1Department of Medical Physics, University of Wisconsin-Madison, Madison, WI. 2Department of Radiology, University of Wisconsin-Madison, Madison, WI ...
Yinsheng Li1, Jie Tang1, Guang-Hong Chen1,2 1Department 2Department
of Medical Physics, University of Wisconsin-Madison, Madison, WI of Radiology, University of Wisconsin-Madison, Madison, WI
Introduction and Motivation
Problem Setup: From 1-norm to p-norm
Phantom Studies and Performance Evaluation
In vivo Studies and Performance Evaluation
Conclusions 2
Introduction and Motivation
Problem Setup: From 1-norm to p-norm
Phantom Studies and Performance Evaluation
In vivo Studies and Performance Evaluation
Conclusions 3
PICCS
Ù
{
x = arg min a ||y (x - x p ) ||1 +(1- a ) ||y (x) ||1
}
s.t. Ax = y
*
ìl ü** T x = arg min í (Ax - y) D(Ax - y) + a ||y (x - x p ) ||1 +(1- a ) ||y (x) ||1 ý î2 þ Ù
*G.-H. Chen, J. Tang, and S. Leng, Med. Phys. (2008) Vol. 35 p660. **PT Lauzier and G-.H Chen. Medical Physics 39(10) 2012
PICCS
*
Limited view angle range problem
Few view problem
Noise/dose reduction
Cardiac CT (TRI-PICCS)
Respiratory gated CBCT in IGRT
CT perfusion
Time-resolved interventional CT
Cardiac gated CBCT
General CT application (DR-PICCS)
Dual energy CT *G.-H. Chen, J. Tang, and S. Leng, Med. Phys. (2008) Vol. 35 p660
• 4DCBCT in IRGT: time-resolved cone beam CT imaging 1 • Challenge: ~1 min data acquisition time v.s. 2-5 s breathing motion period • Retrospective data sorting is used to acquire the simultaneous surrogate signal.
1. Keall, P.J., et al. Med Phys 33(10), 2006.
t
Projection data
…
FBP single phase (undersampled))
…
Prior
…
FBP from all views (time average)
PICCS
information HighTemporal SNR Poor SNR Temporal Information Streak artifacts
High SNR No temporal information
Clinical Scenario #2: Time-resolved dynamic Prior Image Prior PriorImage Image contrast imaging Volume Volume Volume
N Time Frames
FBP
Prior Image
frame Timeframe frame Time 1 11
Time Frame 1
PICCS CCS PICCS PICCS
Time frame Time frame frame Time 2 22
Time Frame 2
PICCS PICCS PICCS PICCS
…
View-Angle View-Angl View-An Time frame Time frame Time frame Undersampled Undersamp Undersam N NN View Angle Data Data Data Undersampled Acquisition Acquisitions Acquisit Time Frame N Data Acquisitions
PICCS PICCS PICCS PICCS
Ultra-low dose Dose red Dose Dose reduced Image Volumes Image Vo Image Image Volumes 10
▪ To date, published studies employed L1-norm in the PICCS function. ▪ Goals of this presentation: 1. How does image quality depend on the norm selection in the objective function? 2. Is there any improvement in image quality when a reweighted scheme is applied?
Introduction and Motivation
Problem Setup: From 1-norm to p-norm
Phantom Studies and Performance Evaluation
In vivo Studies and Performance Evaluation
Conclusions 12
Prior Image Constrained Compressed Sensing (PICCS)1,2
ìl ü T x = arg min í (Ax - y) D(Ax - y) + f piccs (x) ý î2 þ Ù
Data Consistency
PICCS
f piccs (x) = a ||y (x - x p ) ||1 +(1- a ) ||y (x) ||1 Prior image term
Compressed Sensing term
From 1-norm to P-norm
fnd- piccs (x) = a ||y (x - x p ) || pp +(1- a ) ||y (x) || pp
1. Chen et al. Medical Physics 2008
[ 0 , 1]
p [1, 2 ]
2. PT Lauzier and G-.H Chen. Medical Physics 39(10) 2012
When the selected norm deviates from 1, it has been
suggested that a reweighted scheme may be applied to asymptotically approach the result as if the l1norm was used. Thus, an iterative reweighted / FOCUSS technique is
also applied to study the performance of PICCS in this work.
1. Gorodnitsky and Rao, IEEE Tran. Signal Processing, Vol.45:600 (1997) 2. Jung, Ye, and Kim, Phys. Med. Biol., Vol. 52:3201(2007) 3. Candes, Wakin, and Boyd, J. Fourier Anal. Applications, Vol.14:877(2008)
14
f Value
f = (xi+1 - xi )2 + (xi+ M - xi )2
|| f ||
1 1
kth iteration :
f
kth iteration :
åf i
p
p-1 k -1
f
åf i
2 Iteration
k -1
|| f ||
1 1
(k-1)th iteration image
(k-1)th iteration image 15
Introduction and Motivation
Problem Setup: From 1-norm to p-norm
Phantom Studies and Performance Evaluation
In vivo Studies and Performance Evaluation
Conclusions 16
Numerical simulations : Intensity
Time frame
Prior image was reconstructed from interleaved undersampled projection data sets from all time frames.
The degree of undersampling was varied.
Reconstruct images using Norm dependent PICCS w/ and w/o the reweighted scheme.
17
Ground truth
P=1.25
80-views FBP
Prior Image
P=1.5
P=1.75
P=1.0
P=2.0 18
P=1.0
P=1.5
P=1.25
P=1.75
P=2.0 19
Ground truth
P=1.25
80-views FBP
P=1.5
Prior Image
P=1.75
P=1.0
P=2.0 20
P=1.0
P=1.5
P=1.25
P=1.75
P=2.0 21
P=1.5 W/O REWEIGHTED SCHEME
P=1.5 W/ REWEIGHTED SCHEME
22
P=1.5 W/O REWEIGHTED SCHEME
P=1.5 W/ REWEIGHTED SCHEME
23
P=2.0 W/O REWEIGHTED SCHEME
P=2.0 W/ REWEIGHTED SCHEME
24
P=2.0 W/O REWEIGHTED SCHEME
P=2.0 W/ REWEIGHTED SCHEME
25
Quantification of reconstruction accuracy of numerical phantom data was performed using the relative Root Mean Square Error (rRMSE): 1
rR M SE (x ) N
ROI
x p ,q x p ,q ref
( p , q ) R O I
(
ref
)
2
x p ,q
The ROI was selected in the dynamic enhancement region.
Prior image
Target image
26
Reconstruction accuracy is degraded as p increases. Reconstruction accuracy is better w/ reweighted scheme Reconstruction accuracy is optimal for alpha = 0.5
W/O REWEIGHTED SCHEME p=1.0
0.14
p=1.25
p=1.5
W/ REWEIGHTED SCHEME
p=1.75
p=2.0
(%)
0.1
0.08
rRMSE
(%)
p=1.25
p=1.5
p=1.75
p=2.0
0.12
0.12
rRMSE
p=1.0
0.14
0.06
0.1
0.08
0.06
0.04
0.04
0.02
0.02
0
0.1
0.2
0.3
0.4
0.5
alpha
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
alpha 27
Reconstruction accuracy is higher when we use more views
W/ REWEIGHTED SCHEME
W/O REWEIGHTED SCHEME p=1.0
0.14
p=1.25
p=1.5
p=1.75
p=2.0
p=1.25
p=1.5
p=1.75
p=2.0
(%)
0.12
0.1
0.08
rRMSE
rRMSE
(%)
0.12
0.06
0.1
0.08
0.06
0.04
0.04
0.02
0.02
20
p=1.0
0.14
30
40
50
60
70
80
90
number of views
100
110
120
20
30
40
50
60
70
80
90
100
110
120
number of views 28
Quantification of undersampled streaks in numerical phantom data :
relative Streak Artifacts Level (rSAL) :
rSA L
| T V ( I ) T V ( tr u th ) | T V ( tr u th )
T V (x)
( x i 1, j x i , j ) ( x i , j 1 x i , j ) 2
2
i, j
|| Ñ
||=
å i
|| Ñ
||=
-å
å
i
= 0.8733
i
29
Quantification of undersampled streaks in numerical phantom data :
relative Streak Artifacts Level (rSAL) :
rSA L
| T V ( I ) T V ( tr u th ) | T V ( tr u th )
T V (x)
( x i 1, j x i , j ) ( x i , j 1 x i , j ) 2
2
i, j
|| Ñ
||=
å i
|| Ñ
||=
-å
å
i
= 0.2522
i
30
Undersampling streaks are increasing as p increases Undersampling streaks are better controlled w/ reweighted scheme.
rSAL W/O REWEIGHTED SCHEME
rSAL W/ REWEIGHTED SCHEME
1
1
p=1.25
p=1.5
p=1.75
p=2.0
p=1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
rSAL
rSAL
p=1.0
0.5
0.4
p=1.5
p=1.75
p=2.0
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
p=1.25
100
200
300
400
500
600
lambda
700
800
900
1000
0
100
200
300
400
500
600
700
800
900
1000
lambda 31
The streak artifacts level is lower when we use more views Norm can be relaxed to 1.5 with nearly comparable performance w/ reweighted scheme
rSAL W/O REWEIGHTED SCHEME
rSAL W/ REWEIGHTED SCHEME
1
1
p=1.25
p=1.5
p=1.75
p=2.0
p=1.0
0.9
0.9
0.8
0.8
0.7
0.7
rrsal (%)
rrsal (%)
p=1.0
0.6
0.5
0.4
0.2
0.1
0.1
50
60
70
80
90
number of views
100
110
120
p=2.0
0.4
0.2
40
p=1.75
0.5
0.3
30
p=1.5
0.6
0.3
0 20
p=1.25
0 20
30
40
50
60
70
80
90
100
110
120
number of views 32
Introduction and Motivation
Problem Setup: From 1-norm to p-norm
Phantom Studies and Performance Evaluation
In vivo Studies and Performance Evaluation
Conclusions 33
In Vivo Data Acquisition: Animal model: 59 kg swine Acquired on a 64-slice GE Discovery 750 HD Tube potential:120 kVp Tube current: 500 mA Acquisition time: 0.4 s Short scan range: 234o Undersampled rate: 107/642 views
Prior image: reconstructed from interleaved undersampled projection data of all frames.
34
PRIOR
642 VIEWS REFERENCE
35
P=1.0 W/ STAT W/ REWEIGHTED SCHEME
P=1.5 W/ STAT W/ REWEIGHTED SCHEME
P=2.0 W/ STAT W/ REWEIGHTED SCHEME
36
P=1.0 W/ STAT W/ REWEIGHTED SCHEME
P=1.5 W/ STAT W/ REWEIGHTED SCHEME
P=2.0 W/ STAT W/ REWEIGHTED SCHEME
37
Universal Quality Index (UQI) of in vivo data: UQI
4 x ref (
2 x
1
N
ROI
1
2 ref
) ( x ref ) 2
( p , q ) N
2
( x p ,q x ) ( x p ,q ref ) ref
ROI
UQI can be used to measure how the reconstructed image looks like the reference image with respect to the noise texture, mean intensity and contrast level.
38
UQI decreases when p increases. UQI is higher w/ reweighted scheme.
39
Introduction and Motivation
Problem Setup: From 1-norm to p-norm
Phantom Studies and Performance Evaluation
In vivo Studies and Performance Evaluation
Conclusions 40
A relaxed norm in the PICCS objective function may introduce streak artifacts in the few view reconstruction problem.
When the reweighted scheme is used, the image quality of PICCS with 1.5-norm is nearly comparable to that of the 1-norm with respect to reconstruction accuracy, relative streak artifact level, and universal image quality.
41
The work is partially supported by NIH R01 EB009699 The work is partially supported by Varian Medical System
e
Thanks for your attention !!! 42
Universal Quality Index (UQI) of in vivo data: UQI
4 x ref (
2 x
1
N
ROI
1
2 ref
) ( x ref ) 2
( p , q ) N
2
( x p ,q x ) ( x p ,q ref ) ref
ROI
Very similar !!!
UQI
x ref
2 x
2 ref
2
1 .0
43
Universal Quality Index (UQI) of in vivo data: UQI
4 x ref (
2 x
1
N
ROI
1
2 ref
) ( x ref ) 2
( p , q ) N
2
( x p ,q x ) ( x p ,q ref ) ref
ROI
Extremely different!!! UQI
0 .0 7 UQI can be used to measure how the reconstructed image looks like the reference image with respect to the noise texture, mean intensity and contrast level. 44
Advanced treatment techniques have been proposed to reduce motion induced treatment margin1. Breath holding, beam gating, tumor tracking, …
Tumor tracking delivery would be the ideal technology. Goal: static tumor from beam’s eye view (BEV)
Tumor tracking can be achieved by Dynamic MLC2,3 Robotic linear accelerator (Cyberknife) Couch motion4 …. [1] Keall, P.J., et al. Med Phys 33(10), 2006. [2] Keall, P. J., et al. Phys. Med. Biol. 46(1), 2001. [3] Neicu, T., et al. Phys. Med. Biol. 48(5), 2003. [4] D’Souaz, W. D., et al. Phys. Med. Biol. 50(17), 2005.
In order to adapt the treatment for the best outcome, tumor motion profile plays a pivotal role in the entire game. Motion field (from 4D imaging 4D-CT, 4D-MRI etc.)
Treatment planning
Register images between phases for dose calculation
Prior to the treatment delivery
Check tumor motion trajectory to verify the plan and reposition/re-plan if necessary
During the treatment delivery
Synchronize the beam with the tumor
1 minute scan with 640 views of projection data. RPM recorded respiratory signal 20 phase bins; 25 views per phase
The relaxed norm in PICCS framework may enhance numerical stability and efficiency.
The purpose of this study is two-fold: 1. How dose image quality depend on the norm selection in the object function ? 2. Is there any improvement in image quality when a reweighted scheme is applied ?
48
The iterative reweighted scheme aims at approximating the L1 norm solution even when p is higher than 1.
Define the normalization matrix iteratively : u W
Introduce an variable :
1
W k d ia g { | x k 1 | } q
x
Determine q value by forcing the P-norm solution approaching to the 1-norm solution: || u || p || W p
1
p
q
x || p
||| x |
p (1 q ) 1
p (1 q )
x || p || x || p (1 q ) p
q 1
1 p
49
Step 1. Minimize an UC-PICCS function with respective to
u
:
T u k 1 a r g m in ( W k u y ) D ( W k u y ) f n d p ic c s ( u ) 2
Step 2. Normalize u to get the image update : x k 1 W k u k 1
Step 3. Update the normalization matrix : 1
W k 1 d ia g { | x k 1 |
1 p
}
50
Standard Deviation (SD) v.s. Ensemble Variance (EV):
For most streak-free images, EV and SD should be quite similar, such as fully sampled FBP images
EV
SD 51
Correlation between EV and SD are decreasing when p is increasing. Streaks make lots of contribution to SD since all curves deviate from the FBP line. -3
x 10
4
4
3.5
3.5
(mm -1 )
(mm -1 )
x 10
3
3
2.5
Standard Deviation
Standard Deviation
2.5
-3
2
1.5
p=1.0 w/o reweighted p=1.25 w/o reweighted
1
2
1.5
p=1.0 w/ reweighted p=1.25 w/ reweighted
1
p=1.5 w/o reweighted
p=1.5 w/ reweighted
p=1.75 w/o reweighted
0.5
p=1.75 w/ reweighted
0.5
p=2.0 w/o reweighted
p=2.0 w/ reweighted
FBP 0
0
0.5
1
1.5
2
2.5
3
FBP 3.5
-1
Ensemble Variance (mm )
4
4.5 x 10
-3
0
0
0.5
1
1.5
2
2.5
3.5
3
-1
Ensemble Variance (mm )
4
4.5 x 10
-3
52
Undersampled streaks are better inhibited w/ rewieghted since SD w/ is lower than SD w/o when matching the EV.
EV / SD P=1.0 -3
2.5
x 10
EV / SD P=2.0
-3
5
3
2
-3
4 3.5
1
0.5
2.5
Standard Deviation
1.5
2
1.5
1
p=1.5 w/o reweighted
p=1.0 w/o reweighted 0
1
2
3
4
5
-1
Ensemble Variance (mm )
0.5 6
x 10
3 2.5 2 1.5
p=2.0 w/o reweighted p=2.0 w/ reweighted
p=1.5 w/ reweighted
p=1.0 w/ reweighted 0
x 10
4.5
3.5
(mm -1 )
(mm -1 )
4
Standard Deviation
Standard Deviation
(mm -1 )
3
x 10
EV / SD P=1.5
-4
0
0.5
1
1.5
-1
Ensemble Variance (mm )
x 10
-3
1
0
0.5
1
1.5
-1
Ensemble Variance (mm )
2 x 10
-3
53
P=1.75 W/O REWEIGHTED
P=1.75 W/ REWEIGHTED
54
LAMBDA=30 P=1.25 W/ STAT W/O REWEIGHT
LAMBDA=30 P=1.25 W/ STAT W/ REWEIGHT
55
LAMBDA=5 P=1.75 W/ STAT W/O REWEIGHT
LAMBDA=5 P=1.75 W/ STAT W/ REWEIGHT
56
LAMBDA=30 P=1.0 W/ STAT W/O REWEIGHT
LAMBDA=30 P=1.0 W/ STAT W/ REWEIGHT
57
LAMBDA=10 P=1.5 W/ STAT W/O REWEIGHT
LAMBDA=10 P=1.5 W/ STAT W/ REWEIGHT
58
LAMBDA=1 P=2.0 W/ STAT W/O REWEIGHT
LAMBDA=1 P=2.0 W/ STAT W/ REWEIGHT
59
W/O REWEIGHTED p=1.0
p=1.25
p=1.5
p=1.75
p=2.0
0.12
0.12
0.1
0.1
0.08
0.06
0.04
0.04
0.02
0.02 0.1
0.2
0.3
0.4
0.5
alpha
0.6
0.7
0.8
0.9
1
p=1.25
p=1.5
p=1.75
p=2.0
0.08
0.06
0
p=1.0
0.14
rRMSE (%)
rRMSE (%)
0.14
W/ REWEIGHTED
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
alpha
60
W/O REWEIGHTED p=1.25
p=1.5
p=1.75
p=2.0
p=1.0
0.14
0.14
0.12
0.12
0.1
0.1
rRMSE (%)
rRMSE (%)
p=1.0
W/ REWEIGHTED
0.08
0.06
0.04
0.04
0.02
0.02
0.1
0.2
0.3
0.4
0.5
alpha
0.6
0.7
0.8
0.9
1
p=1.5
p=1.75
p=2.0
0.7
0.9
0.08
0.06
0
p=1.25
0
0.1
0.2
0.3
0.4
0.5
0.6
0.8
1
alpha
61
W/ REWEIGHTED
0.14
0.14
0.12
0.12
0.1
0.1
rRMSE (%)
rRMSE (%)
W/O REWEIGHTED
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0.02
p=1.0 0
0.1
0.2
p=1.25 0.3
0.4
p=1.5 0.5
alpha
0.6
p=1.75 0.7
0.8
p=2.0 0.9
p=1.0 1
0
0.1
0.2
p=1.25 0.3
0.4
p=1.5 0.5
0.6
p=1.75
p=2.0
0.7
0.9
0.8
1
alpha
62
RSAL W/O REWEIGHTED
RSAL W/ REWEIGHTED
1
1
p=1.25
p=1.5
p=1.75
p=2.0
p=1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
rSAL
rSAL
p=1.0
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
100
200
300
400
500
lambda
600
700
800
900
1000
p=1.5
p=1.75
p=2.0
0.5
0.4
0
p=1.25
0
100
200
300
400
500
600
700
800
900
1000
lambda
63
RSAL W/O REWEIGHTED
RSAL W/ REWEIGHTED 1
1
p=1.25
p=1.5
p=1.75
p=1.0
p=2.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
rSAL
rSAL
p=1.0
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
100
200
300
400
500
lambda
600
700
800
900
1000
p=1.5
p=1.75
p=2.0
700
900
0.5
0.4
0
p=1.25
0
100
200
300
400
500
600
800
1000
lambda
64
RSAL W/O REWEIGHTED
RSAL W/ REWEIGHTED
1
1
p=1.25
p=1.5
p=1.75
p=2.0
p=1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
rSAL
rSAL
p=1.0
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
100
200
300
400
500
lambda
600
700
800
900
1000
p=1.5
p=1.75
p=2.0
0.5
0.4
0
p=1.25
0
100
200
300
400
500
600
700
800
900
1000
lambda
65
