image restoration 影像復原 - diunitofarid/teaching/dip/liu - ch 5 - image restoration and... ·...

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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14-May-18 21

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