Dual Energy CT Image Reconstruction
Algorithms and Performance
Joseph A. O’Sullivan
Samuel C. Sachs Professor
Dean, UMSL/WU
/
Joint Undergraduate
d
d
Engineering Program
Professor of Electrical and Systems Engineering,
Biomedical Engineering
Engineering, and Radiology
jao@wustl.edu
Supported by Washington University and NIH-NCI grant
R01CA75371 (J. F. Williamson, VCU, PI).
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
Collaborators
Faculty
Students
Bruce R. Whiting
David G. Politte
Jeffrey F. Williamson, VCU
Donald L. Snyder
Chemical Engineers
M. Dudukovic
M. Al-Dahhan
Al Dahhan
R. Varma
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
Norbert Agbeko
Liangjun
gj Xie
Daniel Keesing
Josh Evans, VCU
Debashish Pal
Jasenka Benac
Giovanni Lasio
Lasio, VCU
Special thanks to Bruce R
R. Whiting
and N. Agbeko
Dual Energy CT Image Reconstruction
• Data Models Î Reconstruction
Algorithms
• Image Reconstruction Approaches
SOMATOM Definition CT Scanner
ccir.wustl.edu
– “Linear” Approaches
– Statistical Iterative Reconstruction
• Simulation Study off the Dual Energy
Alternating Minimization Algorithm
• Performance
P f
Q
Quantification:
tifi ti
the
th Cramer-Rao
C
R
Lower Bound
• Conclusions
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
Motivation
• Dual Source, Fast kVp Switching, Phton
C
Counting,
ti
and
d Energy
E
Selective
S l ti Detectors
D t t
A
Are
Available
– Dual energy image reconstruction algorithms are
needed now
– Imaging III session yesterday; this session
• System Selection and Algorithm Design
– Basis for comparing systems based on
performance
– Fundamental approach that extends to new
systems (multiple energies,
energies photon counting,
counting etc.)
etc )
– Quantifying the impact of modeling errors
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
Dual Energy Image Reconstruction
d1(y )
1.4
1.2
1
Preprocess
N
Normalize
li
d2 (y )
Post-process/
Display
Joint
Processing
4
x 10
10
c1(x )
0.8
06
0.6
0.4
0.2
0
0.2
9
0.18
8
0.16
7
6
5
4
Preprocess
Normalize
3
Post-process/
Post
process/
Display
0 14
0.14
0.12
c2 (x )
0.1
0.08
0.06
0.04
0.02
2
0
1
• Multiple inputs with different spectral sensitivities
• Some standard normalizations (e.g., relative to air scans)
• Joint processing combines the data sets
• Post
Post-processing
processing can extract the desired image(s)
– Components
– Estimated images
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
X-Ray Transmission Tomography—
S (E)
Back to Basics
• Source Spectrum, Energy E
• Beer
Beer’ss Law and Attenuation
S ( E )e ∫l
− μ ( x , E ) dx
• Mean
ea Photons/Detector
oto s/ etecto I0
I 0 S ( E )e ∫l
− μ ( x , E ) dx
• Beam-hardening
• Detector Sensitivity Spectrum
• Mean (Photon) Flux
J. L. Prince and J. Links, Medical Imaging Signals and Systems.
D( E )), Φ ( E ) = S ( E ) D( E )
I 0 Φ ( E )e ∫l
− μ ( x , E ) dx
kVp
I 0 Φ ( E )e ∫l
− μ ( x , E ) dx
dE
∫
P-20 Seminar, 3/12/05 R. M. Arthur
0
J. A. O’Sullivan, AAPM 07/28/09
Pearson Education: Upper Saddle River
River, NJ
NJ, 2006
2006.
“Linear” Image Reconstruction
•
•
•
•
Normalize relative to an air scan
Beam hardening correction to target energy
Beam-hardening
Negative log to estimate line integrals at target energy
Linear inversion, normalize (e.g. water
water-equivalent)
equivalent)
− ∫ μ ( x , E ) dx
⎡
⎛ kVp
⎞⎤
l( y)
I 0 Φ ( E )e
dE ⎟ ⎥
⎢
−1 ⎜ ∫0
kVp
⎟⎥
∫l ( y ) μ ( x, E )dx ≈ − log ⎢ BH ⎜
I 0 Φ ( E )dE
⎜
⎟⎥
⎢
∫
0
⎝
⎠⎦
⎣
μˆ ( x, E )
μˆ ( x, E ) = FBP ∫ μ ( x, E )dx , cˆ( x) =
l( y)
μ water ( E )
)
(
d (y )
1.4
1.2
1
Normalize
& correct
Negative
logarithm
FBP
normalize
0.8
0.6
0.4
0.2
0
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
c(x )
X-Ray Transmission Tomography—
Back to Basics
• Detector Sensitivity D(E)
– Probability that a photon of energy E is detected
– Mean response to a photon of energy E:
• p
photon counting
g Î response
p
=1
• energy integrating Î response = E
• other
• Detector
D t t St
Statistics
ti ti
– Photon counting, Beer’s Law as survival probability
Î Poisson distribution
– Energy integrating or other Î compound
Poisson
k
λ −λ
Poisson mean λ:
P( N = k ) =
e , k ≥0
k!
Log-likelihood function: l (λ ) = k ln λ − λ
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
X-Ray Transmission Tomography—
Reconstruction Principles
• Deterministic Model
– Data equal a function of the desired image
– Approximately invert that function to reconstruct
image (minimize a measure of error between the
data and the model)
2
min d − g (⋅ : μ )
μ
• Random Model
– Find the log-likelihood
g
function for the data
– Maximize the (possibly penalized) log-likelihood
function over possible images
max l (d | g (⋅ : μ ))
μ
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
Statistical Image
g Reconstruction
• Source-detector pairs indexed by y; voxels indexed by x
• Data dj(y) Poisson,
Poisson means gj(y:μ),
(y:μ) log
log-likelihood
likelihood function
l (d j | g j (⋅ : μ )) = ∑ d j ( y ) ln g j ( y : μ ) − g j ( y : μ )
y∈Y
⎛
⎞
g j ( y : μ ) = ∑ I j Φ j ( y, E ) exp ⎜ − ∑ h( y, x) μ ( x, E ) ⎟ + β j ( y )
E
⎝ x∈X
⎠
• Mean unattenuated counts Ij, mean background βj
• Attenuation function μ(x,E
, ), E energies
g
I
μ ( x, E ) = ∑ ci ( x) μi ( E )
i =1
• Maximize over μ or ci
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
Attenuation Function Approximation
• Voxels as function approximation
– Co
Constant
sta t atte
attenuation
uat o o
over
e a ssmall
a volume,
o u e, o
or
– Linear combination of basis functions
I
• Energy dependence
– Water equivalent
l
– Linear combination of basis functions
μ ( x, E ) = ∑ ci ( x) μi ( E )
i =1
• Basis functions
–
–
–
–
Physics (photoelectric and Compton scatter)
Physiological (e.g., fat and bone)
Signal processing (e
(e.g.,
g SVD)
Hand selected (e.g., CaCl and styrene)
• Constrained system
y
– Dual Energy, 3 images
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
ci ( x) ≥ 0, c1 ( x) + c2 ( x) + c3 ( x) = 1
3
μ ( x, E ) = ∑ ci ( x) μi ( E )
i =1
Statistical Image Reconstruction—
AM Algorithm
• Alternating Minimization Algorithm
– Several papers from our group (O’S
(O S & Benac, TMI 2007)
– Monotonically increasing log-likelihood
I
⎛
⎞
g j ( y : μ ) = ∑ I j Φ j ( y, E ) exp ⎜ − ∑ h( y, x)∑ μi ( E )ci ( x) ⎟ + β j ( y ),
E
i =1
⎝ x∈X
⎠
2
2 ⎡
⎤
max ∑ l (d j | g j (⋅ : μ )) = ∑ ⎢ ∑ d j ( y ) ln g j ( y : μ ) − g j ( y : μ ) ⎥
{ci ( x )} j =1
j =1 ⎣ y∈Y
⎦
g j ( y ) = ∑ E q j ( y, E )
• Compare using backprojection of data, estimated means
⎛ ∑ b% ( x) ⎞
⎜
⎟
1
d
(k )
i, j
cˆi( k +1) = cˆi( k +1) ( x) −
ln ⎜ j ( k )
⎟
Z i ( x) ⎜ ∑ bˆi , j ( x) ⎟
⎝ j
⎠
c
Compare
U
d t
Update
bˆi(,kj ) ( x) = ∑ h( y, x) μi ( E )q (jk ) ( y, E )
y ,E
b%i(,kj ) ( x) = ∑ h( y, x) μi ( E )
(k )
j
(k )
j
d j ( y )q ( y, E )
q ( y, E ')
∑
P-20 Seminar, 3/12/05 R. M. Arthur
y ,E
J. A. O’Sullivan, AAPM 07/28/09
E'
g
Forward
Model
Dual Energy CT Image Reconstruction
• Data Models Î Reconstruction
Algorithms
• Image Reconstruction Approaches
SOMATOM Definition CT Scanner
ccir.wustl.edu
– “Linear” Approaches
– Statistical Iterative Reconstruction
• Simulation Study off the Dual Energy
Alternating Minimization Algorithm
• Performance
P f
Q
Quantification:
tifi ti
the
th Cramer-Rao
C
R
Lower Bound
• Conclusions
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
Dual Energy Image Reconstruction
d1(y )
1.4
1.2
1
Preprocess
N
Normalize
li
d2 (y )
Post-process/
Display
Joint
Processing
4
x 10
10
c1(x )
0.8
06
0.6
0.4
0.2
0
0.2
9
0.18
8
0.16
7
6
5
4
Preprocess
Normalize
3
Post-process/
Post
process/
Display
0 14
0.14
0.12
c2 (x )
0.1
0.08
0.06
0.04
0.02
2
0
1
• Multiple inputs with different spectral sensitivities
• Some standard normalizations (e.g., relative to air scans)
• Joint processing combines the data sets
• Post
Post-processing
processing can extract the desired image(s)
– Components
– Estimated images
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
Dual Energy Image Reconstruction
d1(y )
1.4
1.2
1
Preprocess
N
Normalize
li
d2 (y )
Post-process/
Display
Joint
Processing
4
x 10
10
c1(x )
0.8
06
0.6
0.4
0.2
0
0.2
9
0.18
8
0.16
Post-process/
Post
process/
Display
7
6
5
4
Preprocess
Normalize
0 14
0.14
0.12
c2 (x )
0.1
0.08
0.06
3
0.04
0.02
2
0
1
• Multiple inputs with different spectral sensitivities
Selected Technologies
4
x 10
10
9
8
7
6
5
SOMATOM Definition CT Scanner
Dual Source
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
4
3
2
1
• Multiple sources
• Fast kVpp switchingg
• Multiple detectors
• Energy selective photon counting
Selected Issues
• Data quality versus dose
• Clinical issues, including motion
• Image use (application)
Dual Energy Image Reconstruction
d1(y )
1.4
1.2
1
Preprocess
N
Normalize
li
- Logarithm
d2 (y )
4
x 10
10
9
8
7
6
5
4
3
2
Preprocess
Normalize
- Logarithm
éa11 a12 ù
êa 21 a 22 ú
ë
û
Reconstruct
Image
0.8
06
0.6
0.4
0.2
0
0.2
0.18
0.16
Reconstruct
Image
0 14
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
1
• Joint processing combines the data sets
– Linear combination of attenuations, then
reconstruct individual images independently
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
c1(x )
c2 (x )
Dual Energy Image Reconstruction
L1(y )
d1(y )
1.4
1.2
1
Preprocess
N
Normalize
li
Reconstruct
Image
Solve
c1(x )
0.8
06
0.6
0.4
0.2
0
d2 (y )
4
x 10
10
9
8
7
6
5
4
Preprocess
Normalize
d1 = g1(L1, L2 ) L 2 (y )
d2 = g2 (L1, L2 )
Reconstruct
Image
0.2
0.18
0.16
0 14
0.14
0.12
c2 (x )
0.1
0.08
0.06
3
0.04
0.02
2
0
1
• Joint processing combines the data sets
– Nonlinear inversion of raw data, then
reconstruct individually
g1(L1, L2 ) =
å
I 1F 1(E ) exp (- L1m1(E ) - L2 m2 (E ))
E
g2 (L1, L2 ) =
å
I 2F 2 (E ) exp (- L1m1(E ) - L2 m2 (E ))
E
Kalendar, Klotz, Kostaridou, TMI 1988.
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
Dual Energy Image Reconstruction
d1(y )
1.4
1.2
Preprocess
Reconstruct
Image
d2 (y )
4
x 10
10
9
8
7
6
5
4
3
Preprocess
Reconstruct
Image
1
0.8
éa11 a12 ù
êa 21 a 22 ú
ë
û
06
0.6
0.4
0.2
0
0.2
0.18
0.16
0 14
0.14
0.12
0.1
0.08
0.06
0.04
0.02
2
0
1
• Joint processing combines the data sets
– Reconstruct individual images from each data
set, then (linearly) combine the resulting
attenuation images
g to estimate desired
outputs
Williamson Li
Williamson,
Li, Whiting
Whiting, Lerma
Lerma, Med
Med. Phys
Phys. 2006
2006.
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
c1(x )
c2 (x )
Dual Energy Image Reconstruction
d1(y )
1.4
1.2
1
Preprocess
N
li
Normalize
d
c
Compare
Post-process/
Display
c1(x )
0.8
06
0.6
0.4
0.2
Update
0
d2 (y )
4
x 10
10
g
9
8
7
6
5
4
Preprocess
Normalize
0.2
Forward
Model
0.18
0.16
Post-process/
Post process/
Display
3
0 14
0.14
0.12
0.1
0.08
0.06
0.04
0.02
2
0
1
• Joint processing combines the data sets
– Use knowledge of source spectrum and detector
sensitivity spectrum
– Use a model of jjoint data dependence
p
on unknown
underlying attenuation maps
– Jointly estimate component images based on that
model (e.g., statistical iterative reconstruction)
See also: Sukovic and Clinthorne, TMI 2000.
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
c2 (x )
Dual Energy CT Image Reconstruction
• Data Models Î Reconstruction
Algorithms
• Image Reconstruction Approaches
SOMATOM Definition CT Scanner
ccir.wustl.edu
– “Linear” Approaches
– Statistical Iterative Reconstruction
• Simulation Study off the Dual Energy
Alternating Minimization Algorithm
• Performance
P f
Q
Quantification:
tifi ti
the
th Cramer-Rao
C
R
Lower Bound
• Conclusions
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
Simulations: Post-reconstruction vs.
Joint Statistical Image Reconstruction
• Large Phantom: 20cm in diameter; thin outer lucite shell;
water; four rods in inner 60mm lucite cylinder.
– C
Calibration
lib ti rods:
d calcium
l i
chloride,
hl id ethanol,
th
l teflon
t fl and
d polystyrene
l t
in
i
the 12, 3, 6, and 10 o’clock positions, resp.
– Test phantom rods: muscle, ethanol, teflon and substance X (a
p
, resp.
p
bonelike material)) in the 12,, 3,, 6,, and 10 o’clock positions,
• Small Phantom: 60mm diameter lucite cylinder with rods
• FBP-BVM (Basis Vector Method; JF Williamson, et al. 2006)
– Water-equivalent
Water equivalent beam hardening correction
– Requires calibration data to estimate linear transformation that
generates the component images
– FBP uses a ramp filter
• AM-DE
AM DE (AM Dual
D l Energy
E
Algorithm)
Al ith )
– NO pre-correction of data
– NO calibration data
– NO regularization
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
Materials Used - Fractions
Substance
Styrene
Ca. Chloride
Ethanol
Lucite
Teflon
Water
Muscle
Substance X
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
Styrene
F
Fraction
ti
1
0
0.79904
1.14
1.4194
0.90357
0.93995
0.03
Ca. Chloride
F
Fraction
ti
0
1
0.03369
0.05834
0.48799
0.1357
0.13904
2.8613
Large Phantom (I0=105) – Component Images
Styrene
Calcium
Chloride
Truth
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
FBP-BVM
AM-DE
Large Phantom – Synthesized Images
20 keV
65 keV
Truth
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
FBP-BVM
AM-DE
Large Phantom Profiles
Profiles through column 256 of 20keV image(left) and 65keV image(right)
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
Mini CT – Small Phantom
• 360 Source positions
• 92 detectors
• Based on Siemens Somatom Plus 4
scanner
• 64 by 64 image
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
Small Phantom – Component Images
Styrene
Calcium
Chloride
Truth
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
FBP-BVM
AM-DE
Small Phantom – Synthesized Images
20 keV
65 keV
Truth
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
FBP-BVM
AM-DE
Small Phantom – 20keV profiles
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
Small Phantom – 20keV ratio
images and profiles
FBP-BVM
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
AM-DE
Small Phantom: Relative Errors
I0 = 10,000
10 000
FBP-BVM
AM-DE
• Lucite – Center: A region of interest in the center of the phantom.
• Lucite – Edge: A region of interest near the edge of the phantom,
between Substance X and Teflon
Teflon.
• Relative error equals absolute difference divided by truth
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
Small Phantom: Relative Errors
I0 = 100,000
100 000
FBP-BVM
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
AM-DE
Small Phantom: Relative Errors
I0 = 1,000,000
FBP-BVM
AM-DE
• AM-DE relative errors decrease consistently as the
number of counts increases
• AM-DE performs at least an order of magnitude better
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
Dual Energy CT Image Reconstruction
• Data Models Î Reconstruction
Algorithms
• Image Reconstruction Approaches
SOMATOM Definition CT Scanner
ccir.wustl.edu
– “Linear” Approaches
– Statistical Iterative Reconstruction
• Simulation Study off the Dual Energy
Alternating Minimization Algorithm
• Performance
P f
Q
Quantification:
tifi ti
the
th Cramer-Rao
C
R
Lower Bound
• Conclusions
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
Small Phantom – Ensemble mean and variance (from 15
samples)
p ) I0 = 100,000 Styrene
y
Component
p
AM-DE
FBP-BVM
Truth
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
Mean
Variance
Small Phantom – Ensemble mean and variance (from 15
samples)
p ) I0 = 100,000 Ca. Chloride Component
p
Variance is inversely proportional to I0
AM-DE
FBP-BVM
Truth
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
Mean
Variance
Predicting Performance: Fisher
Information and the Cramer
Cramer-Rao
Rao Bound
• The variance of an unbiased estimate
is g
greater than or equal
q
to the
Cramer-Rao lower bound (CRLB)
• CRLB is conditioned on a model
• CRLB is independent of the algorithm
• AM-DE is biased (in part due to
nonnegativity constraint)
• CRLB is derived from the inverse of
Fisher information
• Fisher information measures the joint
dependence of values within a
component image and across images
•P-20Fisher
information is proportional to I0
Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
Fisher Information: Within a
Component Image
Calcium chloride
Maximum 1.2E5
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
Styrene
Maximum 1.6E4
Fisher Information:
Across Images
Maximum 4.2E4
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
CRLB Images: Diagonals of Inverse
of Fisher Information
Ratio of variance images to CRLB. Different display windows.
Styrene
Calcium
Chl id
Chloride
CRLB
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
FBP-BVM
AM-DE
CRLB Images: Diagonals of Inverse
of Fisher Information
Ratio of variance images to CRLB. SAME display windows.
Styrene
Calcium
Chl id
Chloride
CRLB
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
FBP-BVM
AM-DE
Conclusions
• Reviewed Models for CT Image Reconstruction
• Outlined
O tli d Various
V i
Approaches
A
h for
f Dual
D l Energy
E
CT
Image Reconstruction
• Summarized a Simulation Study Showing
Performance of a Dual Energy Alternating
Minimization Algorithm
• Outlined
O tli d and
d Applied
A li d a Method
M th d for
f Quantifying
Q
tif i
Achievable Performance Based on Fisher Information
• Quantitative Analysis Shows Benefit of AM-DE
AM DE
algorithm
• AM-DE Algorithm is Computationally Demanding
• Other Work: Quantify Impact of Modeling Errors
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
References
• J. F. Williamson et al., “Prospects for quantitative computed
tomography imaging in the presence of foreign metal bodies
using statistical image reconstruction,
reconstruction ” Med.
Med Phys.
Phys 2910,
2910 24042404
2418 2002
• J. F. Williamson, S. Li, B. R. Whiting, and F. A. Lerma, “On twoparameter representations of photon cross section: Application
t d
to
duall energy CT imaging,”
i
i ” Medical
M di l Physics
Ph i , vol.
l 33,
33 pp. 41154115
4129, November 2006
• J. A. O’Sullivan and J. Benac, “Alternating minimization
g
for transmission tomography,”
g p y, IEEE Transactions on
algorithms
Medical Imaging, vol. 26, pp. 283-297, March 2007
• J. Benac, “Alternating minimization algorithms for x-ray
computed tomography: multigrid acceleration and dual energy.”
PhD thesis
thesis, Washington University in St.
St Louis,
Louis St.
St Louis
Louis, 2005
• J. Benac, J. A. O’Sullivan, and J. F. Williamson, “Alternating
minimization algorithm for dual energy X-ray CT,” in Proc. IEEE
Int. Symp. Biomedical Imag., Arlington, VA, Apr. 2004, pp.
579582
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
Small Phantom – Ensemble mean and variance (from
15 samples) I0 = 100,000
20keV Image
AM-DE
FBP-BVM
Truth
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
Mean
Variance
Small Phantom – Ensemble mean and variance (from
15 samples) I0 = 100,000
65keV Image
AM-DE
FBP-BVM
Truth
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
Mean
Variance
CRLB images for 20keV (above)
and 65keV (below) images
0.016
1.6
0.55
0.014
0.5
1.4
0.45
0.012
1.2
0.4
0.01
0.35
1
0.3
0.008
0.8
0.25
0.006
06
0.6
02
0.2
0.15
0.004
0.4
0.1
0.002
0.2
0.05
0
−3
x 10
1.6
3
0.55
1.4
0.5
2.5
0.45
1.2
0.4
2
1
0.35
0.3
0.8
1.5
0.25
0.6
0.2
1
0.15
0.4
0.5
0.1
0.2
0.05
CRLB
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09
FBP-BVM
AM-DE
Var / CRLB
Model-Based Imaging:
Principled Algorithm Development
• Model as much of the underlying physics, biology, chemistry
as possible
• Derive an objective function based on physical model,
problem definition, and implementation constraints
((complexity,
p
y, robustness)) Æ loglikelihood
g
function
• Derive algorithms to optimize objective function Æ
maximum likelihood; mathematical considerations
• Predict and evaluate performance using simulations and
phantom experiments
• Revisit physical models, algorithms
• Transition to clinical settings Æ improve algorithms,
implementations, connect to imaging sensors or scanners,
identify key applications
P-20 Seminar, 3/12/05 R. M. Arthur
J. A. O’Sullivan, AAPM 07/28/09