antal nagy department of image processing and computer graphics university of szeged 17th ssip 2009,...
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Antal NagyDepartment of Image Processing
and Computer GraphicsUniversity of Szeged
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Human perception Image degradation Convolution, Furier Transform Noise Image operations
◦ Frequency filters◦ Spatial filtering
Inverse filtering Wiener filtering
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Aim◦ to improve the perception
of information images for human viewers
◦ to provide ‘better’ input for other automated image
processing techniques
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No general theory for determining what is good image enhancement◦ If it looks good, it is good!?
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http://www.youtube.com/watch?v=_d_l5nsnIvM
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Focus◦ Noise reduction techniques
Quantitative measures can determine which techniques are most appropriate◦ How does it improve e.g.
the result of the next automated image processing step? E.g. image segmentation
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Non-linear mapping◦ E.g., non-linear sensitivity, image of the straight
line is not straight e.t.c. Blurring
◦ Image of a point is blob Moving during the image acquisition Probabilistic noise
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),(),(),(),(
),(),(),(),(
vuNvuFvuHvuG
yxyxfyxhyxg
Frequency domain
Spatial domain
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◦ Multiplication point by point
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)()()*( hFfFhfF
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* =
=·
Multiplication
Convolution
Fourier transf.Inverse
Fourier transf.
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dvduvyuxgvufyxgf
duuxgufxgf
,),(),)((
)( ))((
Definition
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ationdifferenti '*'*)'*(
vitydistributi )*()*()(*
ityassociativ *)*()*(*
multipl.scalar ity withassociativ )(**)()*(
itycommutativ **
gfgfgf
hfgfhgf
hgfhgf
gafgfagfa
fggf
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otherwise,0
,2/1if,1)(
xxg
2/1
2/1
)()(x
x
dfxgf
x
x
x
)(xf
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Even functions that are not periodic can be expressed as the integrals of sines and/or cosines multiplied by a weighting function.
The formulation in this case is the Fourier transform.
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∑∑ ==
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Taught mathematics in Paris
Eventually traveled to Egypt with Napoleon to become the secretary of the Institute of Egypt
After fall of Napoleon worked at Bureau of Statistics
Elected to National Academy of Sciences in 1817
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dueuFxf
dxexfuF
ixu
ixu
2
2
)()(
)()(
),(1 Lf
(invers transform)(invers transform)
(continous)(continous)
base-functionsbase-functions
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dudvevuFyxf
dydxeyxfvuF
yvxui
yvxui
)(2
)(2
),(),(
),(),(
base-functionsbase-functions
(invers transform)(invers transform)
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))(2cos(
Re )(2
vyuxi
e vyuxi
u=0, v=0 u=1, v=0 u=2, v=0u=-2, v=0 u=-1, v=0
u=0, v=1 u=1, v=1 u=2, v=1u=-2, v=1 u=-1, v=1
u=0, v=2 u=1, v=2 u=2, v=2u=-2, v=2 u=-1, v=2
u=0, v=-1 u=1, v=-1 u=2, v=-1u=-2, v=-1 u=-1, v=-1
u=0, v=-2 u=1, v=-2 u=2, v=-2u=-2, v=-2 u=-1, v=-2
u
v
wavelength:
22/1 vu
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F(0,0) - value is by far the largest component of the image,
Other frequency components are usually much smaller,
The magnitude of F(X,Y) decreases quickly◦ Instead of displaying the |F(u,v)| we display
log( 1 + |F(u,v)| ) real function usually
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x
y
v
u
),(1log vuF),( yxf
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The 2D Fourier transform can be separated
The edges on the image appears as point series in perpendicular direction in Fourier transform of the image and vice versa.
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Image-spaceImage-space
Frequency spaceFrequency spaceooriginal rotation linearity riginal rotation linearity shiftshift scale scale
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Noise unknown subtraction not possible Periodic noise
◦ N(u,v) can be estimated from G(u,v)
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),(),(),(
and
),(),(),(
vuNvuFvuG
yxyxfyxg
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Gaussian◦ In an image due to
factors Electronic circuit
noise Sensor noise due to
poor illumination High temperature
Rayleigh◦ Range imaging
Exponential and gamma◦ Laser imaging
Impulse◦ Faulty switching
Uniform density◦ Practical situations
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22 2/)(
2
1)(
noiseGaussian
zzezp
azfor 0
for )(2
)(
noiseRayleigh
/)( 2
azeazbzp
baz
4
)4(
and
4/
2
b
baz
0zfor 0
0for )!1(
ap(z)
noise (gamma) Erlang1b
zeb
z azb
22
and
a
b
a
bz
0a where
0zfor 0
0for )(
noise lExponentia
zae
zpaz
22 1
and
1
a
az
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otherwise 0
if 1
)(
Noise Uniform
bzaabzp
12
and2
22 ab
baz
otherwise 0
for
for
)(
noise peper)-and-(salt Impulse
bzP
azP
zp b
a
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Electrical and electromechanical interference
Spatial dependent noise Can be reduced via
frequency domain filtering ◦ Pair of impulses
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T: A=[a(i,j)] → B=[b(i,j)]
b(i,j)=T{a(i,j), S(i,j), i, j}
intensity enviroment position
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Global: b(i,j)=T{A} (S(i,j)=A) (e.g. Fourier-transformation)
Local: T{a(i,j), S} given size of S and independent from the position (e.g. convolution with a mask)
Local, adaptive: T{a(i,j), S(i,j), i, j} the size of S(i,j) is independent from the size of image (e.g. adaptive thresholding)
Point operation: T{a(i,j)} (e.g. gamma-correction, histogram-equalization)
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D0: cutoff frequency
All frequencies less than D0 will be passed,
Other frequencies will be filtered out.Bluring and ringing propertiesScope: noise filtering
otherwise,0
if,1),(
20
22 DYXYXH ILPF
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F
F-1.
Input image
Frequency-mask
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Original 5
15 30
80 230
Cutoff frequencies
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n: order of the filter
Properties: Smooth transition in blurring No ring effect (continouos filter) Smoothed edges
nBLPF
DYX
YXH 2
0
22
1
1),(
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Original 5
15 30
80 230
Cutoff frequenciesCutoff frequencies
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022 2/)(),( Dvu
GLPF evuH
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Original 5
15 30
80 230
Cutoff frequenciesCutoff frequencies
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),(1),( vuLPFvuHPF
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Bandreject and Bandpass Filters Notch Filters
◦ Rejects or passes frequencies in a predefined neighborhood about the frequency rectangle
◦ Zero-phase-shift filters Symmetric about the origin
(u0,v0) (-u0,-v0)
◦ Product of highpass filters whose centers have been translated to the centers of the notches.
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Q
kkkNR vuHvuHvuH
1
),(),(),(
),(1),( vuHvuH NRNP
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noisyimage
frequencymask
frequencyimage
filteredimage
11
00
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Mean Filter
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where g input image, S(x,y) neighborhood of (s,t) point,
mn number of pixels in neighborhood.
3x3 neighborhood
1
1
1
1
),(9
1),(ˆ
u v
vyuxgyxf
xySts
tsgmn
yxf),(
),(1
),(ˆ
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111
111
111
9
1
:where
),,)((
),(),(
),(9
1),(ˆ
1
1
1
1
1
1
1
1
h
jihf
vuhvyuxg
vyuxgyxf
u v
u v
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AvAveerraagingging ◦ same weight for every pixels in neighborhood,same weight for every pixels in neighborhood,
Weighted averageWeighted average ◦ weights for pixels in the neighborhood (generally decreasing with weights for pixels in the neighborhood (generally decreasing with
the distance).the distance).
The sum of the Noise Filtering/smoothing The sum of the Noise Filtering/smoothing masks elements is 1! masks elements is 1!
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121
242
121
16
1
111
111
111
9
1
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Smooth when the difference between the intensity value of the given pixel and the mean of the neighborhood is larger than threshold value defined in advance.
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otherwise),,(
,),(),( if),,(),('
jif
Tjifjigjigjig
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Arithmetic mean filter
Geometric mean filter◦ Lose less image details
Harmonic mean filter◦ Works well for
salt noise, Gaussian◦ Fails for pepper noise
Contraharmonic mean filter◦ Q>0 pepper, Q<0 salt
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xySts
tsgmn
yxf),(
),(1
),(ˆ
mn
Sts xy
tsgyxf
1
),(
),(),(ˆ
xySts tsg
mnyxf
),( ),(1
),(ˆ
xy
xy
Sts
Q
Sts
Q
tsg
tsg
yxf
),(
),(
1
),(
),(
),(ˆ
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Median filter
Max and min filters
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Behavior changes based on statistical characteristic in the filter region◦ Improved filtering power◦ Increase in filter complexity◦ Noise only!
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Adaptive, Local noise reduction filter◦ Mean◦ Variance
Local region Sxy
◦ g(x,y) intensity value◦ the variance of the noise corrupting f(x,y) to
form g(x,y) ◦ mL local mean◦ local variance
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2
2L
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Adaptive, Local noise reduction filter1. If is zero, the filter should return the value of
g(x,y) Zero noise case
2. If the local variance is high relative to the filter should return a value close to g(x,y)
Edges should be preserved3. If the variances are equal the filter should
return the arithmetic mean value Local noise averaging
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2
2
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Simplest approach to restoration is direct inverse filtering
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experience:H: quickly decreasing function,N: not (so quickly) decreasinglet us cut the high frequencies
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No additive noise
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If we only know the degradation function◦ Can not recover the undegraded image
H(u,v ) is not known Get around the zero or small value problem
◦ To limit the filter frequencies to values near the origin H(0,0) is usually the highest value of H(u,v) in the
frequency domain
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Inverse filtering makes no explicit provision for handling noise
Approach incorporates◦ Degradation function◦ Statistical characteristics of noise
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Method◦ Considering images and noise as random variable◦ Objective is to find an estimate of the
uncorrupted image f such that the mean square error between them is minimized.
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f̂
argument theof value
expected theis where
ˆ 22
E
ffEe
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spectrum
Wiener filter
input
result
original
spectrum of the result
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Edge preserving filters◦ E.g. anisotropic diffusion
Wavelet denoising Point operations
◦ E.g. gamma correction, histogram operations Iterative filtering E.t.c.
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We always should consider some kind of noise model◦ Even when working on phantom data
Should do in automatic way◦ Have to chose carefully the method
Depends on the given task◦ Determining the parameters
What we gain?◦ Less problem afterwards◦ Better final result
Even no other technique will be applied
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Digital Image Processing◦ Gonzalez and Woods◦ www.ImageProcessingPlace.com
Course on Image Processing at University of Szeged◦ Attila Kuba, Kálmán Palágyi
Image Restoration presentation◦ Attila Kuba◦ SSIP 2006
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