Post on 01-Jan-2016
Chapter 5 Image Restoration
Image enhancement◦ Subjective process
Image restoration◦ Objective process to recover the original
image. Involving formula or criteria that will yield an
optimal result.
5. 1 Model of Image degradation and restoration5. 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.
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
Gaussian noise
◦ : mean; : variance
◦ 70% [(-), (+)]◦ 95 % [(-2),
(+2)]
22 2/)(
2
1)(
zezp
•Because of its tractability, Gaussian (normal) noise model is often applicable at best.
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.
Rayleigh noise
= a+(b/4)1/2
= b(4-)/4
azfor
azforeazbzp
baz
0
)(2
)(/)( 2
Erlang (gamma) noise
=b/a; =b/a2
00
0)!1()(
1
zfor
zforeb
zazp
azbb
Exponential noise
=1/a; =1/a2
• A special case Erlang noise model when b = 1.
00
0)(
zfor
zforaezp
az
Uniform noise
=(a+b)/2; =(b-a)2/12
otherwise
bzaforabzp
0
1)(
Impulse noise (salt and pepper noise)
otherwise
bzfor
azfor
P
P
zp b
a
0
)(
ExampleExample
Results of adding noiseResults of adding noise
Results of adding noiseResults of adding noise
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.
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.
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.
Estimation of Noise ParametersEstimation of Noise Parameters
Used when only additive noise is present.
Mean filters◦Arithmetic mean filter:
The new pixel value is resulted from averaging the pixels in the area by Sxy.
◦Geometric mean filter:
Achieves smoothing comparable to arithmetic approach, but tends to lose less image detail.
xySts
tsgmn
yxf),(
),(1
),(ˆ
mn
Sts xy
tsgyxf
/1
),(
),(),(ˆ
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.
),(),(ˆ),(
tsgmedianyxfxySts
),(max),(ˆ),(
tsgyxfxySts
),(min),(ˆ),(
tsgyxfxySts
),(min),(max
2
1),(ˆ
),(),(tsgtsgyxf
xyxy StsSts
Example 5.3Example 5.3Iteratively applying median filter to an image corrupted by impulse noise.
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
xyStsr tsg
dmnyxf
),(
),(1
),(ˆ
Filters whose behavior changes based on statistical characteristics of the image.
Two adaptive filters are considered:(1) Adaptive, local noise reduction filter.(2) Adaptive median filter.
Two parameters are considered:◦ Mean: measure of average gray level.◦ Variance: measure of average contrast.
Four measurements:(1) noisy image at (x, y ): g(x, y )(2) The variance of noise 2
(3) The local mean mL in Sxy
(4) The local variance 2L
Given the corrupted image g(x, y), find f(x, y).Conditions:
(a) 2 is zero (Zero-noise case)
– Simply return the value of g(x, y).(b) If 2
L is higher than 2
– Could be edge and should be preserved.– Return value close to g(x, y).
(c) 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:
]),([),(),(ˆ2
2
LL
myxgyxgyxf
Adaptive, Local Noise Reduction FilterAdaptive, Local Noise Reduction 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
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 Belse increase the window sizeIf window size Smax repeat level
Aelse output zxy
Level B: B1=zxy-zmin B2=zxy-zmax
If B1>0 AND B2<0, output zxy
Else output zmed.
Example 5.5Example 5.5