Quantitative Comparison of Deep LearningBased Image Reconstruction Methods for LowDose and SparseAngle CT Applications
Abstract
:1. Introduction
1.1. Goal of This Study
1.1.1. Reconstruction of LowDose Medical CT Images
1.1.2. Reconstruction of SparseAngle CT Images
2. Dataset Description
2.1. LoDoPaBCT Dataset
2.2. Apple CT Datasets
3. Algorithms
3.1. Learned Reconstruction Methods
3.1.1. PostProcessing
3.1.2. Fully Learned
3.1.3. Learned Iterative Schemes
3.1.4. Generative Approach
3.1.5. Unsupervised Methods
3.2. Classical Reconstruction Methods
4. Evaluation Methodology
4.1. Evaluation Metrics
4.1.1. Peak SignaltoNoise Ratio
 PSNR: In this case $L=max\left({x}^{\u2020}\right)min\left({x}^{\u2020}\right)$, that is, the difference between the highest and lowest entry in ${x}^{\u2020}$. This allows for a PSNR value that is adapted to the range of the current ground truth image. The disadvantage is that the PSNR is imagedependent in this case.
 PSNRFR: The same fixed L is chosen for all images. It is determined as the maximum entry computed over all training ground truth images, that is, $L=1.0$ for LoDoPaBCT and $L=0.0129353$ for the Apple CT datasets. This can be seen as an (empirical) upper limit of the intensity range in the ground truth. In general, a fixed L is preferable because the scaling of the metric is imageindependent in this case. This allows for a direct comparison of PSNR values calculated on different images. The downside for most CT applications is, that high values ($\widehat{=}$ dense material) are not present in every scan. Therefore, the results can be too optimistic for these scans. However, based on Equation (7), all mean PSNRFR values can be directly converted for another fixed choice of L.
4.1.2. Structural Similarity
4.1.3. Data Discrepancy
Poisson Regression Loss on LoDoPaBCT Dataset
Mean Squared Error on Apple CT Data
4.2. Training Procedure
5. Results
5.1. LoDoPaBCT Dataset
5.1.1. Reconstruction Performance
5.1.2. Visual Comparison
5.1.3. Data Consistency
5.2. Apple CT Datasets
5.2.1. Reconstruction Performance
5.2.2. Visual Comparison
5.2.3. Data Consistency
6. Discussion
6.1. Computational Requirements and Reconstruction Speed
Transfer to 3D Reconstruction
6.2. Impact of the Datasets
6.2.1. Number of Training Samples
6.2.2. Observations on LoDoPaBCT and Apple CT
6.2.3. Robustness to Changes in the Scanning Setup
6.2.4. Generalization to Other CT Setups
6.3. Conformance of Image Quality Scores and Requirements in Real Applications
6.4. Impact of Data Consistency
6.5. Recommendations and Future Work
7. Conclusions
Supplementary Materials
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Learned Reconstruction Methods
Appendix A.1. Learned PrimalDual
Algorithm A1 Learned PrimalDual. 
Given learned proximal dual and primal operators ${\mathsf{\Gamma}}_{{\theta}_{k}^{d}},{\mathsf{\Lambda}}_{{\theta}_{k}^{p}}$ for $k=1,\dots ,K$ the reconstruction from noisy measurements ${y}_{\delta}$ is calculated as follows.

Appendix A.2. UNet
Appendix A.3. UNet++
Appendix A.4. MixedScale Dense Convolutional Neural Network
Appendix A.5. Conditional Invertible Neural Networks
Algorithm A2 Conditional Invertible Neural Network (CINN). 
Given a noisy measurement, ${y}_{\delta}$, an invertible neural network F and a conditioning network C. Let $K\in \mathbb{N}$ be the number of random samples that should be drawn from a normal distribution $\mathcal{N}\left(\right)open="("\; close=")">0,\mathrm{I}$. The algorithm calculates the mean and variance of the conditioned reconstructions.

Appendix A.6. ISTA UNet
Algorithm A3 ISTA UNet. 
Given a noisy input ${y}_{\delta}$, learned dictionaries ${D}_{\kappa},{D}_{\theta},{D}_{\gamma}$ and learned step sizes η and λ the reconstruction using the ISTA UNet can be computed as follows.

Appendix A.7. Deep Image Prior with TV Denoising
Algorithm A4 Deep Image Prior + Total Variation (DIP + TV). 
Given a noisy measurement ${y}_{\delta}$, a neural network ${F}_{\mathsf{\Theta}}$ with initial parameterization ${\mathsf{\Theta}}^{\left[0\right]}$, forward operator $\mathcal{A}$ and a fixed random input z. The reconstruction $\widehat{x}$ is calculated iteratively over a number of $K\in \mathbb{N}$ iterations:

Appendix A.8. iCTUNet
Appendix B. Classical Reconstruction Methods
Appendix B.1. Filtered BackProjection (FBP)
Window  Frequency Scaling  

LoDoPaBCT Dataset  Hann  0.641  
Apple CT Dataset A (Noisefree)  50 angles  Cosine  0.11 
10 angles  Cosine  0.013  
5 angles  Hann  0.011  
2 angles  Hann  0.011  
Apple CT Dataset B (Gaussian noise)  50 angles  Cosine  0.08 
10 angles  Cosine  0.013  
5 angles  Hann  0.011  
2 angles  Hann  0.011  
Apple CT Dataset C (Scattering)  50 angles  Cosine  0.09 
10 angles  Hann  0.018  
5 angles  Hann  0.011  
2 angles  Hann  0.009 
Appendix B.2. Conjugate Gradient Least Squares
Algorithm A5 Conjugate Gradient Least Squares (CGLS). 
Given a geometry matrix, A, a data vector ${y}_{}$ and a zero solution vector ${\widehat{x}}^{\left[0\right]}=0$ (a black image) as the starting point, the algorithm below gives the solution at k^{th} iteration.

Appendix B.3. Total Variation Regularization
Discrepancy  Iterations  Step Size  $\mathit{\alpha}$  

LoDoPaBCT Dataset  ${\ell}_{\mathrm{Pois}}$  5000  0.001  20.56  
Apple CT Dataset A (Noisefree)  50 angles  MSE  600  $3\times {10}^{2}$  $2\times {10}^{12}$ 
10 angles  MSE  75,000  $3\times {10}^{3}$  $6\times {10}^{12}$  
5 angles  MSE  146,000  $1.5\times {10}^{3}$  $1\times {10}^{11}$  
2 angles  MSE  150,000  $1\times {10}^{3}$  $2\times {10}^{11}$  
Apple CT Dataset B (Gaussian noise)  50 angles  MSE  900  $3\times {10}^{4}$  $2\times {10}^{10}$ 
10 angles  MSE  66,000  $2\times {10}^{5}$  $6\times {10}^{10}$  
5 angles  MSE  100,000  $1\times {10}^{5}$  $3\times {10}^{9}$  
2 angles  MSE  149,000  $1\times {10}^{5}$  $4\times {10}^{9}$  
Apple CT Dataset C (Scattering)  50 angles  MSE  400  $5\times {10}^{3}$  $1\times {10}^{11}$ 
10 angles  MSE  13,000  $2\times {10}^{3}$  $4\times {10}^{11}$  
5 angles  MSE  149,000  $1\times {10}^{3}$  $4\times {10}^{11}$  
2 angles  MSE  150,000  $4\times {10}^{4}$  $6\times {10}^{11}$ 
Algorithm A6 Total Variation Regularization (TV). 
Given a noisy measurement ${y}_{\delta}$, an initial reconstruction ${\widehat{x}}^{\left[0\right]}$, a weight $\alpha >0$ and a maximum number of iterations K.

Appendix C. Further Results
NoiseFree  Standard Deviation of PSNR  Standard Deviation of SSIM  

Number of Angles  50  10  5  2  50  10  5  2 
Learned PrimalDual  1.51  1.63  1.97  2.58  0.022  0.016  0.014  0.022 
ISTA UNet  1.40  1.77  2.12  2.13  0.018  0.018  0.022  0.037 
UNet  1.56  1.61  2.28  1.63  0.021  0.019  0.025  0.031 
MSDCNN  1.51  1.65  1.81  2.09  0.021  0.020  0.024  0.022 
CINN  1.40  1.64  1.99  2.17  0.016  0.019  0.023  0.027 
iCTUNet  1.68  2.45  1.92  1.93  0.024  0.027  0.030  0.028 
TV  1.60  1.29  1.21  1.49  0.022  0.041  0.029  0.023 
CGLS  0.69  0.48  2.94  0.70  0.014  0.027  0.029  0.039 
FBP  0.80  0.58  0.54  0.50  0.021  0.023  0.028  0.067 
Gaussian Noise  Standard Deviation of PSNR  Standard Deviation of SSIM  
Number of Angles  50  10  5  2  50  10  5  2 
Learned PrimalDual  1.56  1.63  2.00  2.79  0.021  0.018  0.021  0.022 
ISTA UNet  1.70  1.76  2.27  2.12  0.025  0.021  0.022  0.038 
UNet  1.66  1.59  1.99  2.22  0.023  0.020  0.025  0.026 
MSDCNN  1.66  1.75  1.79  1.79  0.025  0.024  0.019  0.022 
CINN  1.53  1.51  1.62  2.06  0.023  0.017  0.017  0.020 
iCTUNet  1.98  2.06  1.89  1.91  0.031  0.032  0.039  0.027 
TV  1.38  1.26  1.09  1.62  0.036  0.047  0.039  0.030 
CGLS  0.78  0.49  1.76  0.68  0.014  0.026  0.029  0.037 
FBP  0.91  0.58  0.54  0.50  0.028  0.023  0.028  0.067 
Scattering Noise  Standard Deviation of PSNR  Standard Deviation of SSIM  
Number of Angles  50  10  5  2  50  10  5  2 
Learned PrimalDual  1.91  1.80  1.71  2.47  0.017  0.016  0.016  0.060 
ISTA UNet  1.48  1.59  2.05  1.81  0.023  0.019  0.019  0.038 
UNet  1.76  1.56  1.81  1.47  0.015  0.021  0.027  0.024 
MSDCNN  2.04  1.78  1.85  2.03  0.023  0.022  0.015  0.020 
CINN  1.82  1.92  2.32  2.25  0.019  0.024  0.029  0.030 
iCTUNet  1.91  2.09  1.78  2.29  0.030  0.031  0.033  0.040 
TV  2.53  2.44  1.86  1.59  0.067  0.076  0.035  0.062 
CGLS  2.38  1.32  1.71  0.95  0.020  0.020  0.026  0.032 
FBP  2.23  0.97  0.80  0.68  0.044  0.025  0.023  0.058 
NoiseFree  PSNRFR  SSIMFR  

Number of Angles  50  10  5  2  50  10  5  2 
Learned PrimalDual  45.33  42.47  37.41  28.61  0.971  0.957  0.935  0.872 
ISTA UNet  45.48  41.15  34.93  27.10  0.967  0.944  0.907  0.823 
UNet  46.24  40.13  34.38  26.39  0.975  0.917  0.911  0.830 
MSDCNN  46.47  41.00  35.06  27.17  0.975  0.936  0.898  0.808 
CINN  46.20  41.46  34.43  26.07  0.975  0.958  0.896  0.838 
iCTUNet  42.69  36.57  32.24  25.90  0.957  0.938  0.920  0.861 
TV  45.89  35.61  28.66  22.57  0.976  0.904  0.746  0.786 
CGLS  39.66  28.43  19.22  21.87  0.901  0.744  0.654  0.733 
FBP  37.01  23.71  22.12  20.58  0.856  0.711  0.596  0.538 
Gaussian Noise  PSNRFR  SSIMFR  
Number of Angles  50  10  5  2  50  10  5  2 
Learned PrimalDual  43.24  40.38  36.54  28.03  0.961  0.944  0.927  0.823 
ISTA UNet  42.65  40.17  35.09  27.32  0.956  0.942  0.916  0.826 
UNet  43.09  39.45  34.42  26.47  0.961  0.924  0.904  0.843 
MSDCNN  43.28  39.82  34.60  26.50  0.962  0.932  0.886  0.797 
CINN  43.39  38.50  33.19  26.60  0.966  0.904  0.878  0.816 
iCTUNet  39.51  36.38  31.29  26.06  0.939  0.932  0.905  0.867 
TV  38.98  33.73  28.45  22.70  0.939  0.883  0.770  0.772 
CGLS  33.98  27.71  21.52  21.73  0.884  0.748  0.668  0.734 
FBP  34.50  23.70  22.12  20.58  0.839  0.711  0.596  0.538 
Scattering Noise  PSNRFR  SSIMFR  
Number of Angles  50  10  5  2  50  10  5  2 
Learned PrimalDual  44.42  40.80  33.69  27.60  0.967  0.954  0.912  0.760 
ISTA UNet  42.55  38.95  34.03  26.57  0.959  0.922  0.887  0.816 
UNet  41.58  39.52  33.55  25.56  0.932  0.910  0.877  0.828 
MSDCNN  44.66  40.13  34.34  26.81  0.969  0.927  0.889  0.796 
CINN  45.18  40.69  34.66  25.76  0.976  0.952  0.936  0.878 
iCTUNet  32.88  29.46  27.86  24.93  0.931  0.901  0.896  0.873 
TV  27.71  26.76  24.48  21.15  0.903  0.799  0.674  0.743 
CGLS  27.46  24.89  20.64  20.80  0.896  0.738  0.659  0.736 
FBP  27.63  22.42  20.88  19.68  0.878  0.701  0.589  0.529 
NoiseFree  Standard Deviation of PSNRFR  Standard Deviation of SSIMFR  

Number of Angles  50  10  5  2  50  10  5  2 
Learned PrimalDual  1.49  1.67  2.03  2.54  0.007  0.006  0.010  0.019 
ISTA UNet  1.37  1.82  2.21  2.21  0.005  0.010  0.020  0.034 
UNet  1.53  1.66  2.33  1.68  0.006  0.012  0.019  0.026 
MSDCNN  1.46  1.71  1.90  2.15  0.006  0.011  0.021  0.015 
CINN  1.35  1.65  2.09  2.21  0.004  0.007  0.023  0.025 
iCTUNet  1.82  2.54  2.03  1.91  0.014  0.017  0.020  0.023 
TV  1.54  1.32  1.28  1.36  0.006  0.023  0.026  0.018 
CGLS  0.71  0.51  2.96  0.56  0.009  0.029  0.033  0.045 
FBP  0.77  0.46  0.38  0.41  0.011  0.015  0.029  0.088 
Gaussian Noise  Standard Deviation of PSNRFR  Standard Deviation of SSIMFR  
Number of Angles  50  10  5  2  50  10  5  2 
Learned PrimalDual  1.52  1.68  2.04  2.83  0.006  0.008  0.013  0.016 
ISTA UNet  1.65  1.78  2.36  2.17  0.008  0.010  0.018  0.034 
UNet  1.61  1.62  2.05  2.24  0.007  0.012  0.019  0.024 
MSDCNN  1.62  1.80  1.84  1.84  0.008  0.011  0.015  0.014 
CINN  1.50  1.59  1.65  2.09  0.007  0.016  0.017  0.019 
iCTUNet  2.07  2.12  1.93  1.90  0.020  0.021  0.026  0.024 
TV  1.30  1.26  1.15  1.50  0.014  0.027  0.030  0.019 
CGLS  0.63  0.45  1.76  0.53  0.012  0.028  0.034  0.043 
FBP  0.83  0.46  0.38  0.41  0.014  0.015  0.029  0.088 
Scattering Noise  Standard Deviation of PSNRFR  Standard Deviation of SSIMFR  
Number of Angles  50  10  5  2  50  10  5  2 
Learned PrimalDual  1.92  1.85  1.81  2.51  0.005  0.007  0.014  0.038 
ISTA UNet  1.56  1.68  2.17  1.89  0.010  0.014  0.014  0.035 
UNet  1.72  1.63  1.91  1.59  0.010  0.012  0.024  0.024 
MSDCNN  2.02  1.84  1.96  2.08  0.008  0.012  0.016  0.019 
CINN  1.74  1.97  2.41  2.21  0.005  0.011  0.016  0.022 
iCTUNet  1.96  2.14  1.79  2.32  0.016  0.023  0.022  0.030 
TV  2.43  2.35  1.80  1.49  0.048  0.074  0.040  0.051 
CGLS  2.28  1.24  1.67  0.83  0.016  0.021  0.030  0.035 
FBP  2.14  0.87  0.66  0.55  0.028  0.016  0.020  0.078 
Noise Free  MSE $\times {10}^{9}$  

Number of Angles  50  10  5  2 
Learned PrimalDual  $0.083\pm 0.027$  $0.405\pm 0.156$  $1.559\pm 0.543$  $2.044\pm 1.177$ 
ISTA UNet  $0.323\pm 0.240$  $0.633\pm 0.339$  $2.672\pm 1.636$  $17.840\pm 12.125$ 
UNet  $0.097\pm 0.093$  $1.518\pm 0.707$  $5.011\pm 3.218$  $31.885\pm 17.219$ 
MSDCNN  $0.117\pm 0.088$  $0.996\pm 0.595$  $3.874\pm 2.567$  $20.879\pm 12.038$ 
CINN  $0.237\pm 0.259$  $1.759\pm 0.348$  $3.798\pm 2.176$  $33.676\pm 16.747$ 
iCTUNet  $2.599\pm 3.505$  $6.686\pm 8.469$  $14.508\pm 16.694$  $18.876\pm 12.553$ 
TV  $0.002\pm 0.000$  $0.001\pm 0.000$  $0.000\pm 0.000$  $0.001\pm 0.000$ 
CGLS  $1.449\pm 0.299$  $29.921\pm 6.173$  $752.997\pm 722.151$  $22.507\pm 13.748$ 
FBP  $12.229\pm 3.723$  $89.958\pm 9.295$  $159.746\pm 15.596$  $273.054\pm 114.552$ 
Ground truth  $0.000\pm 0.000$  $0.000\pm 0.000$  $0.000\pm 0.000$  $0.000\pm 0.000$ 
Gaussian Noise  MSE$\times {\mathbf{10}}^{\mathbf{9}}$  
Number of Angles  50  10  5  2 
Learned PrimalDual  $19.488\pm 5.923$  $19.813\pm 5.851$  $20.582\pm 5.690$  $32.518\pm 4.286$ 
ISTA UNet  $19.438\pm 5.943$  $20.178\pm 6.060$  $21.167\pm 6.052$  $32.435\pm 9.782$ 
UNet  $19.802\pm 6.247$  $22.114\pm 6.364$  $23.645\pm 6.527$  $38.895\pm 17.211$ 
MSDCNN  $19.348\pm 5.921$  $20.056\pm 5.930$  $23.080\pm 5.959$  $47.625\pm 18.133$ 
CINN  $19.429\pm 5.891$  $21.069\pm 5.663$  $29.517\pm 7.296$  $42.876\pm 15.471$ 
iCTUNet  $25.645\pm 9.602$  $25.421\pm 9.976$  $38.179\pm 22.887$  $41.956\pm 15.942$ 
TV  $18.760\pm 5.674$  $18.107\pm 5.395$  $20.837\pm 5.510$  $18.514\pm 5.688$ 
CGLS  $87.892\pm 23.312$  $71.526\pm 17.600$  $262.616\pm 151.655$  $98.520\pm 18.245$ 
FBP  $31.803\pm 9.558$  $109.430\pm 14.107$  $179.260\pm 19.744$  $292.692\pm 109.223$ 
Ground truth  $19.538\pm 6.029$  $19.505\pm 6.019$  $19.551\pm 6.028$  $19.483\pm 6.086$ 
Scattering Noise  MSE$\times {\mathbf{10}}^{\mathbf{9}}$  
Number of Angles  50  10  5  2 
Learned PrimalDual  $541.30\pm 311.82$  $579.14\pm 317.59$  $549.30\pm 328.41$  $435.07\pm 260.02$ 
ISTA UNet  $553.64\pm 355.14$  $557.03\pm 342.67$  $575.94\pm 338.82$  $522.33\pm 365.58$ 
UNet  $629.62\pm 353.54$  $635.91\pm 343.31$  $550.54\pm 340.27$  $642.20\pm 295.46$ 
MSDCNN  $579.86\pm 332.39$  $585.18\pm 331.93$  $533.35\pm 331.21$  $606.55\pm 365.25$ 
CINN  $638.80\pm 355.24$  $619.47\pm 353.47$  $603.53\pm 362.96$  $649.30\pm 409.83$ 
iCTUNet  $622.51\pm 348.32$  $622.63\pm 335.28$  $652.18\pm 359.00$  $573.46\pm 324.00$ 
TV  $3.35\pm 5.02$  $3.19\pm 4.83$  $2.96\pm 4.47$  $2.55\pm 6.33$ 
CGLS  $6.40\pm 6.39$  $34.71\pm 8.16$  $286.20\pm 205.42$  $19.92\pm 14.01$ 
FBP  $12.48\pm 6.88$  $73.53\pm 10.19$  $144.70\pm 15.82$  $221.79\pm 59.71$ 
Ground truth  $610.47\pm 355.25$  $610.40\pm 355.16$  $611.23\pm 354.51$  $620.11\pm 386.79$ 
Appendix D. Training Curves
References
 Liguori, C.; Frauenfelder, G.; Massaroni, C.; Saccomandi, P.; Giurazza, F.; Pitocco, F.; Marano, R.; Schena, E. Emerging clinical applications of computed tomography. Med. Devices 2015, 8, 265. [Google Scholar]
 National Lung Screening Trial Research Team. Reduced lungcancer mortality with lowdose computed tomographic screening. N. Engl. J. Med. 2011, 365, 395–409. [Google Scholar] [CrossRef][Green Version]
 Yoo, S.; Yin, F.F. Dosimetric feasibility of conebeam CTbased treatment planning compared to CTbased treatment planning. Int. J. Radiat. Oncol. Biol. Phys. 2006, 66, 1553–1561. [Google Scholar] [CrossRef]
 Swennen, G.R.; Mollemans, W.; Schutyser, F. Threedimensional treatment planning of orthognathic surgery in the era of virtual imaging. J. Oral Maxillofac. Surg. 2009, 67, 2080–2092. [Google Scholar] [CrossRef]
 De Chiffre, L.; Carmignato, S.; Kruth, J.P.; Schmitt, R.; Weckenmann, A. Industrial applications of computed tomography. CIRP Ann. 2014, 63, 655–677. [Google Scholar] [CrossRef]
 Mees, F.; Swennen, R.; Van Geet, M.; Jacobs, P. Applications of Xray Computed Tomography in the Geosciences; Special Publications; Geological Society: London, UK, 2003; Volume 215, pp. 1–6. [Google Scholar]
 Morigi, M.; Casali, F.; Bettuzzi, M.; Brancaccio, R.; d’Errico, V. Application of Xray computed tomography to cultural heritage diagnostics. Appl. Phys. A 2010, 100, 653–661. [Google Scholar] [CrossRef]
 Coban, S.B.; Lucka, F.; Palenstijn, W.J.; Van Loo, D.; Batenburg, K.J. Explorative Imaging and Its Implementation at the FleXray Laboratory. J. Imaging 2020, 6, 18. [Google Scholar] [CrossRef][Green Version]
 McCollough, C.H.; Bartley, A.C.; Carter, R.E.; Chen, B.; Drees, T.A.; Edwards, P.; Holmes, D.R., III.; Huang, A.E.; Khan, F.; Leng, S.; et al. Lowdose CT for the detection and classification of metastatic liver lesions: Results of the 2016 Low Dose CT Grand Challenge. Med Phys. 2017, 44, e339–e352. [Google Scholar] [CrossRef] [PubMed][Green Version]
 Radon, J. On the determination of functions from their integral values along certain manifolds. IEEE Trans. Med Imaging 1986, 5, 170–176. [Google Scholar] [CrossRef]
 Natterer, F. The mathematics of computerized tomography (classics in applied mathematics, vol. 32). Inverse Probl. 2001, 18, 283–284. [Google Scholar]
 Boas, F.E.; Fleischmann, D. CT artifacts: Causes and reduction techniques. Imaging Med. 2012, 4, 229–240. [Google Scholar] [CrossRef][Green Version]
 Wang, G.; Ye, J.C.; Mueller, K.; Fessler, J.A. Image Reconstruction is a New Frontier of Machine Learning. IEEE Trans. Med. Imaging 2018, 37, 1289–1296. [Google Scholar] [CrossRef] [PubMed]
 Sidky, E.Y.; Pan, X. Image reconstruction in circular conebeam computed tomography by constrained, totalvariation minimization. Phys. Med. Biol. 2008, 53, 4777. [Google Scholar] [CrossRef][Green Version]
 Niu, S.; Gao, Y.; Bian, Z.; Huang, J.; Chen, W.; Yu, G.; Liang, Z.; Ma, J. Sparseview Xray CT reconstruction via total generalized variation regularization. Phys. Med. Biol. 2014, 59, 2997. [Google Scholar] [CrossRef] [PubMed]
 Hestenes, M.R.; Stiefel, E. Methods of conjugate gradients for solving linear systems. J. Res. Natl. Bur. Stand. 1952, 49, 409–436. [Google Scholar] [CrossRef]
 Arridge, S.; Maass, P.; Öktem, O.; Schönlieb, C.B. Solving inverse problems using datadriven models. Acta Numer. 2019, 28, 1–174. [Google Scholar] [CrossRef][Green Version]
 Lunz, S.; Öktem, O.; Schönlieb, C.B. Adversarial Regularizers in Inverse Problems. In Advances in Neural Information Processing Systems; Bengio, S., Wallach, H., Larochelle, H., Grauman, K., CesaBianchi, N., Garnett, R., Eds.; Curran Associates, Inc.: Red Hook, NY, USA, 2018; Volume 31, pp. 8507–8516. [Google Scholar]
 Adler, J.; Öktem, O. Learned PrimalDual Reconstruction. IEEE Trans. Med. Imaging 2018, 37, 1322–1332. [Google Scholar] [CrossRef][Green Version]
 Jin, K.H.; McCann, M.T.; Froustey, E.; Unser, M. Deep Convolutional Neural Network for Inverse Problems in Imaging. IEEE Trans. Image Process. 2017, 26, 4509–4522. [Google Scholar] [CrossRef][Green Version]
 Pelt, D.M.; Batenburg, K.J.; Sethian, J.A. Improving Tomographic Reconstruction from Limited Data Using MixedScale Dense Convolutional Neural Networks. J. Imaging 2018, 4, 128. [Google Scholar] [CrossRef][Green Version]
 Chen, H.; Zhang, Y.; Zhang, W.; Liao, P.; Li, K.; Zhou, J.; Wang, G. Lowdose CT via convolutional neural network. Biomed. Opt. Express 2017, 8, 679–694. [Google Scholar] [CrossRef]
 Chen, H.; Zhang, Y.; Kalra, M.K.; Lin, F.; Chen, Y.; Liao, P.; Zhou, J.; Wang, G. Lowdose CT with a residual encoderdecoder convolutional neural network. IEEE Trans. Med. Imaging 2017, 36, 2524–2535. [Google Scholar] [CrossRef]
 Yang, Q.; Yan, P.; Kalra, M.K.; Wang, G. CT image denoising with perceptive deep neural networks. arXiv 2017, arXiv:1702.07019. [Google Scholar]
 Yang, Q.; Yan, P.; Zhang, Y.; Yu, H.; Shi, Y.; Mou, X.; Kalra, M.K.; Zhang, Y.; Sun, L.; Wang, G. Lowdose CT image denoising using a generative adversarial network with Wasserstein distance and perceptual loss. IEEE Trans. Med. Imaging 2018, 37, 1348–1357. [Google Scholar] [CrossRef]
 Feng, R.; Rundle, D.; Wang, G. Neuralnetworksbased PhotonCounting Data Correction: Pulse Pileup Effect. arXiv 2018, arXiv:1804.10980. [Google Scholar]
 Zhu, B.; Liu, J.Z.; Cauley, S.F.; Rosen, B.R.; Rosen, M.S. Image reconstruction by domaintransform manifold learning. Nature 2018, 555, 487–492. [Google Scholar] [CrossRef][Green Version]
 He, J.; Ma, J. Radon inversion via deep learning. IEEE Trans. Med. Imaging 2020, 39, 2076–2087. [Google Scholar] [CrossRef] [PubMed][Green Version]
 Li, Y.; Li, K.; Zhang, C.; Montoya, J.; Chen, G.H. Learning to reconstruct computed tomography images directly from sinogram data under a variety of data acquisition conditions. IEEE Trans. Med Imaging 2019, 38, 2469–2481. [Google Scholar] [CrossRef]
 European Society of Radiology (ESR). The new EU General Data Protection Regulation: What the radiologist should know. Insights Imaging 2017, 8, 295–299. [Google Scholar] [CrossRef] [PubMed][Green Version]
 Kaissis, G.A.; Makowski, M.R.; Rückert, D.; Braren, R.F. Secure, privacypreserving and federated machine learning in medical imaging. Nat. Mach. Intell. 2020, 2, 305–311. [Google Scholar] [CrossRef]
 Leuschner, J.; Schmidt, M.; Baguer, D.O.; Maass, P. The LoDoPaBCT Dataset: A Benchmark Dataset for LowDose CT Reconstruction Methods. arXiv 2020, arXiv:1910.01113. [Google Scholar]
 Armato, S.G., III; McLennan, G.; Bidaut, L.; McNittGray, M.F.; Meyer, C.R.; Reeves, A.P.; Zhao, B.; Aberle, D.R.; Henschke, C.I.; Hoffman, E.A.; et al. The Lung Image Database Consortium (LIDC) and Image Database Resource Initiative (IDRI): A Completed Reference Database of Lung Nodules on CT Scans. Med. Phys. 2011, 38, 915–931. [Google Scholar] [CrossRef]
 Baguer, D.O.; Leuschner, J.; Schmidt, M. Computed tomography reconstruction using deep image prior and learned reconstruction methods. Inverse Probl. 2020, 36, 094004. [Google Scholar] [CrossRef]
 Armato, S.G., III; McLennan, G.; Bidaut, L.; McNittGray, M.F.; Meyer, C.R.; Reeves, A.P.; Zhao, B.; Aberle, D.R.; Henschke, C.I.; Hoffman, E.A.; et al. Data From LIDCIDRI; The Cancer Imaging Archive: Frederick, MD, USA, 2015. [Google Scholar] [CrossRef]
 Buzug, T. Computed Tomography: From Photon Statistics to Modern ConeBeam CT; Springer: Berlin/Heidelberg, Germany, 2008. [Google Scholar] [CrossRef]
 Coban, S.B.; Andriiashen, V.; Ganguly, P.S. Apple CT Data: Simulated ParallelBeam Tomographic Datasets; Zenodo: Geneva, Switzerland, 2020. [Google Scholar] [CrossRef]
 Coban, S.B.; Andriiashen, V.; Ganguly, P.S.; van Eijnatten, M.; Batenburg, K.J. Parallelbeam Xray CT datasets of apples with internal defects and label balancing for machine learning. arXiv 2020, arXiv:2012.13346. [Google Scholar]
 Leuschner, J.; Schmidt, M.; Ganguly, P.S.; Andriiashen, V.; Coban, S.B.; Denker, A.; van Eijnatten, M. Source Code and Supplementary Material for “Quantitative comparison of deep learningbased image reconstruction methods for lowdose and sparseangle CT applications”. Zenodo 2021. [Google Scholar] [CrossRef]
 Ronneberger, O.; Fischer, P.; Brox, T. Unet: Convolutional networks for biomedical image segmentation. In International Conference on Medical Image Computing and ComputerAssisted Intervention; Springer: Heidelberg, Germany, 2015; pp. 234–241. [Google Scholar]
 Zhou, Z.; Siddiquee, M.M.R.; Tajbakhsh, N.; Liang, J. Unet++: A nested unet architecture for medical image segmentation. In Deep Learning in Medical Image Analysis and Multimodal Learning for Clinical Decision Support; Springer: Heidelberg, Germany, 2018; pp. 3–11. [Google Scholar]
 Liu, T.; Chaman, A.; Belius, D.; Dokmanić, I. Interpreting UNets via TaskDriven Multiscale Dictionary Learning. arXiv 2020, arXiv:2011.12815. [Google Scholar]
 Comelli, A.; Dahiya, N.; Stefano, A.; Benfante, V.; Gentile, G.; Agnese, V.; Raffa, G.M.; Pilato, M.; Yezzi, A.; Petrucci, G.; et al. Deep learning approach for the segmentation of aneurysmal ascending aorta. Biomed. Eng. Lett. 2020, 1–10. [Google Scholar]
 Dashti, M.; Stuart, A.M. The Bayesian Approach to Inverse Problems. In Handbook of Uncertainty Quantification; Springer International Publishing: Cham, Switzerland, 2017; pp. 311–428. [Google Scholar] [CrossRef][Green Version]
 Adler, J.; Öktem, O. Deep Bayesian Inversion. arXiv 2018, arXiv:1811.05910. [Google Scholar]
 Goodfellow, I.; PougetAbadie, J.; Mirza, M.; Xu, B.; WardeFarley, D.; Ozair, S.; Courville, A.; Bengio, Y. Generative adversarial nets. arXiv 2014, arXiv:1406.2661. [Google Scholar]
 Ardizzone, L.; Lüth, C.; Kruse, J.; Rother, C.; Köthe, U. Guided image generation with conditional invertible neural networks. arXiv 2019, arXiv:1907.02392. [Google Scholar]
 Denker, A.; Schmidt, M.; Leuschner, J.; Maass, P.; Behrmann, J. Conditional Normalizing Flows for LowDose Computed Tomography Image Reconstruction. arXiv 2020, arXiv:2006.06270. [Google Scholar]
 Hadamard, J. Lectures on Cauchy’s Problem in Linear Partial Differential Equations; Dover: New York, NY, USA, 1952. [Google Scholar]
 Nashed, M. A new approach to classification and regularization of illposed operator equations. In Inverse and IllPosed Problems; Engl, H.W., Groetsch, C., Eds.; Academic Press: Cambridge, MA, USA, 1987; pp. 53–75. [Google Scholar] [CrossRef]
 Natterer, F.; Wübbeling, F. Mathematical Methods in Image Reconstruction; Society for Industrial and Applied Mathematics: Philadelphia, PA, USA, 2001. [Google Scholar]
 Saad, Y. Iterative Methods for Sparse Linear Systems, 2nd ed.; Society for Industrial and Applied Mathematics: Philadelphia, PA, USA, 2003. [Google Scholar]
 Björck, Å.; Elfving, T.; Strakos, Z. Stability of conjugate gradient and Lanczos methods for linear least squares problems. SIAM J. Matrix Anal. Appl. 1998, 19, 720–736. [Google Scholar] [CrossRef]
 Chen, H.; Wang, C.; Song, Y.; Li, Z. Split Bregmanized anisotropic total variation model for image deblurring. J. Vis. Commun. Image Represent. 2015, 31, 282–293. [Google Scholar] [CrossRef]
 Wang, Z.; Bovik, A.C.; Sheikh, H.R.; Simoncelli, E.P. Image quality assessment: From error visibility to structural similarity. IEEE Trans. Image Process. 2004, 13, 600–612. [Google Scholar] [CrossRef] [PubMed][Green Version]
 Goodfellow, I.; Bengio, Y.; Courville, A. Deep Learning; MIT Press: Cambridge, MA, USA, 2016; Available online: http://www.deeplearningbook.org (accessed on 1 March 2021).
 Leuschner, J.; Schmidt, M.; Ganguly, P.S.; Andriiashen, V.; Coban, S.B.; Denker, A.; van Eijnatten, M. Supplementary Material for Experiments in “Quantitative comparison of deep learningbased image reconstruction methods for lowdose and sparseangle CT applications”. Zenodo 2021. [Google Scholar] [CrossRef]
 Leuschner, J.; Schmidt, M.; Baguer, D.O.; Bauer, D.; Denker, A.; Hadjifaradji, A.; Liu, T. LoDoPaBCT Challenge Reconstructions compared in “Quantitative comparison of deep learningbased image reconstruction methods for lowdose and sparseangle CT applications”. Zenodo 2021. [Google Scholar] [CrossRef]
 Leuschner, J.; Schmidt, M.; Ganguly, P.S.; Andriiashen, V.; Coban, S.B.; Denker, A.; van Eijnatten, M. Apple CT Test Reconstructions compared in “Quantitative comparison of deep learningbased image reconstruction methods for lowdose and sparseangle CT applications”. Zenodo 2021. [Google Scholar] [CrossRef]
 Leuschner, J.; Schmidt, M.; Otero Baguer, D.; Erzmann, D.; Baltazar, M. DIVal Library. Zenodo 2021. [Google Scholar] [CrossRef]
 Knoll, F.; Murrell, T.; Sriram, A.; Yakubova, N.; Zbontar, J.; Rabbat, M.; Defazio, A.; Muckley, M.J.; Sodickson, D.K.; Zitnick, C.L.; et al. Advancing machine learning for MR image reconstruction with an open competition: Overview of the 2019 fastMRI challenge. Magn. Reson. Med. 2020, 84, 3054–3070. [Google Scholar] [CrossRef]
 Putzky, P.; Welling, M. Invert to Learn to Invert. In Advances in Neural Information Processing Systems; Wallach, H., Larochelle, H., Beygelzimer, A., dAlch’eBuc, F., Fox, E., Garnett, R., Eds.; Curran Associates, Inc.: Red Hook, NY, USA, 2019; Volume 32, pp. 446–456. [Google Scholar]
 Etmann, C.; Ke, R.; Schönlieb, C. iUNets: Learnable Invertible Up and Downsampling for LargeScale Inverse Problems. In Proceedings of the 30th IEEE International Workshop on Machine Learning for Signal Processing (MLSP 2020), Espoo, Finland, 21–24 September 2020; pp. 1–6. [Google Scholar] [CrossRef]
 Ziabari, A.; Ye, D.H.; Srivastava, S.; Sauer, K.D.; Thibault, J.; Bouman, C.A. 2.5D Deep Learning For CT Image Reconstruction Using A MultiGPU Implementation. In Proceedings of the 2018 52nd Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, CA, USA, 28–31 October 2018; pp. 2044–2049. [Google Scholar] [CrossRef][Green Version]
 Scherzer, O.; Weickert, J. Relations Between Regularization and Diffusion Filtering. J. Math. Imaging Vis. 2000, 12, 43–63. [Google Scholar] [CrossRef]
 Perona, P.; Malik, J. Scalespace and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 1990, 12, 629–639. [Google Scholar] [CrossRef][Green Version]
 Mendrik, A.M.; Vonken, E.; Rutten, A.; Viergever, M.A.; van Ginneken, B. Noise Reduction in Computed Tomography Scans Using 3D Anisotropic Hybrid Diffusion With Continuous Switch. IEEE Trans. Med Imaging 2009, 28, 1585–1594. [Google Scholar] [CrossRef] [PubMed]
 Adler, J.; Lunz, S.; Verdier, O.; Schönlieb, C.B.; Öktem, O. Task adapted reconstruction for inverse problems. arXiv 2018, arXiv:1809.00948. [Google Scholar]
 Boink, Y.E.; Manohar, S.; Brune, C. A partiallylearned algorithm for joint photoacoustic reconstruction and segmentation. IEEE Trans. Med. Imaging 2019, 39, 129–139. [Google Scholar] [CrossRef]
 Handels, H.; Deserno, T.M.; Maier, A.; MaierHein, K.H.; Palm, C.; Tolxdorff, T. Bildverarbeitung für die Medizin 2019; Springer Fachmedien Wiesbaden: Wiesbaden, Germany, 2019. [Google Scholar] [CrossRef]
 Mason, A.; Rioux, J.; Clarke, S.E.; Costa, A.; Schmidt, M.; Keough, V.; Huynh, T.; Beyea, S. Comparison of objective image quality metrics to expert radiologists’ scoring of diagnostic quality of MR images. IEEE Trans. Med. Imaging 2019, 39, 1064–1072. [Google Scholar] [CrossRef] [PubMed]
 Coban, S.B.; Lionheart, W.R.B.; Withers, P.J. Assessing the efficacy of tomographic reconstruction methods through physical quantification techniques. Meas. Sci. Technol. 2021. [Google Scholar] [CrossRef]
 Goodfellow, I.J.; Shlens, J.; Szegedy, C. Explaining and Harnessing Adversarial Examples. In Proceedings of the 3rd International Conference on Learning Representations (ICLR 2015), San Diego, CA, USA, 7–9 May 2015. [Google Scholar]
 Antun, V.; Renna, F.; Poon, C.; Adcock, B.; Hansen, A.C. On instabilities of deep learning in image reconstruction and the potential costs of AI. Proc. Natl. Acad. Sci. USA 2020. [Google Scholar] [CrossRef]
 Gottschling, N.M.; Antun, V.; Adcock, B.; Hansen, A.C. The troublesome kernel: Why deep learning for inverse problems is typically unstable. arXiv 2020, arXiv:2001.01258. [Google Scholar]
 Schwab, J.; Antholzer, S.; Haltmeier, M. Deep null space learning for inverse problems: Convergence analysis and rates. Inverse Probl. 2019, 35, 025008. [Google Scholar] [CrossRef]
 Chambolle, A.; Pock, T. A firstorder primaldual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 2011, 40, 120–145. [Google Scholar] [CrossRef][Green Version]
 Kingma, D.P.; Ba, J. Adam: A Method for Stochastic Optimization. In Proceedings of the 3rd International Conference on Learning Representations (ICLR 2015), San Diego, CA, USA, 7–9 May 2015. [Google Scholar]
 Hinton, G.; Srivastava, N.; Swersky, K. Neural networks for machine learning lecture 6a overview of minibatch gradient descent. Lect. Notes 2012, 14, 1–31. [Google Scholar]
 Winkler, C.; Worrall, D.; Hoogeboom, E.; Welling, M. Learning likelihoods with conditional normalizing flows. arXiv 2019, arXiv:1912.00042. [Google Scholar]
 Dinh, L.; SohlDickstein, J.; Bengio, S. Density estimation using Real NVP. In Proceedings of the 5th International Conference on Learning Representations (ICLR 2017), Toulon, France, 24–26 April 2017. [Google Scholar]
 Dinh, L.; Krueger, D.; Bengio, Y. NICE: Nonlinear Independent Components Estimation. In Proceedings of the 3rd International Conference on Learning Representations (ICLR 2015), San Diego, CA, USA, 7–9 May 2015. [Google Scholar]
 Kingma, D.P.; Dhariwal, P. Glow: Generative Flow with Invertible 1x1 Convolutions. In Proceedings of the Advances in Neural Information Processing Systems 31: Annual Conference on Neural Information Processing Systems 2018 (NeurIPS 2018), Montréal, QC, Canada, 3–8 December 2018; Bengio, S., Wallach, H.M., Larochelle, H., Grauman, K., CesaBianchi, N., Garnett, R., Eds.; pp. 10236–10245. [Google Scholar]
 Daubechies, I.; Defrise, M.; De Mol, C. An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun. Pure Appl. Math. 2004, 57, 1413–1457. [Google Scholar] [CrossRef][Green Version]
 Gregor, K.; LeCun, Y. Learning fast approximations of sparse coding. In Proceedings of the 27th International Conference on International Conference on Machine Learning, Haifa, Israel, 21–24 June 2010; pp. 399–406. [Google Scholar]
 Lempitsky, V.; Vedaldi, A.; Ulyanov, D. Deep Image Prior. In Proceedings of the 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, Salt Lake City, UT, USA, 18–23 June 2018; pp. 9446–9454. [Google Scholar] [CrossRef]
 Dittmer, S.; Kluth, T.; Maass, P.; Otero Baguer, D. Regularization by Architecture: A Deep Prior Approach for Inverse Problems. J. Math. Imaging Vis. 2019, 62, 456–470. [Google Scholar] [CrossRef][Green Version]
 Chakrabarty, P.; Maji, S. The Spectral Bias of the Deep Image Prior. arXiv 2019, arXiv:1912.08905. [Google Scholar]
 Heckel, R.; Soltanolkotabi, M. Denoising and Regularization via Exploiting the Structural Bias of Convolutional Generators. Int. Conf. Learn. Represent. 2020. [Google Scholar]
 Adler, J.; Kohr, H.; Ringh, A.; Moosmann, J.; Banert, S.; Ehrhardt, M.J.; Lee, G.R.; Niinimäki, K.; Gris, B.; Verdier, O.; et al. Operator Discretization Library (ODL). Zenodo 2018. [Google Scholar] [CrossRef]
 Van Aarle, W.; Palenstijn, W.J.; De Beenhouwer, J.; Altantzis, T.; Bals, S.; Batenburg, K.J.; Sijbers, J. The ASTRA Toolbox: A platform for advanced algorithm development in electron tomography. Ultramicroscopy 2015, 157, 35–47. [Google Scholar] [CrossRef] [PubMed][Green Version]
 Coban, S. SophiaBeads Dataset Project Codes. Zenodo. 2015. Available online: http://sophilyplum.github.io/sophiabeadsdatasets/ (accessed on 10 June 2020).
 Wang, T.; Nakamoto, K.; Zhang, H.; Liu, H. Reweighted Anisotropic Total Variation Minimization for LimitedAngle CT Reconstruction. IEEE Trans. Nucl. Sci. 2017, 64, 2742–2760. [Google Scholar] [CrossRef]
 Paszke, A.; Gross, S.; Massa, F.; Lerer, A.; Bradbury, J.; Chanan, G.; Killeen, T.; Lin, Z.; Gimelshein, N.; Antiga, L.; et al. PyTorch: An Imperative Style, HighPerformance Deep Learning Library. In Advances in Neural Information Processing Systems 32; Wallach, H., Larochelle, H., Beygelzimer, A., dAlch’eBuc, F., Fox, E., Garnett, R., Eds.; Curran Associates, Inc.: Red Hook, NY, USA, 2019; pp. 8024–8035. [Google Scholar]
Property  LoDoPaBCT  Apple CT 

Subject  Human thorax  Apples 
Scenario  low photon count  sparseangle 
Challenge  3678 reconstructions  100 reconstructions 
Image size  $362\mathrm{px}\times 362\mathrm{px}$  $972\mathrm{p}\mathrm{x}\times 972\mathrm{p}\mathrm{x}$ 
Angles  1000  50, 10, 5, 2 
Detector bins  513  1377 
Sampling ratio  ≈3.9  ≈0.07–$0.003$ 
Model  PSNR  PSNRFR  SSIM  SSIMFR  Number of Parameters 

Learned P.D.  36.25 ± 3.70  40.52 ± 3.64  0.866 ± 0.115  0.926 ± 0.076  874,980 
ISTA UNet  36.09 ± 3.69  40.36 ± 3.65  0.862 ± 0.120  0.924 ± 0.080  83,396,865 
UNet  36.00 ± 3.63  40.28 ± 3.59  0.862 ± 0.119  0.923 ± 0.079  613,322 
MSDCNN  35.85 ± 3.60  40.12 ± 3.56  0.858 ± 0.122  0.921 ± 0.082  181,306 
UNet++  35.37 ± 3.36  39.64 ± 3.40  0.861 ± 0.119  0.923 ± 0.080  9,170,079 
CINN  35.54 ± 3.51  39.81 ± 3.48  0.854 ± 0.122  0.919 ± 0.081  6,438,332 
DIP + TV  34.41 ± 3.29  38.68 ± 3.29  0.845 ± 0.121  0.913 ± 0.082  hyperp. 
iCTUNet  33.70 ± 2.82  37.97 ± 2.79  0.844 ± 0.120  0.911 ± 0.081  147,116,792 
TV  33.36 ± 2.74  37.63 ± 2.70  0.830 ± 0.121  0.903 ± 0.082  (hyperp.) 
FBP  30.19 ± 2.55  34.46 ± 2.18  0.727 ± 0.127  0.836 ± 0.085  (hyperp.) 
Method  ${\mathit{\ell}}_{\mathbf{Pois}}\left(\mathit{A}\widehat{\mathit{x}}\phantom{\rule{0.166667em}{0ex}}\right\phantom{\rule{0.166667em}{0ex}}{\mathit{y}}_{\mathit{\delta}})/{10}^{9}$ 

Learned PrimalDual  $4.022182\pm 0.699460$ 
ISTA UNet  $4.022185\pm 0.699461$ 
UNet  $4.022185\pm 0.699460$ 
MSDCNN  $4.022182\pm 0.699460$ 
UNet++  $4.022163\pm 0.699461$ 
CINN  $4.022184\pm 0.699460$ 
DIP + TV  $4.022183\pm 0.699466$ 
iCTUNet  $4.022038\pm 0.699430$ 
TV  $4.022189\pm 0.699463$ 
FBP  $4.021595\pm 0.699282$ 
${\ell}_{\mathrm{Pois}}\left(A{x}^{\u2020}\phantom{\rule{0.166667em}{0ex}}\right\phantom{\rule{0.166667em}{0ex}}{y}_{\delta})/{10}^{9}$  
Ground truth  $4.022184\pm 0.699461$ 
NoiseFree  PSNR  SSIM  

Number of Angles  50  10  5  2  50  10  5  2 
Learned PrimalDual  38.72  35.85  30.79  22.00  0.901  0.870  0.827  0.740 
ISTA UNet  38.86  34.54  28.31  20.48  0.897  0.854  0.797  0.686 
UNet  39.62  33.51  27.77  19.78  0.913  0.803  0.803  0.676 
MSDCNN  39.85  34.38  28.45  20.55  0.913  0.837  0.776  0.646 
CINN  39.59  34.84  27.81  19.46  0.913  0.871  0.762  0.674 
iCTUNet  36.07  29.95  25.63  19.28  0.878  0.847  0.824  0.741 
TV  39.27  29.00  22.04  15.95  0.915  0.783  0.607  0.661 
CGLS  33.05  21.81  12.60  15.25  0.780  0.619  0.537  0.615 
FBP  30.39  17.09  15.51  13.97  0.714  0.584  0.480  0.438 
Gaussian Noise  PSNR  SSIM  
Number of Angles  50  10  5  2  50  10  5  2 
Learned PrimalDual  36.62  33.76  29.92  21.41  0.878  0.850  0.821  0.674 
ISTA UNet  36.04  33.55  28.48  20.71  0.871  0.851  0.811  0.690 
UNet  36.48  32.83  27.80  19.86  0.882  0.818  0.789  0.706 
MSDCNN  36.67  33.20  27.98  19.88  0.883  0.831  0.748  0.633 
CINN  36.77  31.88  26.57  19.99  0.888  0.771  0.722  0.637 
iCTUNet  32.90  29.76  24.67  19.44  0.848  0.837  0.801  0.747 
TV  32.36  27.12  21.83  16.08  0.833  0.752  0.622  0.637 
CGLS  27.36  21.09  14.90  15.11  0.767  0.624  0.553  0.616 
FBP  27.88  17.09  15.51  13.97  0.695  0.583  0.480  0.438 
Scattering Noise  PSNR  SSIM  
Number of Angles  50  10  5  2  50  10  5  2 
Learned PrimalDual  37.80  34.19  27.08  20.98  0.892  0.866  0.796  0.540 
ISTA UNet  35.94  32.33  27.41  19.95  0.881  0.820  0.763  0.676 
UNet  34.96  32.91  26.93  18.94  0.830  0.784  0.736  0.688 
MSDCNN  38.04  33.51  27.73  20.19  0.899  0.818  0.757  0.635 
CINN  38.56  34.08  28.04  19.14  0.915  0.863  0.839  0.754 
iCTUNet  26.26  22.85  21.25  18.32  0.838  0.796  0.792  0.765 
TV  21.09  20.14  17.86  14.53  0.789  0.649  0.531  0.611 
CGLS  20.84  18.28  14.02  14.18  0.789  0.618  0.547  0.625 
FBP  21.01  15.80  14.26  13.06  0.754  0.573  0.475  0.433 
Evaluation  50 Angles  10 Angles  5 Angles  2 Angles  

Training  PSNR  SSIM  PSNR  SSIM  PSNR  SSIM  PSNR  SSIM  
50 angles  39.62  0.913  16.39  0.457  11.93  0.359  8.760  0.252  
10 angles  27.59  0.689  33.51  0.803  18.44  0.607  9.220  0.394  
5 angles  24.51  0.708  26.19  0.736  27.77  0.803  11.85  0.549  
2 angles  15.57  0.487  14.59  0.440  15.94  0.514  19.78  0.676 
Model  Reconstruction Error (Image Metrics)  Training Time  Recon Struction Time  GPU Memory  Learned Para Meters  Uses ${\mathcal{D}}_{\mathit{Y}}$ Discre Pancy  Operator Required  

Learned P.D.  $\U0001f7c9\U0001f7c9$  $\U0001f7c9$  $\U0001f7c9\U0001f7c9\U0001f7c9\U0001f7c9$  $\U0001f7c9\U0001f7c9$  $\U0001f7c9\U0001f7c9\U0001f7c9\U0001f7c9$  $\U0001f7c9\U0001f7c9$  no  $\U0001f7c9\U0001f7c9\U0001f7c9$ 
ISTA UNet  $\U0001f7c9\U0001f7c9$  $\U0001f7c9$  $\U0001f7c9\U0001f7c9\U0001f7c9$  $\U0001f7c9\U0001f7c9$  $\U0001f7c9\U0001f7c9\U0001f7c9$  $\U0001f7c9\U0001f7c9\U0001f7c9$  no  $\U0001f7c9\U0001f7c9$ 
UNet  $\U0001f7c9\U0001f7c9$  $\U0001f7c9$  $\U0001f7c9\U0001f7c9$  $\U0001f7c9\U0001f7c9$  $\U0001f7c9\U0001f7c9$  $\U0001f7c9\U0001f7c9$  no  $\U0001f7c9\U0001f7c9$ 
MSDCNN  $\U0001f7c9\U0001f7c9$  $\U0001f7c9$  $\U0001f7c9\U0001f7c9\U0001f7c9\U0001f7c9$  $\U0001f7c9\U0001f7c9$  $\U0001f7c9\U0001f7c9$  $\U0001f7c9$  no  $\U0001f7c9\U0001f7c9$ 
UNet++  $\U0001f7c9\U0001f7c9$    $\U0001f7c9\U0001f7c9$  $\U0001f7c9\U0001f7c9$  $\U0001f7c9\U0001f7c9\U0001f7c9$  $\U0001f7c9\U0001f7c9\U0001f7c9$  no  $\U0001f7c9\U0001f7c9$ 
CINN  $\U0001f7c9\U0001f7c9$  $\U0001f7c9$  $\U0001f7c9\U0001f7c9$  $\U0001f7c9\U0001f7c9\U0001f7c9$  $\U0001f7c9\U0001f7c9\U0001f7c9$  $\U0001f7c9\U0001f7c9\U0001f7c9$  no  $\U0001f7c9\U0001f7c9$ 
DIP + TV  $\U0001f7c9\U0001f7c9\U0001f7c9$      $\U0001f7c9\U0001f7c9\U0001f7c9\U0001f7c9$  $\U0001f7c9\U0001f7c9$  3+  yes  $\U0001f7c9\U0001f7c9\U0001f7c9\U0001f7c9$ 
iCTUNet  $\U0001f7c9\U0001f7c9\U0001f7c9$  $\U0001f7c9\U0001f7c9$  $\U0001f7c9\U0001f7c9$  $\U0001f7c9\U0001f7c9$  $\U0001f7c9\U0001f7c9\U0001f7c9$  $\U0001f7c9\U0001f7c9\U0001f7c9\U0001f7c9$  no  $\U0001f7c9$ 
TV  $\U0001f7c9\U0001f7c9\U0001f7c9$  $\U0001f7c9\U0001f7c9\U0001f7c9$    $\U0001f7c9\U0001f7c9\U0001f7c9$  $\U0001f7c9$  3  yes  $\U0001f7c9\U0001f7c9\U0001f7c9\U0001f7c9$ 
CGLS    $\U0001f7c9\U0001f7c9\U0001f7c9\U0001f7c9$    $\U0001f7c9$  $\U0001f7c9$  1  yes  $\U0001f7c9\U0001f7c9\U0001f7c9\U0001f7c9$ 
FBP  $\U0001f7c9\U0001f7c9\U0001f7c9\U0001f7c9$  $\U0001f7c9\U0001f7c9\U0001f7c9\U0001f7c9$    $\U0001f7c9$  $\U0001f7c9$  2  no  $\U0001f7c9\U0001f7c9\U0001f7c9\U0001f7c9$ 
Legend  LoDoPaB  Apple CT  Rough values for Apple CT Dataset B  
Avg. improv. over FBP  (varying for different setups and datasets)  
$\U0001f7c9\U0001f7c9\U0001f7c9\U0001f7c9$  0%  0–15%  >2 weeks  >10 min  >10 GiB  >${10}^{8}$  Direct  
$\U0001f7c9\U0001f7c9\U0001f7c9$  12–16%  25–30%  >5 days  >30 s  >3 GiB  >${10}^{6}$  In network  
$\U0001f7c9\U0001f7c9$  17–20%  40–45%  >1 day  >0.1 s  >1.5 GiB  >${10}^{5}$  For input  
$\U0001f7c9$  50–60%  ≤0.02 s  ≤1 GiB  ≤${10}^{5}$  Only concept 
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Leuschner, J.; Schmidt, M.; Ganguly, P.S.; Andriiashen, V.; Coban, S.B.; Denker, A.; Bauer, D.; Hadjifaradji, A.; Batenburg, K.J.; Maass, P.; et al. Quantitative Comparison of Deep LearningBased Image Reconstruction Methods for LowDose and SparseAngle CT Applications. J. Imaging 2021, 7, 44. https://doi.org/10.3390/jimaging7030044
Leuschner J, Schmidt M, Ganguly PS, Andriiashen V, Coban SB, Denker A, Bauer D, Hadjifaradji A, Batenburg KJ, Maass P, et al. Quantitative Comparison of Deep LearningBased Image Reconstruction Methods for LowDose and SparseAngle CT Applications. Journal of Imaging. 2021; 7(3):44. https://doi.org/10.3390/jimaging7030044
Chicago/Turabian StyleLeuschner, Johannes, Maximilian Schmidt, Poulami Somanya Ganguly, Vladyslav Andriiashen, Sophia Bethany Coban, Alexander Denker, Dominik Bauer, Amir Hadjifaradji, Kees Joost Batenburg, Peter Maass, and et al. 2021. "Quantitative Comparison of Deep LearningBased Image Reconstruction Methods for LowDose and SparseAngle CT Applications" Journal of Imaging 7, no. 3: 44. https://doi.org/10.3390/jimaging7030044