Bayesian 3D Xray Computed Tomography with a Hierarchical Prior Model for Sparsity in Haar Transform Domain
Abstract
:1. Introduction
2. The SemiSupervised Hierarchical Model
2.1. Forward System Model and Likelihood
2.2. Hierarchical Prior Model and Prior Distributions
2.3. The HHBM Method
2.4. Joint Maximum a Posteriori Estimation
Algorithm 1 The JMAP Algorithm for the HHBM Method 

3. Initialization and Experimental Results
 the Relative Mean Squared Error (RMSE), RMSE = ${\left(\right)}^{\mathit{f}}2$, which shows a relative error of the results;
 the Improvement of the SignaltoNoise Ratio (ISNR), which measures the improvement during iterations;
 the Peak SignaltoNoiseRatio (PSNR), which presents the SNR relative to the peak data value;
 the Structural Similarity of IMage (SSIM) [62], which evaluates the quality of the result approaching human vision.
3.1. Initializations
3.2. Simulation Results with a Limited Number of Projections
3.3. Simulation Results with a Limited Angle of Projections
3.4. Simulation with a Different Forward Model
4. Discussion
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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$256\times 256\times 256$Shepp–Logan Phantom  
180 Projections  90 Projections  
40 dB  20 dB  40 dB  20 dB  
QR  TV  HHBM  QR  TV  HHBM  QR  TV  HHBM  QR  TV  HHBM  
$\mathbf{RMSE}$  $0.0236$  $0.0114$  $\mathbf{0.0069}$  $0.1309$  $\mathbf{0.0209}$  $0.0755$  $0.0401$  $0.0212$  $\mathbf{0.0092}$  $0.1558$  $\mathbf{0.0491}$  $0.1117$ 
$\mathbf{ISNR}$  $5.5584$  $8.7217$  $\mathbf{10.9346}$  $7.2024$  $\mathbf{15.1775}$  $10.2162$  $6.6136$  $9.3832$  $\mathbf{12.9973}$  $8.4583$  $\mathbf{13.4765}$  $9.9056$ 
$\mathbf{PSNR}$  $30.0675$  $33.2308$  $\mathbf{35.4437}$  $22.6318$  $\mathbf{30.6069}$  $25.0209$  $27.7743$  $30.5439$  $\mathbf{34.1579}$  $21.8754$  $\mathbf{26.8937}$  $23.3227$ 
$\mathbf{SSIM}$  $0.9999$  $0.9999$  $\mathbf{1.0000}$  $0.9992$  $\mathbf{0.9999}$  $0.9995$  $0.9997$  $0.9999$  $\mathbf{0.9999}$  $0.9990$  $\mathbf{0.9997}$  $0.9993$ 
60 Projections  45 Projections  
40 dB  20 dB  40 dB  20 dB  
QR  TV  HHBM  QR  TV  HHBM  QR  TV  HHBM  QR  TV  HHBM  
$\mathbf{RMSE}$  $0.0636$  $0.0321$  $\mathbf{0.0107}$  $0.1656$  $\mathbf{0.0753}$  $0.1293$  $0.0904$  $0.0474$  $\mathbf{0.0132}$  $0.1854$  $0.0901$  $\mathbf{0.1414}$ 
$\mathbf{ISNR}$  $9.3826$  $12.3480$  $\mathbf{17.1346}$  $9.1492$  $\mathbf{12.5701}$  $10.2226$  $10.3301$  $13.1308$  $\mathbf{18.6839}$  $10.0137$  $13.1476$  $\mathbf{11.1916}$ 
$\mathbf{PSNR}$  $25.7693$  $28.7347$  $\mathbf{33.5214}$  $21.6116$  $\mathbf{25.0325}$  $22.6849$  $24.2404$  $27.0412$  $\mathbf{32.5942}$  $21.1195$  $24.2535$  $\mathbf{22.2974}$ 
$\mathbf{SSIM}$  $0.9996$  $0.9995$  $\mathbf{0.9999}$  $0.9990$  $\mathbf{0.9995}$  $0.9992$  $0.9994$  $0.9997$  $\mathbf{0.9999}$  $0.9988$  $0.9994$  $\mathbf{0.9991}$ 
36 Projections  18 Projections  
40 dB  20 dB  40 dB  20 dB  
QR  TV  HHBM  QR  TV  HHBM  QR  TV  HHBM  QR  TV  HHBM  
$\mathbf{RMSE}$  $0.1177$  $0.0680$  $\mathbf{0.0169}$  $0.1957$  $\mathbf{0.1116}$  $0.1500$  $0.2581$  $0.2104$  $\mathbf{0.0574}$  $0.2907$  $0.2313$  $\mathbf{0.2014}$ 
$\mathbf{ISNR}$  $10.6591$  $13.0424$  $\mathbf{19.0933}$  $10.8633$  $\mathbf{13.3032}$  $12.0187$  $10.7122$  $11.5992$  $\mathbf{17.2373}$  $10.8088$  $11.8022$  $\mathbf{12.4036}$ 
$\mathbf{PSNR}$  $23.0949$  $25.4783$  $\mathbf{31.5292}$  $20.8865$  $\mathbf{23.3264}$  $22.0420$  $19.6263$  $20.5133$  $\mathbf{26.1514}$  $19.1085$  $20.1020$  $\mathbf{20.7033}$ 
$\mathbf{SSIM}$  $0.9993$  $0.9996$  $\mathbf{0.9999}$  $0.9988$  $\mathbf{0.9993}$  $0.9990$  $0.9983$  $0.9987$  $\mathbf{0.9996}$  $0.9981$  $0.9985$  $\mathbf{0.9987}$ 
RMSE  QR  TV  HHBM 

180 proj  0.0581  0.0540  0.0558 
90 proj  0.0655  0.0573  0.0610 
60 proj  0.0846  0.0675  0.0690 
45 proj  0.1079  0.0830  0.0783 
36 proj  0.1326  0.1027  0.0882 
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Wang, L.; MohammadDjafari, A.; Gac, N.; Dumitru, M. Bayesian 3D Xray Computed Tomography with a Hierarchical Prior Model for Sparsity in Haar Transform Domain. Entropy 2018, 20, 977. https://doi.org/10.3390/e20120977
Wang L, MohammadDjafari A, Gac N, Dumitru M. Bayesian 3D Xray Computed Tomography with a Hierarchical Prior Model for Sparsity in Haar Transform Domain. Entropy. 2018; 20(12):977. https://doi.org/10.3390/e20120977
Chicago/Turabian StyleWang, Li, Ali MohammadDjafari, Nicolas Gac, and Mircea Dumitru. 2018. "Bayesian 3D Xray Computed Tomography with a Hierarchical Prior Model for Sparsity in Haar Transform Domain" Entropy 20, no. 12: 977. https://doi.org/10.3390/e20120977