bioe153:imaging as an inverse problem grant t. gullberg [email protected] 510 486-7483 1
TRANSCRIPT
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BioE153:Imaging As An Inverse Problem
Grant T. [email protected]
http://muti.lbl.gov/jonathan/courses/bioe153-2002
510 486-7483
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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
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Examples• X-ray Computed Tomography• MRI• PET• SPECT• Ultrasonic Tomography• Electrical Source Imaging• Electrical Impedance Tomography• Magnetic Source Imaging• Optical Tomography• Photo-Acoustic Imaging
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X-ray CT Inverse Problem
)sin,(cos
),( spx
ysx ,
s
2
)()(),( xdsxxsp
source
detector
)(xattenuation distribution
4
projection
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MRI Inverse Problem
x
y
2
)()( xdexts tGxi
)(xproton spin density
5
gradient
signalzz
along the bore of the magnet
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PET Inverse Problem
)sin,(cos
),( spx
y
sx ,
s
22
)()()()(exp),( xdsxxcxdsxxsp
)(xc isotope concentration
)(xattenuation distribution
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projectiondetector2
detector1
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SPECT Inverse Problem
)sin,(cos
),( spx
y
sx ,
s
2
)(')'()'(exp)(),( xdsxxdsxxxcspx
)(xc isotope concentration
)(xattenuation distribution
projection
7
detector
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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
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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
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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
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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
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xeII 0
oI I
x
source
Beer’s Law
xII 0ln
detector
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units of length-1
flux of photons
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i
iiII x0ln
oI I
12
3
1x 2x 3x
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different attenuation coefficients
i
iix
eII
0
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Image Matrix
1 2 3
4 5 6
7 8 9
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pixelized array of attenuation coefficients
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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
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•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
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solve for the unknown attenuation coefficients from a set of two projections
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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
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.09
.30
.30
.35.33.01
1 2 3
4 5 6
7 8 9 .0345
.2230
.3465
.0860
0
22
solve for the unknown attenuation coefficients from a set of three projections
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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
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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
24
Matrix Equation
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•Reconstruction is an Inverse Solution
1 2 3
4 5 6
7 8 9
.09
.30
.30
.35.33.01
25
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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
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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
27
solution from two projection measurements
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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
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.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
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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
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Our examples have been two-dimensional. However, X-ray CT imaging is a three-dimensional inverse problem.
Comment:
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