Profs. Brooks and DiMarzio Northeastern University Spring 2004
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Transcript of Profs. Brooks and DiMarzio Northeastern University Spring 2004
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February 2004 Chuck DiMarzio, Northeastern University 10471-12-1
ECEU692Subsurface Imaging
Course NotesPart 12: Imaging with Light (4):Diffusive Optical Tomography
Profs. Brooks and DiMarzio
Northeastern University
Spring 2004
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February 2004 Chuck DiMarzio, Northeastern University 10471-12-2
Topic Outline
• Goal: “Find the Matrix Elements”• A Bit of Radiometry
– Terminology and Units– Radiative Transport
• Approximation to Radiative Transport Equation– Diffusion Approximation– Wave Solution– Generating the Diffusive Waves
• Examples• Adding Ultrasound• Solving for the Matrix Elements
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February 2004 Chuck DiMarzio, Northeastern University 10471-12-3
The Matrix Elements
t
P
t
P
DCAC AmplitudeAC Phase
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February 2004 Chuck DiMarzio, Northeastern University 10471-12-4
Radiometric Quantities
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February 2004 Chuck DiMarzio, Northeastern University 10471-12-5
Radiometry and Photometry, Flux M, Flux/Proj. Area
I, Flux/ L,Flux/AE, Flux/Area Rcd.
Radiant FluxWattsLuminous FluxLumens
Radiant ExitanceWatts/m2
Luminous ExitanceLumens/m2=Lux
A /
RadianceWatts/m2/srLuminanceLumens/m2/sr1 Lambert=(1L/cm2/sr)/
1 ftLambert= (1L/ft2/sr)/1mLambert= (1L/m2/sr)/
Radiant IntensityWatts/srLuminous IntensityLumens/sr
1 Candela=1cd=1L/sr
/
IrradianceWatts/m2
IlluminanceLumens/m2=Lux
1 Ft Candle=1L/ft2
Notes: Spectral x=dx/d or dx/d: Add subscript or , divide units by Hz or m.
1 W is 683 L at 555 nm.
2R
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February 2004 Chuck DiMarzio, Northeastern University 10471-12-6
t
nL
c
ndnnpnLnL
s
nL ˆ''ˆ,ˆ'ˆ
4ˆ
ˆ
What Is Radiative Transport?
• The Radiative Transport Equation
dsd dL L+dL
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February 2004 Chuck DiMarzio, Northeastern University 10471-12-7
Solutions to RTE
• Monte-Carlo
• Low Scattering
• High Scattering – Diffusion Approximation– Usually Valid in Tissue, Except...
• Certain Tissue Types
• Certain Imaging Modalities (eg. Confocal, OCT)
• Close to Source or to Rapid Changes in Parameters
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February 2004 Chuck DiMarzio, Northeastern University 10471-12-8
• Approach– Monte-Carlo
– Reciprocity
– Fourier Transform
• Parameters– Depth 1 cm.
– Thickness 2 cm.
• TransilluminationDunn, Andrew, and Charles A. DiMarzio, “Efficient Computation of Time--Resolved Transfer Functions for Imaging in Turbid Media,” Journal of the Optical Society of America A 13, No. 1, January 1996. Pp. 65--70.
Tissue Parametersa = 0.03 /cms = 200 /cmg = 0.95d = 1 cm
125 150
200-ps Gate
Spatial Frequency, /cm
MT
F
Resolution Limits (M-C)
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February 2004 Chuck DiMarzio, Northeastern University 10471-12-9
t
nL
c
ndnnpnLnL
s
nL ˆ''ˆ,ˆ'ˆ
4ˆ
ˆ
n )/ ( / ) nL( ˆˆ J
qn
c
t/c)(· a
1J
013
gn
c
as
J
Photon Diffusion Approximation• The Radiative Transport Equation
• Taylor Series: is Fluence Rate, J is Flux
• Result
n̂
J
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February 2004 Chuck DiMarzio, Northeastern University 10471-12-10
Fluence Rate?
• Another Radiometric Quantity– Fluence is Energy/Area– Fluence Rate is Energy/Area/Time
• =Power/Area
• Units Like E or M, but Different Meaning
• Relation to Absorbed Power/Volume– A=a
– Used to Determine in Monte-Carlo
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February 2004 Chuck DiMarzio, Northeastern University 10471-12-11
t 0
D as gn
cD
13 n
ca
)(0
tie rk
D
i
nD
ck a
2 Re
Imk2 k
Dispersion Equation• The Diffusion Equation
• Wave Solution
=0
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February 2004 Chuck DiMarzio, Northeastern University 10471-12-12
Dispersion Results
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February 2004 Chuck DiMarzio, Northeastern University 10471-12-13
Spherical Waves
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February 2004 Chuck DiMarzio, Northeastern University 10471-12-14
Different Types of Waves
100
105
1010
1015
102010
-4
10-2
100
102
104
106
108
f, Frequency, Hz.
k/(2
), W
aven
umbe
r, m
-1
Sound
(Real)
DPDW
Light(Real)
(Imag)(Imag)
10059_1
1m
1mm
1km
1m
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February 2004 Chuck DiMarzio, Northeastern University 10471-12-15
Physical Reason for Dispersion
-0.5
0
0.5
Sam
ple
200 MHz.
0 5 10
10
20
30
40
50
-0.5
0
0.5
Sam
ple
500 MHz.
0 5 10
10
20
30
40
50
0 5 10-50
0
50
Sig
nal
Time, ns0 5 10
-50
0
50
Sig
nal
Time, ns
Imaginary partof k increaseswith frequency
Easy to understand in terms of multiple paths.
m100574a.m
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February 2004 Chuck DiMarzio, Northeastern University 10471-12-16
Watch the Photons Migrate!• 20 Photon Tracks • 20,000 Photon Tracks
– Pabs=0.1– Pext=0.3
• Received Photons
0 20 40 60 80 1000
10
20
30
40
50
60
70
80
90
Time Step
Pho
ton
s in
Bo
x
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February 2004 Chuck DiMarzio, Northeastern University 10471-12-17
ExtrapolatedBoundary
Tissue
ImageSource
EffectiveSource
Input
Detector
ImageSource
How Diffuisve Waves Begin?• Generation
– From Light Wave
• Wave Behavior– Absorption
– Reflection
– Refraction
– Diffraction
– Interference
– Scattering
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February 2004 Chuck DiMarzio, Northeastern University 10471-12-18
Noise Issues
0 1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
0.8
1
1.2
Sig
nal
Time, ns
m100574a.m
Noise proportionalto square root ofDC signal.
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February 2004 Chuck DiMarzio, Northeastern University 10471-12-19
TECHNOLOGY•Near-infrared light•Fiber optics•Computed Tomography
ADVANTAGES•Optical contrast•Portable - bedside, ambulance•Continuous•Inexpensive
•DISADVANTAGES•Resolution•Depth penetration
From David A. Boas - MGH NMR Center
DOT Instrumentation at MGH Imaging Center
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February 2004 Chuck DiMarzio, Northeastern University 10471-12-20
DetectorsSources
6 cm
4 cm
Mid-line
Data Set I - 98-05-14
At RestPassive movement of
right armPassive movement of
right arm
From David A. Boas - MGH NMR Center
Functional Imaging of a Neonate
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February 2004 Chuck DiMarzio, Northeastern University 10471-12-21
0 1 2 3 4 5 6
0123456-5-4-3-2-10
X axisY axis
Z a
xis
0 1 2 3 4 5 6-5
-4
-3
-2
-1
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 1 2 3 4 5 6-5
-4
-3
-2
-1
0
0
0.01
0.02
0.03
0.04
0.05
0 2 4 6-5
-4
-3
-2
-1
0
0
0.05
0.1
0.15
Keeping the Matrix Rank UpSource
Detector
Object
y=4z
x
Reconstruction with Reflection only(Top Sources)
Reflection and Transmission(All Sources)
DiMarzio, et. al., Presented at Photonics West, Jan 1999
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February 2004 Chuck DiMarzio, Northeastern University 10471-12-22
UltrasoundUltrasoundFocal PointFocal Point
UltrasoundUltrasoundBeamBeamOpticalOptical
SourceSource
OpticalOpticalReceiverReceiver
OpticalOpticalSourceSource
OpticalOpticalReceiverReceiver
OpticalOpticalSourceSource
OpticalOpticalReceiverReceiver
All Light fromSource Fiber
Light from Source to Receiver
Light from Source to Receiver through Ultrasound Focus
API Virtual Source
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February 2004 Chuck DiMarzio, Northeastern University 10471-12-23
Solving the Wave Equation (1)
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February 2004 Chuck DiMarzio, Northeastern University 10471-12-24
Solving the Wave Equation (2)
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February 2004 Chuck DiMarzio, Northeastern University 10471-12-25
The First Born Approximation
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February 2004 Chuck DiMarzio, Northeastern University 10471-12-26
Why Do We Want a Model?
• Applications– Forward Model
• Will it work?
– Inverse Algorithms• How Much Does k
Change?– ie. Is there a Tumor?
• And Where?
• Understanding– What is k?– See Panel to Right.