mixture based denoising and contrast enhancement in ... · i. frosio, n. a. borghese ais labflb.,...
TRANSCRIPT
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Mixture based denoising and Mixture based denoising and contrast enhancement in digital contrast enhancement in digital contrast enhancement in digital contrast enhancement in digital radiography radiography
I. Frosio, N. A. Borgheseb f lAIS Lab., University of Milan
OverviewOverviewStatistical models and digital radiographyImpulsive noise removal filterImpulsive noise removal filterSoft tissue filterConclusion
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Statistical models...Statistical models...
Principled statistical models as an effective lt ti t li d li filt i alternative to linear and non-linear filtering
(Lucy, 1974; Richardson 1974; Shepp & Vardi, 1982);Maximum likelihood / a posteriori criterions lead to non linear cost functions;Filtering as a computationally intensive iterative procedure:Expectation Maximization (EM) procedure:Expectation Maximization (EM) (Shepp & Vardi, 1982; Geman & Geman, 1984);Since 90s, the necessary computational power is finally available on standard PCs.
Statistical models...Statistical models...A proper statistical model for...
... Image characteristics:◦ Typical distribution of the norm of gradient:
Gaussian – Tikonov regularization;Gibbs – TV regularization;
◦ Typical-a priori grey level distribution:image histogram;
...
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Statistical models...Statistical models...A proper statistical model for...
... Image noise characteristics:◦ Distribution
Gaussian, Poisson, Impulsive, SpeckleMixture...
◦ CorrelationhWhite
PSFSpatially variant...
... And digital radiography... And digital radiographyCephalometric
Intra-oral
Chest
Panoramic
And so on...
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... And digital radiography... And digital radiographyIssues for the radiologist... ???
Low contrastLow visibility of small anatomical details
... And their (partial) solutions from the researchers:
Contrast enhancement algorithms (e.g. γcorrection)Feature enhancement algorithms (e.g. Unsharp Masking, UM)
OverviewOverviewStatistical models and digital radiographyImpulsive noise removal filterImpulsive noise removal filterSoft tissue filterConclusion
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Impulsive Impulsive noise: generationnoise: generation
Sensor
X-ray tube
Patient Cpu & monitor
Doctor
Pixel failures,
A/D converter errors
Impulsive noise: effectImpulsive noise: effectNo filter UM & γ
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Impulsive noise: effectImpulsive noise: effectLow contrast, poor visibilityLow contrast, poor visibility
Raw ImageRaw Image
γ correction + UM
γ correction + UM High contrast,
visibility?High noise
High contrast, visibility?
High noise
DenoiseDenoise
γ correction + UM
γ correction + UM High contrast,
High visibilityLow noise
Impulsive noise generates spikes
Impulsive noise: switching filterImpulsive noise: switching filter
PULSE DETECTION PULSE CORRECTION Two stages filtering
Input pixel
Identity
Median
Output pixel
SWITCH
PULSE DETECTION STAGE
PULSE CORRECTION STAGE
Pulse detector A pulse detectorbased on statistics.
Neighbour pixels
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Impulsive noise: the mixtureImpulsive noise: the mixture
X-ray photons visible photons (scintillator)
X-ray tubep a
electrons (CCD sensor) (ADC converter);Linear sensor:gn,i = G · pn,i
pn,i: noisy number of photons for the ith pixel ( )
Patient
Scintillator
pn,i
e-(Poisson statistics);gn,i: noisy grey level for the ithpixel (??? statistics);G: sensor gain (from photons to grey level - unknown).
e
ADC converter
gn,i
Impulsive noise: the mixtureImpulsive noise: the mixtureChanging the variable…
Mean (unnoisy) number of
( ) [ ]
g
pGg
Poissonp
epppp
gG
g
inin
in
ppi
iin
iin
iin
!|
,,
,,
,
,
⋅=
⋅=
−
Mean (unnoisy) number of photons for the ith pixel.
( ) ( )G
Gg
eGg
GGg
Gg
pdpdp
pppggpin
GGi
iin
in
iniiniin
1
!
1|||,
,
,
,,, ⋅
⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅=⋅⎟⎟
⎠
⎞⎜⎜⎝
⎛=⋅=
−
Mean (unnoisy) number of photons for the ith pixel.
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Impulsive noise: the mixtureImpulsive noise: the mixtureA mixture of photon counting and impulsive noise:
( ) ( ) ( )⎪⎨⎧ ⋅+⋅= ||| iinImpImpiinPCPCiin ggpPggpPggp
pImp(gn,i|gi)=1/Ng (uniform distribution)Ng, number of grey levelsPPC and Pimp, probabilities that a pixel is corrupted by photon counting or impulsive noise.
Unknowns
( ) ( ) ( )⎪⎩
⎪⎨⎧
=+≤≤≤≤ 1,10,10
||| ,,,
ImpPCImpPC
iinImpImpiinPCPCiin
PPPP
ggpggpggp
UnknownsPPC and GAlso gi is unknown!
Supposing that the true grey level gi is given for i=1..N, PPCand G can be computed maximizing the likelihood of the data.
Impulsive noise: the mixtureImpulsive noise: the mixture… A constrained optimization (0<PPC<1, 0<PImp<1, PPC+PImp=1) should be performed.A i l th d t t i th l tiA simple method to constrain the solution:
( ) ( ) [ ] ( )
⎪⎪⎩
⎪⎪⎨
⎧
−=−=
=
⋅−+⋅=
−
−
−−
2
2
22
11
|1|| ,,,
PC
PC
PCPC
ePP
eP
ggpeggpeggp
PCImp
PC
iinImpiinPCiin
γ
γ
γγ
γPC and G can be computed maximizing the likelihood of the data (PPC, PImp are then derived).
⎩
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Impulsive noise: the likelihoodImpulsive noise: the likelihoodLet us write the neg log likelihood of the measured data (grey levels of the pixels):measured data (grey levels of the pixels):
( ) ( )[ ] ( ) ( )[ ]
( ) [ ] ( ){ }∑
∑∏
=
−−
==
⎫⎧⎤⎡ ⎞⎛
⎤⎡
=⋅−+⋅−=
=−=⎥⎦
⎤⎢⎣
⎡−=−=
g
n
iiinImpiinPC
N
iiin
N
iiinPCPC
ggpeggpe
ggpggpGLGf
in
PCPC
1,,
1,
1,
|1|ln
|ln|ln,ln,
,
22 γγ
γγ
[ ] ( )∑=
−−−
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧⋅−+⎥
⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⎟⎠⎞
⎜⎝⎛⋅⋅−=
n
iiinImp
inGg
Gi ggpeGg
eGg
Ge PC
i
PC
1,
, |1!1ln2
,
2 γγ
Impulsive noise: the likelihoodImpulsive noise: the likelihoodWhat about the factorial term?It can be approximated using the Stirling’s
( ) ( ) ( ) ( )π
π
2ln21ln
21ln!ln
2!
++−≈
⋅⋅≈ −
nnnnn
ennn nn
an app ma d ng S ngapproximation:
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Impulsive noise: the likelihoodImpulsive noise: the likelihoodWith the Stirling’s approximation:
[ ]+
⎪⎫
⎪⎧
⎥⎤
⎢⎡
⎞⎛⎤⎡
⎞⎛g
gg
gi
inin 1,,
For simplicity, let us define:
( ) [ ] ( )
( ) [ ] ( )∑
∑
=
−−−−−
=
−−−−
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⋅−+⋅⋅⋅⋅⋅
⋅−
=⎪⎭
⎪⎬
⎪⎩
⎪⎨ ⋅−+
⎥⎥⎥
⎦⎢⎢⎢
⎣⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⎟⎠⎞
⎜⎝⎛⋅⋅−≈
n
iiinImp
Ggggin
gi
in
n
iiinImp
GgG
inGgG
iPC
ggpeeggGg
e
ggpeeGg
eGg
GeGf
PCiinininPC
PC
ini
PC
1,
1
,21
,
1,
2,
|12
1ln
|121ln,
2,,,
2
2,
2
γγ
γγ
π
πγ
⎪⎧H 21
⎪⎩
⎪⎨⎧
⋅⋅=
⋅=−− iininin ggg
ingii
ini
eggQ
gH,,,
,
,21 π
Impulsive noise: the likelihoodImpulsive noise: the likelihoodSome numerical problem for Qi:
iininin gggin
gii eggQ −− ⋅⋅= ,,,
,
Xx for x<127 overflow!!!Better computing Qi as follows:
( ) ( )[ ]{ }iininiinQ gggggeQi i −+−⋅== ,,,
)ln( lnlnexp
( ) ( )[ ] iiiiii gggggK −+−⋅= lnln( ) ( )[ ] iininiini gggggK + ,,, lnln
iKi eQ =
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Impulsive noise: the likelihoodImpulsive noise: the likelihoodWe finally have the negative log likelihood:likelihood:
A non linear function of G and γPC
It can be efficiently minimized through
( ) [ ] ( )∑=
−−−
⎭⎬⎫
⎩⎨⎧
⋅−+⋅⋅⋅−≈n
iiinImp
GK
iPC ggpeeGHeGf PCi
PC
1,
21
|1ln,22 γγγ
It can be efficiently minimized through EM (a few seconds required for a 4Mpixels image @ 12bpp)
Impulsive noise: what Impulsive noise: what about gi?about gi?The unnoisy image gi i=1..N is unknownunknown…By application of a 3x3 median filter, we obtain an image which is free from impulsive noise;Experimental results demonstrate that,
fusing this image for gi, a reliable and efficient pulse detector can be built.
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Impulsive noise: pulse Impulsive noise: pulse detectordetectorFrom γPC, PPC and PImpcan be computed; PPCpPC(gn,i|gi)pEach pixel which satisfies:
is recognized as a pulse and corrected by the
it hi filt
( )[ ] ( )[ ]iinImpImpiinPCPC ggpPggpP || ,, ⋅<⋅
PImppImp(gn,i|gi)
switching filter.The classification rule is chosen to minimize the classification error.
gign,ign,i(pulse)