2/4/2011

1

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

2/4/2011

2

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;

...

2/4/2011

3

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...

2/4/2011

4

... 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

2/4/2011

5

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

2/4/2011

6

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

2/4/2011

7

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.

2/4/2011

8

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).

⎩

2/4/2011

9

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:

2/4/2011

10

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 =

2/4/2011

11

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.

2/4/2011

12

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)