Image Restoration 影像復原 - DiUniTofarid/teaching/DIP/Liu - Ch 5 - Image Restoration and... ·...
Transcript of Image Restoration 影像復原 - DiUniTofarid/teaching/DIP/Liu - Ch 5 - Image Restoration and... ·...
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Image Restoration 影像復原
Spring 2005, Jen-Chang Liu
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Outline A model of the image degradation / restoration
process Noise models Restoration in the presence of noise only – spatial
filtering Periodic noise reduction by frequency domain
filtering Linear, position-invariant degradations Estimating the degradation function Inverse filtering
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A model of the image degradation/restoration process
g(x,y)=f(x,y)*h(x,y)+η(x,y)
G(u,v)=F(u,v)H(u,v)+N(u,v)
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Noise models 雜訊模型
Source of noise Image acquisition (digitization) Image transmission
Spatial properties of noise Statistical behavior of the gray-level values of
pixels correlation with the image
Frequency properties of noise Fourier spectrum e.g., white noise (a constant Fourier spectrum)
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Noise probability density functions
Noises are taken as random variables Random variables
Probability density function (PDF)
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Gaussian noise Math. tractability in spatial and frequency
domain Electronic circuit noise and sensor noise
22 2/)(
21)( σµ
σπ−−= zezp
mean variance
∫∞
∞−
=1)( dzzpNote:
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Gaussian noise (PDF)
70% in [(µ−σ), (µ+σ)]95% in [(µ−2σ), (µ+2σ)]
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Uniform noise
≤≤
−=otherwise 0
if 1)( bza
abzp
12)(
22
2 ab
ba
−=
+=
σ
µMean:
Variance:
Less practical, used for random number generator
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Uniform PDF
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Impulse (salt-and-pepper) nosie
==
=otherwise 0for for
)( bzPazP
zp b
a
If either Pa or Pb is zero, it is called unipolar.Otherwise, it is called bipoloar.
•In practical, impulses are usually stronger than imagesignals. Ex., a=0(black) and b=255(white) in 8-bit image.
Quick transients, such as faulty switching during imaging
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Impulse (salt-and-pepper) nosie PDF
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Rayleigh Noise
2( ) /
The PDF of Rayleigh noise is given by2 ( ) for
( )0 for
z a bz a e z ap z b
z a
− − − ≥= <
2
The mean and variance of this density are given by
/ 4(4 )
4
z a bb
ππσ
= +−
=
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Rayleigh Noise
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Erlang (Gamma) Noise
1
The PDF of Erlang noise is given by
for 0 ( ) ( 1)!
0 for
b baza z e z
p z bz a
−−
≥= − <
2 2
The mean and variance of this density are given by
/ /
z b ab aσ
=
=
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Erlang (Gamma) Noise
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Exponential Noise
The PDF of exponential noise is given by
for 0 ( )
0 for
azae zp z
z a
− ≥=
<
2 2
The mean and variance of this density are given by
1/ 1/
z aaσ
=
=
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Exponential Noise
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Test for noise behavior Test pattern
Its histogram:
0 255
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Examples of Noise: Noisy Images(2)
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Periodic noise Arise from electrical or electromechanical
interference during image acquisition Spatial dependence Observed in the frequency domain
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Sinusoidal noise:Complex conjugatepair in frequencydomain
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Estimation of noise parameters Periodic noise
Observe the frequency spectrum Random noise with unknown PDFs
Case 1: imaging system is available Capture images of “flat” environment
Case 2: noisy images available Take a strip from constant area Draw the histogram and observe it Measure the mean and variance
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Observe the histogram
Gaussian uniform
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Histogram is an estimate of PDF
Measure the mean and variance
∑∈
=Sz
iii
zpz )(µ
∑∈
−=Sz
iii
zpz )()( 22 µσ
Gaussian: µ, σUniform: a, b
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Outline A model of the image degradation / restoration
process Noise models Restoration in the presence of noise only – spatial
filtering Periodic noise reduction by frequency domain
filtering Linear, position-invariant degradations Estimating the degradation function Inverse filtering
![Page 28: Image Restoration 影像復原 - DiUniTofarid/teaching/DIP/Liu - Ch 5 - Image Restoration and... · A model of the image degradation / restoration process Noise models Restoration](https://reader034.fdocuments.us/reader034/viewer/2022050412/5f88f2657d2b40275856059a/html5/thumbnails/28.jpg)
Additive noise only
g(x,y)=f(x,y)+η(x,y)
G(u,v)=F(u,v)+N(u,v)
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Spatial filters for de-noising additive noise
Skills similar to image enhancement Mean filters Order-statistics filters Adaptive filters
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Mean filters Arithmetic mean
Geometric mean
∑∈
=xySts
tsgmn
yxf),(
),(1),(ˆ
Window centered at (x,y)
mn
Ststsgyxf
xy
/1
),(),(),(ˆ
∏=
∈
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original NoisyGaussian
µ=0σ=20
Arith.mean
Geometricmean
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Mean filters (cont.) Harmonic mean filter
Contra-harmonic mean filter
∑
∑
∈
∈
+
=
xy
xy
Sts
QSts
Q
tsg
tsgyxf
),(
),(
1
),(
),(),(ˆ
∑∈
=
xySts tsg
mnyxf
),( ),(1),(ˆ
Q=-1, harmonic
Q=0, airth. mean
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PepperNoise黑點
SaltNoise白點
Contra-harmonicQ=1.5
Contra-harmonic
Q=-1.5
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Wrong sign in contra-harmonic filtering
Q=-1.5 Q=1.5
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Order-statistics filters Based on the ordering(ranking) of pixels
Suitable for unipolar or bipolar noise (salt and pepper noise)
Median filters Max/min filters Midpoint filters Alpha-trimmed mean filters
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Order-statistics filters Median filter
Max/min filters
)},({),(ˆ),(
tsgmedianyxfxySts ∈
=
)},({max),(ˆ),(
tsgyxfxySts ∈
=
)},({min),(ˆ),(
tsgyxfxySts ∈
=
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bipolarNoisePa = 0.1Pb = 0.1
3x3Median
FilterPass 1
3x3MedianFilterPass 2
3x3Median
FilterPass 3
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Peppernoise
Saltnoise
Maxfilter
Minfilter
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Order-statistics filters (cont.) Midpoint filter
Alpha-trimmed mean filter Delete the d/2 lowest and d/2 highest gray-level
pixels
+=
∈∈)},({min)},({max
21),(ˆ
),(),(tsgtsgyxf
xyxy StsSts
∑∈−
=xySts
r tsgdmn
yxf),(
),(1),(ˆMiddle (mn-d) pixels
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Uniform noiseµ=0σ2=800
Left +Bipolar NoisePa = 0.1Pb = 0.1
5x5Arith. Mean
filter
5x5Geometric
mean
5x5Medianfilter
5x5Alpha-trim.
Filterd=5
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Adaptive filters Adapted to the behavior based on the
statistical characteristics of the image inside the filter region Sxy
Improved performance v.s increased complexity
Example: Adaptive local noise reduction filter
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Adaptive local noise reduction filter
Simplest statistical measurement Mean and variance
Known parameters on local region Sxy g(x,y): noisy image pixel value σ2
η: noise variance (assume known a prior) mL : local mean σ2
L : local variance
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Adaptive local noise reduction filter (cont.)
Analysis: we want to do If σ2
η is zero, return g(x,y) If σ2
L> σ2η , return value close to g(x,y)
If σ2L= σ2
η , return the arithmetic mean mL
Formula
[ ]LL
myxgyxgyxf −−= ),(),(),(ˆ2
2
σση
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µ=0σ2=1000
Gaussiannoise Arith.
mean7x7
Geometricmean7x7
adaptive
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Outline A model of the image degradation / restoration
process Noise models Restoration in the presence of noise only – spatial
filtering Periodic noise reduction by frequency domain
filtering Linear, position-invariant degradations Estimating the degradation function Inverse filtering
![Page 46: Image Restoration 影像復原 - DiUniTofarid/teaching/DIP/Liu - Ch 5 - Image Restoration and... · A model of the image degradation / restoration process Noise models Restoration](https://reader034.fdocuments.us/reader034/viewer/2022050412/5f88f2657d2b40275856059a/html5/thumbnails/46.jpg)
Periodic noise reduction (cont.) Bandreject filters Bandpass filters Notch filters Optimum notch filtering
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Bandreject filters
* Reject an isotropic frequency
ideal Butterworth Gaussian
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noisy spectrum
bandreject
filtered
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Bandpass filters Hbp(u,v)=1- Hbr(u,v)
{ }),(),(1 vuHvuG bp−ℑ
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Notch filters Reject(or pass) frequencies in predefined
neighborhoods about a center frequency
ideal
Butterworth Gaussian
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HorizontalScan lines
NotchpassDFT
Notchpass
Notchreject
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Outline A model of the image degradation / restoration
process Noise models Restoration in the presence of noise only – spatial
filtering Periodic noise reduction by frequency domain
filtering Linear, position-invariant degradations Estimating the degradation function Inverse filtering
![Page 53: Image Restoration 影像復原 - DiUniTofarid/teaching/DIP/Liu - Ch 5 - Image Restoration and... · A model of the image degradation / restoration process Noise models Restoration](https://reader034.fdocuments.us/reader034/viewer/2022050412/5f88f2657d2b40275856059a/html5/thumbnails/53.jpg)
A model of the image degradation /restoration process
g(x,y)=f(x,y)*h(x,y)+η(x,y)
G(u,v)=F(u,v)H(u,v)+N(u,v)
If linear, position-invariant system
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Linear, position-invariant degradation
Linear system H[af1(x,y)+bf2(x,y)]=aH[f1(x,y)]+bH[f2(x,y)]
Position(space)-invariant system H[f(x,y)]=g(x,y) H[f(x-α, y-β)]=g(x-α, y-β)
Properties of the degradation function H
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Estimating the degradation function
1. Estimation by Image observation2. Estimation by experimentation3. Estimation by modeling
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Estimation by image observation
Take a window in the image Simple structure Strong signal content
Estimate the original image in the window
),(ˆ),(),(
vuFvuGvuH
s
ss =
known
estimate
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Estimation by experimentation If the image acquisition system is ready Obtain the impulse response
impulse Impulse response
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Estimation by modeling (1) Ex. Atmospheric model
6/522 )(),( vukevuH +−=
original k=0.0025
k=0.001 k=0.00025
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Estimation by modeling (2) Derive a mathematical model Ex. Motion of image
∫ −−=T
dttyytxxfyxg0 00 ))(),((),(
Planar motionFouriertransform
∫ +−=T tvytuxj dtevuFvuG0
)]()([2 00),(),( π
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Estimation by modeling: example
original Apply motion model
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Inverse filtering With the estimated degradation function
H(u,v)G(u,v)=F(u,v)H(u,v)+N(u,v)
=>),(),(),(
),(),(),(ˆ
vuHvuNvuF
vuHvuGvuF +==
Estimate oforiginal image
Problem: 0 or small values
Unknownnoise
Sol: limit the frequency around the origin
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Fullinversefilterfor
CutOutside40%
CutOutside70%
CutOutside85%
k=0.0025