neighborhood, discard the most atypical elements and calculate mean value using the rest of them.

65. Alpha-Trimmed Mean Filter
Combining the advantages of mean filter and order-statistics filter.
Suppose delete d/2 lowest and d/2 highest gray-level value in the neighborhood of Sxy and average the remaining mn-d pixel, denoted by gr(s, t).
where d=0 ~ mn-1

68. Two adaptive filters are considered:
Filters whose behavior changes based on statistical characteristics of the image.
Two adaptive filters are considered:
Adaptive, local noise reduction filter.
Adaptive median filter.

69. Adaptive, Local Noise Reduction Filter
Two parameters are considered:
Mean: measure of average gray level.
Variance: measure of average contrast.
Four measurements:
noisy image at (x, y ): g(x, y )
The variance of noise 2
The local mean mL in Sxy
The local variance 2L

70. Adaptive, Local Noise Reduction Filter
Given the corrupted image g(x, y), find f(x, y).
Conditions:
2 is zero (Zero-noise case)
Simply return the value of g(x, y).
If 2L is higher than 2
Could be edge and should be preserved.
Return value close to g(x, y).
If 2L = 2
when the local area has similar properties with the overall image.
Return arithmetic mean value of the pixels in Sxy.
General expression:

72. Adaptive, Local Noise Reduction Filter

74. Adaptive Median Filter
Adaptive median filter can handle impulse noise with larger probability (Pa and Pb are large).
This approach changes window size during operation (according to certain criteria).
First, define the following notations:
zmin=minimum gray-level value in Sxy
zmax=maximum gray-level value in Sxy
zmed=median gray-level value in Sxy
zxy= gray-level at (x,y)
Smax=maximum allowed size of Sxy

75. Adaptive Median Filter
The adaptive median filter algorithm works in two levels: A and B
Level A: A1=zmed-zmin
A2=zmed-zmax
If A1>0 and A2<0 goto level B
else increase the window size
If window size  Smax repeat level A
else output zxy
Level B: B1=zxy-zmin
B2=zxy-zmax
If B1>0 AND B2<0, output zxy
Else output zmed.

76. Example 5.5

77. Standard Median output
Salt and pepper noise
Salt and Pepper
Adaptive median Output
Standard Median output

78. With Non-impulsive noise
Adaptive Median Output
Gaussian Noise
Standard Median output

79. With both types of noise
Adaptive Median output
Gaussian and impulsive Noise
Standard Median output

80. Conclusions
The adaptive median filter successfully removes impulsive noise from images. It does a reasonably good job of smoothening images that contain non-impulsive noise.
When both types of noise are present, the algorithm is not as successful in removing impulsive noise and its performance deteriorates.
Overall, the performance is as expected and the successful implementation of the adaptive median filter is presented.

81. The End