TOMOGRAPHIC IMAGE RECONSTRUCTION TOMOGRAPHIC IMAGE RECONSTRUCTION FOR FOR
PARTIALLY-KNOWN SYSTEMS AND IMAGE PARTIALLY-KNOWN SYSTEMS AND IMAGE SEQUENCESSEQUENCES
M.S. Thesis Defense : Jovan Brankov
Project GoalsProject Goals
• New reconstruction algorithms
• Image reconstruction with Partially-Known system model• Applicable for PET
• Spatially adaptive temporal smoothing for image sequence reconstruction
• Applicable for dynamic PET and gated SPECT
Single Photon Emission Tomography Single Photon Emission Tomography
(SPECT(SPECT))
• Radiotracers are gamma emitters
• Isotopes Tc-99, I-123 and Ga-67
• Metal collimators
• NaI(T1) Scintillator
• Photo Multiplier Tubes (PMT)
• Drawback:
• Low sensitivity
• Advantage:
• Inexpensive
• Cyclotron not required
Positron Emission Tomography (PET)Positron Emission Tomography (PET)
• Radiotracers are positron emitters
• Isotopes 11C, 13N, 18F
• Electronic collimation
• NaI(T1) Scintillator
• Photo Multiplier Tubes (PMT)
• Drawback:
• Requires a cyclotron
• Advantage:
• High sensitivity
Image sequenceImage sequence
• Gated study
• Synchronized with a periodic process in the body
• Like stroboscopy
• Dynamic study
• Not synchronized
Partially-known systems:Partially-known systems:System modelingSystem modeling
• The behavior of the system is not exactly known• object dependent (scattering)
• errors in modeling PSFs
• errors in measurement of PSFs
• System is modeled as the sum of:• a known deterministic part
• an unknown random part
o- Actual PSF+- Assumed PSF
-5 -3 -1 1 3 5 0
0.1
0.2
0.3
0.4
2 4 6-4 -2 0-6
-3 -2 -1 0 1 2 3-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Assumption
Added noise
Different variance
Laplasian
Partially-known systems:Partially-known systems:Imaging modelImaging model
• Idealized discrete model
• Discrete model based on distance-independent blur (not suitable for SPECT)
• Discrete model with PSF uncertainty
data imageSystemmatrix
E[ ]g Pf
E[ ]g APf
E[ ]g (A A)Pf
J g( ) ( ) ( )f g APf C g APf Qf T 1 2
PWLS cost function:
C SS Ig a 2 2T
nwhere
• S = circulant matrix composed of sinogram elements 2
a = PSF error variance,
2n = additive noise variance,
• = regularization parameter• Q = circulant Laplacian high-pass operator
Partially-known systems:Partially-known systems:Cost functionalCost functional
Partially-known systems:Partially-known systems:Cost Functional in DFT domainCost Functional in DFT domain
In discrete Fourier transform (DFT) domain:
Jn
f Qfa
bg
FHGG
IKJJ
1
2
20
2
2 2N
G(i) A(i)S(i)
S(i)i
N-1
A(i), S(i) , and G(i) are DFT coefficients of the blurring kernel a, the computed sinogram s, and the observed sinogram g, respectively.
convolution
Partially-known systems :Partially-known systems : Conjugate gradient minimizationConjugate gradient minimization
Conjugate gradient minimizationConjugate gradient minimization
• modified conjugate gradient method
• application to non-convex cost functional • quadratic interpolation for the line-search procedure
• nonnegativity constraint
Partially-known systems:Partially-known systems:Functional gradient in DFT domainFunctional gradient in DFT domain
Functional gradient :
Pn (i) is the ith coefficient of the DFT of the nth column of the projection matrix P.
grf
fn
n a n
J
FHG
af a fc h22 2 2
0N
A (i)P (i) G(i) A(i)S(i)
S(i)n
i
N-1 Re
IKJJ
a
a n
2 2
2 2 2 2 2G(i) A(i)S(i) Re P (i)S (i)
S(i)
n
d i Q QfT
Partially-known systems:Partially-known systems:ExperimentExperiment
o-True PSF+-Assumed PSF
0
0.1
0.2
0.3
0.4
Source Image Point spread functions
Forward problem:
Degrade the sinogram using the true PSF
Inverse problem:
Reconstruct using the incorrect (assumed) PSF
0
1
2
3
4
5
6
7
8
9
10
Partially-known system:Partially-known system: Evaluation criteriaEvaluation criteria
Spatial mean squared error (MSE)
MSE = E1
N1 f fLNM
OQP
2
MSE = E2 LNMOQPe j2
MSE2 of Region of interest (ROI) estimates
true image reconstructed image
true value estimated value
Partially-known systems:Partially-known systems: Evaluation criteriaEvaluation criteria cont. cont.
MSE1 Vs.
0
2000
4000
6000
8000
10000
12000
0 0.005 0.01 0.015 0.02
MSE as a function of for different values of the PSF noise variance 2
a assumed by the reconstruction algorithm.
Conclusions: 1. Accounting for PSF error helps. 2. Not very sensitive to variance estimate
Partially-known systems:Partially-known systems: Evaluation criteriaEvaluation criteria cont. cont.
MSE2 for hot spots Vs.
0
50
100
150
200
250
300
350
400
0 0.005 0.01 0.015 0.02
MSE2 for cold spots Vs.
0
50
100
150
200
0 0.005 0.01 0.015 0.02
MSE2 : Hot spots
MSE2 : cold spots
Partially-known systems:Partially-known systems: Image resultsImage results
0 5 10 15 20 25 30-2
0
2
4
6
8
10
12
14
0 5 10 15 20 25 30-2
0
2
4
6
8
10
12
14
0
1
2
3
4
5
6
7
8
9
10
Figure 1. Original image
0 5 10 15 20 25 30-2
0
2
4
6
8
10
12
14
0 5 10 15 20 25 30-2
0
2
4
6
8
10
12
14
0
2
4
6
8
10
12
14
16
Figure 3. Image reconstructed without modeling PSF uncertainty using: =0.013 and 2
n=100.MSE1=3642.81
0
2
4
6
8
10
12
0 5 10 15 20 25 30-2
0
2
4
6
8
10
12
14
0 5 10 15 20 25 30-2
0
2
4
6
8
10
12
14
Figure 4. Image reconstructed with model of PSF uncertainty using: =0.013 2
A =1.3e-5 and 2n=100.
MSE1 =1187.23.
0
1
2
3
4
5
6
7
8
0 5 10 15 20 25 30-2
0
2
4
6
8
10
12
14
0 5 10 15 20 25 30-2
0
2
4
6
8
10
12
14
Figure 2. Image reconstructed from blurred noisy sinogram using filtered back-projection.
MSE1 =5428.68
Partially-known systems:Partially-known systems: Point response
0
1
2
3
4
5
5 10 15 20 25 30-2
0
2
4
6
8
10
12
14
16
18
5 10 15 20 25 30-2
0
2
4
6
8
10
12
14
16
18
Figure 6. Image reconstructed from blurred noisy sinogram using filtered back-projection
MSE1 =5428.68
Figure 5. Original image
0
1
2
3
4
5
6
7
8
9
10
0 5 10 15 20 25 30-2
0
2
4
6
8
10
12
14
0 5 10 15 20 25 30-2
0
2
4
6
8
10
12
14
0
2
4
6
8
10
12
14
16
Figure 7. Image reconstructed without modeling PSF uncertainty using: =0.013 and 2
n=100. MSE1 =3642.81
5 10 15 20 25 30-2
0
2
4
6
8
10
12
14
16
18
5 10 15 20 25 30-2
0
2
4
6
8
10
12
14
16
18
0 5 10 15 20 25 30-2
0
2
4
6
8
10
12
14
0 5 10 15 20 25 30-2
0
2
4
6
8
10
12
14
1
2
3
4
5
6
7
8
9
Figure 8. Image reconstructed with model of PSF uncertainty using: =0.013 2
A=1.3e-5 and 2n=100.
MSE1 =1187.23.
0
Partially-known systems:Partially-known systems:Conclusion and future work
Future work Increase the rate of computation speed and reduce required
memory Use more realistic model Evaluate with different types of uncertainties Develop automatic estimation of algorithm parameters
Conclusion Improvements in the reconstructed image
• visually
• quantitatively
Spatially adaptive temporal smoothing:Spatially adaptive temporal smoothing:Karhunen-Loève transformation (KL)
1st 2nd 9th 16th 23rd
Original
sinograms
KL transformed
sinograms
• The Maximum noise fraction transform
• noise in all frame are equal - KL/PCAGreen et al. 1988
Spatially adaptive temporal smoothing:Spatially adaptive temporal smoothing:
Karhunen-Loève transformation (KLT)
Steps:
1. Karhunen-Loève transformation
2. Discard components
3. Inverse KLT
4. Reconstruct
Kao et al. IEEE TMI, 1998
Spatially adaptive temporal smoothing:Spatially adaptive temporal smoothing: k-mean algorithm (Generalized Lloyd Algorithm)
y x, y xy
i i iE d C* arg min b g
C d d j iin
i j x R x, y x, y: ( ),b gn s
Step 3. Given yi recalculate cluster assignment according to :
Step 2. Given Ci calculate Centroids yi according to:
Step 1. Initialization
(random cluster assignment - Ci, i=1..k)
Step 4. Repeat steps 2 and 3 until no reassignment occurs
Spatially adaptive temporal smoothing:Spatially adaptive temporal smoothing:
Compare results of three reconstruction procedures:
• no sinogram preprocessing
• sinogram presmoothing by using KL transform (KL)
• sinogram presmoothing by KL transform taking into account different statistics of pixels (KL/Clustering)
• all three reconstructed with Expectation Maximization algorithm (EM).
Tested for three possible applications:• Kinetic study of the brain
• Lesion detection in dynamic PET
• Gated SPECT
Spatially adaptive temporal smoothing:Spatially adaptive temporal smoothing:Brain phantomBrain phantom
Realistic MRI voxel-based numerical brain phantom developed by Zubal et al.
Spatially adaptive temporal smoothing: Spatially adaptive temporal smoothing: Compartment Kinetic ModelCompartment Kinetic Model
dC t
dtk C t k C t k C t k k k C tf
p b n f
( )( ) ( ) + ( ) - ( ) ( ) 1 4 6 2 3 5
dC t
dtk C t k C tb
f b
( )( ) - ( ) 3 4
dC t
dtk C t k C tn
f n
( )( ) ( ) 5 6
C t C t C t C tb n f( ) ( ) ( ) ( )
C t e L e C tdec
t
t
iR t
pi
ni( ) ( )
( )
12
1
2
1
ln
C tL e t t
c e t tp
pmt
iD t
i
i( )
( ) ( )
( )
RS|T|
1 0 0
00
3
Equilibrium solution:
Blood curve model:
The blood sample values obtained in a PET study conducted by the Department of Radiology at the University of Chicago.
Four Compartment Kinetic Model
Spatially adaptive temporal smoothing:Spatially adaptive temporal smoothing: Kinetic brain modelKinetic brain model
Brain region Tree compartment Four compartment
Thalamus - 1Caudate - 0.87
Front Cortex - 0.805Ant. Temporal +Sup. Temporal
Cortex
- 0.805
White matter - 0.1Occipital Cortex 1 -
[11C] Carfentanil Study JJ Frost et al.1990
Brain phantom
Time activation curves
Source image Cluster map in sinogram domain
Filterback projection of each cluster position separately
Spatially adaptive temporal smoothing:Spatially adaptive temporal smoothing: Cluster map
Spatially adaptive temporal smoothing:Spatially adaptive temporal smoothing:Time activity curves (TAC)
Estimated TAC’s for thalamus and occipital cortex
Difference between the original and estimated TAC’s
thalamus and occipital cortex
Spatially adaptive temporal smoothing:Spatially adaptive temporal smoothing:Image resultsImage results
Third frame from dynamic brain study reconstructed with different presmoothing techniques
Reconstructed images
Differences images
Spatially adaptive temporal smoothing:Spatially adaptive temporal smoothing: Lesion detection in dynamic PETLesion detection in dynamic PET
Yu et al. 1997
Time activation curves for different region in lesion dynamic studyPhantom
Spatially adaptive temporal smoothing:Spatially adaptive temporal smoothing: Time activity curves
Estimated TAC’s for small lesion
Difference between the original and estimated TAC’s for small lesion
Spatially adaptive temporal smoothing:Spatially adaptive temporal smoothing:Image resultsImage results
Some frames from dynamic lesion study reconstructed with different presmoothing techniques
Spatially adaptive temporal smoothing:Spatially adaptive temporal smoothing:Torso phantomTorso phantom
The 4D gated mathematical cardiac-torso gMCAT (D1.01 version- fixed anatomy, dynamic (beating heart)) phantom.
University of Massachusetts Medical School, Worcester, MA
Spatially adaptive temporal smoothing:Spatially adaptive temporal smoothing:
Gated SPECT
• The goal is to preserve heart motion
• Difficult to evaluate quantitatively
• ROI on the heart wall
Spatially adaptive temporal smoothing:Spatially adaptive temporal smoothing: Time activation curves
TAC of the ROI without presmooting TAC of the ROI with KL presmooting
TAC of the ROI with KL/Clustering presmooting
Spatially adaptive temporal smoothing:Spatially adaptive temporal smoothing: Conclusion and future work
Future work Apply filtering before KL coefficients estimation
(Manoj et al. 1998) Evaluate on real SPECT/PET data Evaluate for clinical use
Conclusion Possible improvement in estimation of time activation
curves for ROI’s which leads to better:• kinetic model parameters estimation
• delectability of the lesion
• observation of the heart motion and abnormalities