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
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
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)
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)
Noise probability density functions
Noises are taken as random variables Random variables
Probability density function (PDF)
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:
Gaussian noise (PDF)
70% in [(µ−σ), (µ+σ)]95% in [(µ−2σ), (µ+2σ)]
Uniform noise
≤≤
−=otherwise 0
if 1)( bza
abzp
12)(
22
2 ab
ba
−=
+=
σ
µMean:
Variance:
Less practical, used for random number generator
Uniform PDF
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
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)
Periodic noise Arise from electrical or electromechanical
interference during image acquisition Spatial dependence Observed in the frequency domain
Sinusoidal noise:Complex conjugatepair in frequencydomain
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
Observe the histogram
Gaussian uniform
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
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
Additive noise only
g(x,y)=f(x,y)+η(x,y)
G(u,v)=F(u,v)+N(u,v)
Spatial filters for de-noising additive noise
Skills similar to image enhancement Mean filters Order-statistics filters Adaptive filters
Mean filters Arithmetic mean
Geometric mean
∑∈
=xySts
tsgmn
yxf),(
),(1),(ˆ
Window centered at (x,y)
mn
Ststsgyxf
xy
/1
),(),(),(ˆ
∏=
∈
original NoisyGaussian
µ=0σ=20
Arith.mean
Geometricmean
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
PepperNoise黑點
SaltNoise白點
Contra-harmonicQ=1.5
Contra-harmonic
Q=-1.5
Wrong sign in contra-harmonic filtering
Q=-1.5 Q=1.5
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
Order-statistics filters Median filter
Max/min filters
)},({),(ˆ),(
tsgmedianyxfxySts ∈
=
)},({max),(ˆ),(
tsgyxfxySts ∈
=
)},({min),(ˆ),(
tsgyxfxySts ∈
=
bipolarNoisePa = 0.1Pb = 0.1
3x3Median
FilterPass 1
3x3MedianFilterPass 2
3x3Median
FilterPass 3
Peppernoise
Saltnoise
Maxfilter
Minfilter
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
Uniform noiseµ=0σ2=800
Left +Bipolar NoisePa = 0.1Pb = 0.1
5x5Arith. Mean
filter
5x5Geometric
mean
5x5Medianfilter
5x5Alpha-trim.
Filterd=5
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
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
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
µ=0σ2=1000
Gaussiannoise Arith.
mean7x7
Geometricmean7x7
adaptive
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
Periodic noise reduction (cont.) Bandreject filters Bandpass filters Notch filters Optimum notch filtering
Bandreject filters
* Reject an isotropic frequency
ideal Butterworth Gaussian
noisy spectrum
bandreject
filtered
Bandpass filters Hbp(u,v)=1- Hbr(u,v)
{ }),(),(1 vuHvuG bp−ℑ
Notch filters Reject(or pass) frequencies in predefined
neighborhoods about a center frequency
ideal
Butterworth Gaussian
HorizontalScan lines
NotchpassDFT
Notchpass
Notchreject
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
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
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
Estimating the degradation function
1. Estimation by Image observation2. Estimation by experimentation3. Estimation by modeling
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
Estimation by experimentation If the image acquisition system is ready Obtain the impulse response
impulse Impulse response
Estimation by modeling (1) Ex. Atmospheric model
6/522 )(),( vukevuH +−=
original k=0.0025
k=0.001 k=0.00025
Estimation by modeling (2) Derive a mathematical model Ex. Motion of image
∫ −−=T
dttyytxxfyxg0 00 ))(),((),(
Planar motionFouriertransform
∫ +−=T tvytuxj dtevuFvuG0
)]()([2 00),(),( π
Estimation by modeling: example
original Apply motion model
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
Fullinversefilterfor
CutOutside40%
CutOutside70%
CutOutside85%
k=0.0025