BioE153:Imaging As An Inverse Problem

Grant T. Gullberggtgullberg@lbl.gov

http://muti.lbl.gov/jonathan/courses/bioe153-2002

510 486-7483

1

Introduction

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Mathematics and Physics of Emerging Biomedical Imaging, Mathematics and Physics of Emerging Biomedical Imaging, National Academy Press, Washington, D.C., 1996National Academy Press, Washington, D.C., 1996

Examples• X-ray Computed Tomography• MRI• PET• SPECT• Ultrasonic Tomography• Electrical Source Imaging• Electrical Impedance Tomography• Magnetic Source Imaging• Optical Tomography• Photo-Acoustic Imaging

3

X-ray CT Inverse Problem

)sin,(cos

),( spx

ysx ,

s

2

)()(),( xdsxxsp

source

detector

)(xattenuation distribution

4

projection

MRI Inverse Problem

x

y

2

)()( xdexts tGxi

)(xproton spin density

5

gradient

signalzz

along the bore of the magnet

PET Inverse Problem

)sin,(cos

),( spx

y

sx ,

s

22

)()()()(exp),( xdsxxcxdsxxsp

)(xc isotope concentration

)(xattenuation distribution

6

projectiondetector2

detector1

SPECT Inverse Problem

)sin,(cos

),( spx

y

sx ,

s

2

)(')'()'(exp)(),( xdsxxdsxxxcspx

)(xc isotope concentration

)(xattenuation distribution

projection

7

detector

Ultrasound Inverse Problem

)(xvvelocity

traducer/receiver

')|'()'()'|()|( 02

0 rrrrrrrr dPGkPPDbb

kb – reference wavenumberG – reference Green’s function – index of refractionPb – background pressure

Pressure

traducer receiver

Fredholm integral equation( Lipmann-Schwinger )

8

2

2

1bv

v

dppr

prpnpvrvrv

m

iS iiji

13

)()(4

1)()(

3)(

4

1)(

qr

qrqrv

Electrical Source Inverse Problem

potential measurement

9

rr

v – potentialv – potentialn – surface normaln – surface normal - dipole- dipole - conductivity terms- conductivity terms

,

)(q

I

g

current

voltage

Electrical Impedance Inverse Problem

Scg voltage conductivity

sensitivity matrix

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dppr

prpnpvrbrb

m

iS iii

13

0 )()(4

)()(

dppr

prpnpvrvrv

m

iS iiji

13

)()(4

1)()(

30 )(

4)(

qr

qrqrb

3)(

4

1)(

qr

qrqrv

Magnetic Source Inverse Problem

potential measurement

magnetic field measurement

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v – potentialv – potentialn – surface normaln – surface normal - dipole- dipole - conductivity terms- conductivity termsb – magnetic vectorb – magnetic vector - free space permeability - free space permeability

,

0

)(q

rr

A Simple Example of An Imaging Inverse Problem

• X-ray CT Projections • Reconstruction Problem as a Solution

to a System of Linear Equations • Reconstruction is an Inverse Solution

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•X-ray CT Projections

13

xeII 0

oI I

x

source

Beer’s Law

xII 0ln

detector

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units of length-1

flux of photons

i

iiII x0ln

oI I

12

3

1x 2x 3x

15

different attenuation coefficients

i

iix

eII

0

Image Matrix

1 2 3

4 5 6

7 8 9

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pixelized array of attenuation coefficients

Projections

.01 .03 .05

.15

.15.150

0 .15

.09

.30

.30

.35.33.01

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example of projections for a particular pixelized array of attenuation coefficients

•Reconstruction Problem as a Solution to a System of Linear Equations

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Projections

1 2 3

4 5 6

7 8 9

.09

.30

.30

.35.33.01

19

solve for the unknown attenuation coefficients from a set of two projections

35.

33.

01.

30.

30.

09.

963

852

741

987

654

321

20

the system of linear equations

6 equations in 9 unknowns

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the inclusion of a third projection

.09

.30

.30

.35.33.01

1 2 3

4 5 6

7 8 9 .0345

.2230

.3465

.0860

0

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solve for the unknown attenuation coefficients from a set of three projections

03.

0860.55.7.02.6.05.

3465.75.45.23.78.4.25.75.

2230.25.75.2.4.7.2.

0345.02.6.05.

3500.

3300.

0100.

3000.

3000.

0900.

7

87541

9865421

965321

632

963

852

741

987

654

321

23

the system of linear equations

11 equations in 9 unknowns

PF

:= A

1 1 1 0 0 0 0 0 0

0 0 0 1 1 1 0 0 0

0 0 0 0 0 0 1 1 1

1 0 0 1 0 0 1 0 0

0 1 0 0 1 0 0 1 0

0 0 1 0 0 1 0 0 1

0 .05 .6 0 0 .02 0 0 0

.2 .7 .4 0 .2 .75 0 0 .25

.75 .25 0 .4 .78 .23 0 .45 .75

.05 0 0 .6 .02 0 .7 .55 0

0 0 0 0 0 0 .3 0 0

0

086.

3465.

2230.

0345.

35.

33.

01.

3.

3.

09.

PF

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Matrix Equation

•Reconstruction is an Inverse Solution

1 2 3

4 5 6

7 8 9

.09

.30

.30

.35.33.01

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PF PF

min

PFF T1

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Least Squares Solution to a System of Linear Equations

PF G

generalized inverse

Reconstruction

.09

.30

.30

.35.33.01

-.0433 .0633

.0266

.0700

.1400

.1400

.1333

.1333.0266 .01 .03 .05

.15

.15.150

0 .15

OriginalOriginal

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solution from two projection measurements

with(linalg):with(linalg):A:=array([[1,1,1,0,0,0,0,0,0],[0,0,0,1,1,1,0,0,0],[0,0,0,0,0,0,1,1,1],A:=array([[1,1,1,0,0,0,0,0,0],[0,0,0,1,1,1,0,0,0],[0,0,0,0,0,0,1,1,1],[1,0,0,1,0,0,1,0,0],[0,1,0,0,1,0,0,1,],[0,0,1,0,0,1,0,0,1]]);[1,0,0,1,0,0,1,0,0],[0,1,0,0,1,0,0,1,],[0,0,1,0,0,1,0,0,1]]);B:=array([.09,.30,.30,.01,.33,.35]);B:=array([.09,.30,.30,.01,.33,.35]);leastsqrs(A,b,’optimize’);leastsqrs(A,b,’optimize’);

Maple RoutineMaple Routine

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35.

33.

01.

30.

30.

09.

963

852

741

987

654

321

29

6 equations in 9 unknowns

the system of linear equations

.01 .03 .05

.15

.15.150

0 .15

.09

.30

.30

.35.33.01

.0345

.2230

.3465

.0860

0 Reconstruction30

solution from three projection measurements

03.

0860.55.7.02.6.05.

3465.75.45.23.78.4.25.75.

2230.25.75.2.4.7.2.

0345.02.6.05.

3500.

3300.

0100.

3000.

3000.

0900.

7

87541

9865421

965321

632

963

852

741

987

654

321

31

the system of linear equations

11 equations in 9 unknowns

Our examples have been two-dimensional. However, X-ray CT imaging is a three-dimensional inverse problem.

Comment:

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