1. Nonlocal low-rank and sparse matrix decomposition for spectral CT reconstruction
Shanzhou Niu1, Gaohang Yu2, Jianhua Ma2, and Jing Wang1
1Department of Radiation Oncology
UT Southwestern Medical Center, Dallas TX
2Department of Biomedical Engineering
Southern Medical University, Guangzhou China
Today, I will give a talk on “Nonlocal low-rank and sparse matrix decomposition for spectral CT reconstruction”.

2. Background Spectral CT using photon counting detector (PCD)
Multiple projection data with different energy information can be generated from a single scan
Appropriate selection of energy bins has a critical impact on noise and energy resolution
Increasing the number of energy bins will lead to few available photons in each energy bin
Image quality of spectral CT will be severely degraded because of the limited photon number in narrow energy bin
Noise reduction for spectral CT
Reconstruct each energy bin individually by designing different spatial regularizers: TV, structure tensor TV, total generalized variation, tight frame, etc.
Consider correlation between different energy bins.

3. sparse matrix decomposition
Nonlocal low-rank and
sparse matrix decomposition
Exploit the correlation between different energy bins using the low-rank prior information
Nonlocal self-similarity using similar patches
Nonlocal low-rank and sparse matrix decomposition (NLSMD)
Group the similar patches in multi-energy images to form a matrix
Decompose it into the superposition of a low-rank matrix and a sparse matrix
The sparse matrix represents the distinct intensity features, while the low-rank matrix represents the rest of stationary background that is low-rank.

4. Spectral CT imaging model
The projection of each energy bin is approximately formulated as linear equations:
Combining all energy bins:
Incorporating a regularizer

5. Nonlocal matrix construction
Collect small patches in muliti-energy images and group them into one matrix:
is a patch extraction operator, p and k are the indexes for a specific p patch in the k-th energy bin image.
: low-rank matrix represents the rest of stationary background that is low-rank.
These patches have similar structures, thus they locate in a low-dimension subspace. Then, we model:
: sparse matrix represents the distinct intensity features

6. and can be recovered by solving the following minimization problem
Covert to unconstraint minimization problem:

7. NLSMD-based spectral CT reconstruction
Using NLSMD, the following objective function is constructed for spectral CT:
Alternatively solve , , and
I will present the NLSMD-based spectral CT reconstruction method.

8. Minimization with respect to
Independent to the data fidelity term, and it can be minimized patch by patch:
Single value thresholding algorithm
*A. N. Chambolle, J. Math. Imaging Vis
Chambolle’s algorithm
In this slice, I will present the optimization algorithm for the NLSMD reconstruction method. For minimization with L_p, we can obtain an analysis solution using single value thresholding algorithm. For minimization with S_p, the solution can be obtained using Chambolle’s algorithm. The solution of X can be obtained from the optimal condition, i.e., the linear equations, which can be solved by conjugate gradient method.

9. Minimization with respect to X
Note that each patch is voxel-shifted from a previous patch, and each voxel in the multi-energy images is overlapped B times for the patching grouping.
The inverse transform from the patch to the image domain can be formulated as:
In this slice, I will present the optimization algorithm for the NLSMD reconstruction method. For minimization with L_p, we can obtain an analysis solution using single value thresholding algorithm. For minimization with S_p, the solution can be obtained using Chambolle’s algorithm. The solution of X can be obtained from the optimal condition, i.e., the linear equations, which can be solved by conjugate gradient method.

10. Spectrum of 140 kV with five energy bins.
Simulation Studies
Mono-energetic images obtained with a GE Discovery CT 750HD scanner were used as digital phantom.
* A.M. Hernandez, J. M. Boone, Med. Phys., 2014.
The polychromatic x-ray spectrum at 140 kVp was simulated using the TASMICS*
Spectrum of 140 kV with five energy bins.
Fan-beam geometry: 949 mm SID/ 408 mm SOD/ 88 projection views/ 888 detector bin perview
The polychromatic x-ray spectrum at 140 kVp with relative soft filtration was simulated using the TASMICS. We obtained the monochromatic images at 60, 70, 80, 90, and 100 keV from GE Discovery CT 750 HD scanner. We view these images as golden standard, and simulate the projections in seven energy bins respectively.

11. Influence of parameter selection.
Figure: Surface plot of the RRMSE as a function of the patch size and the penalty parameter (70 keV).
Parameters selection. The accuracy of NLSMD reconstruction at 70 keV energy bin is quantified by the RRMSE (relative root mean square error) in this slice. The optimal parameters with the minimal RRMSE was used in this study.

12. Results: 70 keV energy bin
Phantom
FBP
RPCA
NLSMD
It can be observed that the FPB images contain serious noise and streak artifacts because of few available photons and insufficient angular sampling in narrow energy bin. The noise and streak artifacts have been effectively reduced in RPCA results, however, some streak artifacts still exist around the skull. We confirmed that the presented NLSMD method achieves the best performance in terms of artifacts reduction and structure preservation as indicated by the arrow.
RPCA: robust principal component analysis

13. Profile Comparison
(a) (b)
Figure: Horizon profiles that correspond to the blue line in phantom image of reconstructed images of PRCA (a) and NLSMD (b) methods.
The reconstructed images using RPCA method produce underestimated values whereas NLSMD method can produce better matching results.

14. Zoomed ROI
70 KeV
90 KeV

15. Structural similarity measure
To further display the difference between the RPCA and NLSMD results, the zoomed ROI is shown in this slice. The SSIM curves versus the different energy bins was shown in right figure. We can observe that the average SSIM increases from 0.78 in PRCA results to 0.85 in NLSMD results.

16. Noise-Resolution Trade-off
The noise-resolution tradeoff curves are displayed for 70 energy bin images . The noise level of NLSMD results is lower than that of RPCA results with matched spatial resolution. At the 1.8 mm spatial resolution level in two noise-resolution tradeoff curves, the average noise level of NLSMD results is 8.8% lower than that of RPCA results.

17. Material Decomposition
Phantom FBP RPCA NLSMD
Tissue
Bone
The material decomposition images produced using phantom, FPB, RPCA and NLSMD images are shown in this slice.

18. Difference of material decomposition results
RPCA NLSMD
Tissue
Bone
To further validate the difference between NLSMD and RPCA methods for material decomposition, the corresponding difference images are displayed in this slice. It can be observed that the difference images between NLSMD results and phantom are weaker than that obtained from RPCA results.

19. Zoomed-in views of the material decomposition
Phantom
FBP
RPCA
NLSMD

20. SSIM of material decomposition
The zoomed details of two ROIs are depicted in this slice. It can be clearly observed that many pixels at the tissue gap between bones are incorrectly clarified as containing bone using RPCA method, as indicated by the red arrow. Furthermore, while a few pixels at the nasal cavity are wrongly decomposed as bone using RPCA method, the bone image produced using NLSDM method is correct, as indicated by the red arrow. The SSIM values of FBP, RPCA, and NLSMD results are shown in right figure.

21. Ex vivo Data
A lamb chop scanned by MARS spectral CT with the Medipix3RX PCD.
The distance from the source to the center of rotation is mm and the distance from the detector arrays to the center of rotation is 48 mm.
The x-ray tube was set at 50 kV with the current of 120 µA using four energy thresholds (15, 20, 25, and 30 keV).
The polychromatic x-ray spectrum at 140 kVp with relative soft filtration was simulated using the TASMICS. We obtained the monochromatic images at 60, 70, 80, 90, and 100 keV from GE Discovery CT 750 HD scanner. We view these images as golden standard, and simulate the projections in seven energy bins respectively.

22. FBP-163 view
FBP-full view
RPCA
NLSMD
Top: 20 keV Bottom: 25 keV

23. CNR and SNR comparison

24. Summary We have developed a NLSMD method for spectral CT : NLSMD:
Explore nonlocal self-similarity using similar patches
Consider correlation between different energy bins using the low-rank prior information
NLSMD:
Reduce noise/dose for spetral CT
Outperform RPCA based on a number of measures