醫(yī)學(xué)成像系統(tǒng)-x-平行線重建

TerminologyTomography

–

from

the

Greek

tomos,

meaning

“section”CT：Computerized

transverse

axial

tomographyComputer-assisted

tomographyComputerized

axial

tomographyComputerized

transaxial

transmission

reconstructivetomographyComputerized

tomographyReconstructive

tomographyComputed

tomographyEstablished

by

the

Radiological

Society

of

NorthAmerica

in

their

major

journal

RadiologySuperimposition

of

all

structures

on

the

film,whi akes

it

difficult

and

sometimesimpossible

to

distinguish

a

particular

detail.Radiography

is

a

qualitative

rather

thantative

procedureLow

image

contrastLimitation

of

Radiography4TOMOSYNTHESIS

Imaging最小混疊提高圖像對比度(3)量化The

Goal

of

CTG

N

Hounsfield&1979年生理醫(yī)學(xué)獎X-CTA

M

CormackX-CT工作原理示意圖檢測器陣列Image

reconstructionfrom

projections

is

theprocess

of

producing

animage

of

atwo-dimensional

distribution(usually

somephysicalproperty)

from

estimatesof

its

line

integrals

along

afinite

number

of

lines

ofknown

location.DefinitionProjectionLine

integrals:

(

x,

y)dlI0I00II0I1I23I

I41234Image

reconstruction

from

projections

is

the

process

ofproducing

an

image

of

a

two-dimensional

distribution

(usuallysome

physical

property)

from

estimates

of

its

line

integrals

along

a

finite

number

of

lines

of

known

location.DefinitionddI

I

e1

2

d

1

01

3

d

I3

I0

e2

0I

I

e

d

3

4

…0I0I56I

I60

I

e

d

2

3

IX線源檢測器GenerationCTPencil

beamSingle

detectorX線源檢測器Second

GenerationCTMultiplePencil

beamMultipledetectorX線源檢測器Third

Generation

CTFan

beamMultipledetectorarrayX線源Fourth

Generation

CTFan

beam檢測器Stationaryring

ofdetectorsHelical

/

Spiral

CTVolume

CT

scannerThe

x-ray

tube

anddetectors

rotatecontinuously

as

thepatient

movescontinuously

throughthe

gantry.從投影重建圖像xyR0Integral

Linexcos

ysin

Rf

(x,

y)2D

function從投影函數(shù)

g

(R)

重建密度函數(shù)

f

(x,

y)R(x,

y)xyyxcosysinR

f

(x,

y)dlProjection

g

(R)l中心切片定理g

(R)f

(

x,

y)

f

(r,

)xyR0uvG

(

)1D

FTF

(u,

v)

F

(

,

)2D

FTx

cos

y

sin

RThe

1D

F.T.

of

a

projection

at

angle

θ

forms

a

linein

the

2D

F.T.

plane

at

this

same

angle.f

(

x,

y)

f

(r,

)F

(u,

v)

F

(

,

)g

(

R

)x2D

FTyR0uvG

(

)1D

FTx

cos

y

sin

RThe

1D

F.T.

of

a

projection

at

angle

θ

forms

a

linein

the

2D

F.T.

plane

at

this

same

angle.G

(

)F

(,

)中心切片定理2

f

(r

,

)

[r

cos(

)]

rdrd0

0中心切片定理投影函數(shù)：

g

(R)

f

(

x,

y)

(

x

cos

y

sin

R)dxdy

變換：F

(u,

v)

f

(

x,

y)

exp[i2

(ux

vy)]

dxdyF

(

,

)

f

(

x,

y

)

exp[i2

(

x

cos

y

sin

)]

dxdyδ函數(shù)篩選性質(zhì)：e

i

2

(

x

cos

y

sin

)

e

i

2

R

(

x

cos

y

sin

R

)

dR投影線函數(shù)f

(

x,

y)

f

(r,

)F

(u,

v)

F

(

,

)2D

FTg

(R)xyR0uvG

(

)1D

FTx

cos

y

sin

RCentral

section

theoremF

(,

)

f

(x,

y)

ei

2

(

x

cos

y

sin

)

dxdy

f

(x,

y)

(x

cos

y

sin

R)

ei

2R

dxdydR

ei

2R

dR

f

(x,

y)

(x

cos

y

sin

R)

dxdy

g

(R)

ei2R

dR

F1g

(R)δ函數(shù)篩選性質(zhì)投影函數(shù)計(jì)算1D

變換2D變換f

(x,y)

f

(r,

)F

(u,

v)

F

(,

)2D

FTg

(R)xyR0uvG

(

)1D

FTx

cos

y

sin

Rg

(R)

f

(

x,

y)

(

x

cos

y

sin

R)dxdy

2

f

(r

,

)

[r

cos(

)]

rdrd0

0空間域頻域插值G

()g

(R)f

(

x,y)F

(

,

)F

(u,

v)

:(o

~)1D

FT2D

IFTImage

reconstruction

based

on

F.T.yCentral

section

theoremf

(x,

y)F

(u,

v)2D

FTuvg

(R)

0xG

(

)1D

FTG

(

)1D

FTf

(x,

y)yuF

(u,

v)2D

FTvg

(R)x空間域頻域g

(R)

2

sin

c(2R)F

(u,

v)1D

FT2D

IFT

:(o

~)G

(

)

F1

g

R例題

rect

2

插值F

(,

)

circ

f

(x,y)

J1

2

r

/

r空間域G

()g

(R)f

(

x,y)F

(

,

)F

(u,

v)頻域插值

:(o

~)1D

FT2D

IFT費(fèi)時(shí)的二維運(yùn)算插值造成的高頻失真變換方法重建圖像中的問題Back

Projection直接反投影法造成的偽像000010000010000111000001010010+=111141111+010010001+001010100直接反投影重建方法2D圖像校正！重建圖像原始圖像直接反投影法成像系統(tǒng)H圖像恢復(fù)系統(tǒng)1/

Hfb

(x,

y)f

(x,

y)δ函數(shù)hb(x,y)或hb(r)系統(tǒng)沖激響應(yīng)輸入δ函數(shù)直接反投影法成像系統(tǒng)Hhb(x,y)或hb

(r)1、輸入δ函數(shù)

(r)f

(x,

y)

(x,

y)

rg

R2、投影函數(shù)3、反投影重建圖像函數(shù)

hb(x,

y)

或

hb

(r)直接反投影法成像系統(tǒng)H

2

0

0

(r)

r

cos

R

rdrd

rg

(R)

(r)已知輸入δ函數(shù)

(x,

y)

r，求投影函數(shù)

g

Rf

(x,

y)xyR0R

0R

0r

cos

Rxcos

ysin

R換積分限（為計(jì)算δ函數(shù)）δ函數(shù)篩選性質(zhì)(令r

=0)(R)

(r)

r

cos

R

r

drd

0

r

(r)d

r

cos

R

drd

0

0

(R)

(R)

投影線已知投影函數(shù)g

R

(R)，求直接反投影函數(shù)(x,

y)xyyxcosysing

(R)xyx

cos

y

sin

R00R(x,

y)一般情況下，一個(gè)特定角度θ下的反投影對重建的密度函數(shù)的貢獻(xiàn)：b

(x,

y)

g

(x

cos

ysin)說明：處在投影角為θ

時(shí)，二維平面中坐標(biāo)為

(x,y)處被反投影到的值是等于投影函數(shù)

g

R

在坐標(biāo)為

(x

cos

y

sin

)

處的值。g

(x

cos

y

sin

)fb

(x,

y)一般情況下，一個(gè)特定角度θ下的直接反投影對重建的密度函數(shù)的貢獻(xiàn)：b

(x,

y)

g

(x

cos

y

sin)180度直接反投影重建圖像：

b

(x,

y)dgbf

(x,

y)

00dR

x

cos

y

sin

R

dR

R

r

cos

R

dRgbf

(r,

)

0d已知投影函數(shù)g

R

(R)，求直接反投影函數(shù)fb

(x,y)δ函數(shù)篩選性質(zhì)

g

R

x

cos

y

sin

RdRg

(R)

與

b

(x,

y)

的關(guān)系式！g

(R)

與

fb

(x,

y)的關(guān)系式！已知投影函數(shù)，求直接反投影函數(shù)

gR

(R)g

(R)f

(x,

y)xyR0R

0R

0r

cos

Rx

cos

ysin

Rbf

(x,

y)

002

r

cos

0

R

r

cos

R

dR

r

cos

d

bf

(r,

)

0d

2

d

1

dr

21rδ函數(shù)R

=0δ函數(shù)性質(zhì)1r根

gR

r

cos

R

dR

bf

(r,

)

0df

'xi

i

x

x

nf

x

i1重建圖像原始圖像直接反投影法成像系統(tǒng)

Hb

()

1/圖像恢復(fù)系統(tǒng)hb

(r)

1/

r1Hb

()

直接反投影法的圖像校正bF(,

)

F

(,

)

2f

(x,

y)F

,

1

F

函數(shù)f

(x,

y)bFb

(,

)直接反投影空間域頻域f

(x,

y)F

(,

)Fb

(,

)2D

FT2D

IFTg

(R)直接反投影法的圖像校正濾波反投影取投影直接反投影的偽像校正f

(

x,

y

)g

(

R

)直接反投影fb

(

x,

y

)2D濾波f

(

x,

y

)取投影1D濾波直接反投影f

(

x,

y

)g

(

R

)f

(

x,

y

)濾波反投影重建圖像的思路g’

(

R

)000010000010000111000001010010+=111141111+010010001+001010100直接反投影重建方法2D圖像校正！=000040000-1/3-1/3-1/3111-1/3-1/3-1/3-1/31-1/3-1/31-1/3-1/31-1/3++1-1/3-1/3-1/31-1/3-1/3-1/31+-1/3-1/31-1/31-1/31-1/3-1/3濾波反投影重建方法000010000010-1/31-1/3濾波運(yùn)算濾波函數(shù)無需圖像校正！？已知投影函數(shù)g

R，求直接反投影函數(shù)fb

(x,y)g

(R)xy0(x,

y)(x,

y)xyyxcosysing

(x

cos

y

sin

)Rx

cos

y

sin

RdR

g

x

cos

y

sin

d00d

g

R

bf

x,

y

180度直接反投影重建圖像：0

bd

g

R

x

cos

y

sin

R

dR

f

x,

y

直接反投影：

g'

(R)

xcos

+ysin

-R

dRd

02D

FT換積分限δ函數(shù)性質(zhì)

j

2

x

cos

y

sin

002

f x,

y

F

,

e

d

d

F

e

j

2

R

d

xcos

+ysin

-R

dRd

0

F

,

e

j

2

x

cos

y

sin

d

d0

e

j

2

x

cos

y

sin

d

d=F

0F(,

)

F(,

)e

i

2

(

x

cos

y

sin

)

e

i

2

R

(

x

cos

y

sin

R)

dR0

bd

g

R

x

cos

y

sin

R

dR

f

x,

y

直接反投影：

g'

(R)

xcos

+ysin

-R

dRd

02D

FT換積分限δ函數(shù)性質(zhì)

j

2

x

cos

y

sin

002

f x,

y

F

,

e

d

d

F

e

j

2

R

d

xcos

+ysin

-R

dRd

0

F

,

e

j

2

x

cos

y

sin

d

d0

e

j

2

x

cos

y

sin

d

d=F

0F(,

)

F(,

)dj

2RF

e

g

'

0

bd

gR

x

cos

y

sin

R

dRf

x

,

y

直接反投影：f

x

,

y

R

dR

y

sin0x

cosg’

R

d

濾波反投影：g

'

R？原始圖像投影函數(shù)直接反投影g

(R)重建圖像原始圖像g

(R)直接反投影修正后投影函數(shù)重建圖像投影函數(shù)

′?

?

?dj

2RF

eg

'

g'

(R)dj

2RF

e（投影）g

(R)1D

FTF

()（頻率域函數(shù)）F

()

×1D

IFT空間域頻域f

(

x,

y

)1D

FTg

(

R

)g

'

(

R

)1D

IFT

F1{g

(

R

)}

F1{g

(

R

)}濾波反投影法?

???

′?

??

?

??

?

|?|

??????

d?

??

???

??

??

?|?|空間域頻域f

(

x

,

y

)1D

FTg

(

R

)g

'

(

R

)1D

IFTF1{

g

(

R

)}

F1{

g

(

R

)}濾波反投影法f

(x,

y)

C(R)g

(R)g'

(R)卷積反投影法

11C(R)

Ff

(x,

y)C(R)g

(R)g'

(R)卷積反投影法

11C(R)

Fg

'

(R)

g

RC

R???1

/(

e

)1

/

濾波函數(shù)

11C(R)

F

0

lim

e其中

e

H

e

H

e

式中是單位階躍函數(shù)。H

濾波函數(shù)

11C(R)

F

e

H

e

H

e11

011

0

lim

FR

lim

FC

根據(jù)變換的性質(zhì)，假設(shè)

A

為一個(gè)aR

是它的逆變換，則有2F

1

A

i

aR的函數(shù)，A

e

H

e

H

濾波函數(shù)

11C(R)

F

e

H

e

H

11

011

0

lim

F

e

lim

FA

e

H

e

H

2R2e

ei

2R

dd

e

ei

2R1F

A

2

i4R

aR00

11濾波函數(shù)C(R)

F

e

H

e

H

e

2

i

i4

R

2

2

R

a

'

R22F

1

A

i

aR

4

2

R2

222

4

2

R

2

CR

lim

0

2222

2

4

2

R2

4

R2

2=1

/(

e

)1

/

濾波函數(shù)

11C(R)

F

11

1

0

1

0lim

eC(R)

F

lim

F

e22

4

2

R2

lim

0

2

4

2

R22

0

lim

e22

4

2

R2

2

4

2

R2

2C(R)

2

2

4

2

R2

C(R)

lim

0

2

4

2

R2

2R2

/

2

/

23

/

21/

8

2H

R

L

(

)

0

0頻帶有限的濾波函數(shù)：R-L濾波函數(shù)2

0H

(

)

rect

(

)H

R

L

(

)

0

0頻帶有限的濾波函數(shù)02

0

2

2

0

0

rect

H

(

)

rect

(

)

0

1

20

其他當(dāng)0

1

0CR

F020202sinc

2

R

sinc

Rrect

0

20

0

1頻帶有限的濾波函數(shù)2

0H

(

)

rect

(

)

22000

2

sin

c

2

RC

(

R

)

R

sin

cRhR

L

(

R

)H

R

L

(

)

0

0nh

(n

)R

LR-L卷積函數(shù)14T

21n

T2

2C

(

R

)

C

(nT

)

0

2當(dāng)n為偶數(shù)，但

0當(dāng)n為奇數(shù)當(dāng)n

0012

T

Rh

R

L

(

R

)

22000

2

sin

c

2

RC

(

R

)

R

sin

c離散化的卷積函數(shù)S-L濾波函數(shù)HS

L

(

)

0

02

2

S

LH

(

)

rect

(

)

sin

c(

)

20

0

0

20S-L濾波函數(shù)S

LH

(

)

rect

(

)

sin

c(

)2

02

0CR

F201

4

R0

002

21

4

1

4

R

sin2

R

0

0

rect

2

sinc

2

1

HS

L

()

0

20

20

0Rh

S

L

(

R

)(

b

)nh

S

L

(

n

)S-L卷積函數(shù)C

nT

2

2T

2

4n

2

1n

0,

1,

2...012

T

CR201

4

R2

(

a

)1

40

1

40

R

sin2

R2

0

f

(x,

y)g

(R)g'

(R)卷積反投影法

11C(R)

FC(R)g

'

(R)

g

RC

RnnR

-

LS

-

LRadon空間與正弦圖f

(x,y)或f

(r,

)物體空間g

(R)或g

(R,

)Radon

空間F

(u,v)或F

(

,

)空間2R

12R1F

12F

1F2F1中心切片定理物體空間、空間與Radon

空間R02(R,)xyR0g

(R)0探測器通道投影角度平行線投影(R,)特定投影角下取得的投影函數(shù)。Radon

projection

or

sinogramxy0Radon

projection

or

sinogram(r,)f

(x,

y)f

(r,)R

r

cos(

)R2(R,)0探測器通道投影角度r(R',

')

R(R,)(R',

')單點(diǎn)在不同投影角下取得的投影值落在正弦線上。SinogramObject正弦圖空間中的采樣模式（平行線投影）R探測器通道投影角度一次平行線投影g

(R)xyR002g