# A Green Prospective for Learned Post-Processing in Sparse-View Tomographic Reconstruction

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## Abstract

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## 1. Introduction

#### Aim and Contribution of the Paper

## 2. Methods and Materials

#### 2.1. The ResUNet Architecture

#### 2.2. The 3L-SSNet Architecture

#### 2.3. Receptive Field

#### 2.4. Training of the Networks

#### 2.5. Network Comparison

## 3. Experimental Results and Discussion

#### 3.1. Metrics for Image Quality Assessment

#### 3.2. Results on the Test Set

#### 3.3. Tests on Out-of-Domain Data

#### 3.3.1. Test on Unseen Noise

#### 3.3.2. Test on Unseen Image

#### 3.4. Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Arridge, S.; Maass, P.; Öktem, O.; Schönlieb, C.B. Solving inverse problems using data-driven models. Acta Numer.
**2019**, 28, 1–174. [Google Scholar] [CrossRef][Green Version] - McCann, M.T.; Jin, K.H.; Unser, M. Convolutional neural networks for inverse problems in imaging: A review. IEEE Signal Process. Mag.
**2017**, 34, 85–95. [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] - Graff, C.; Sidky, E. Compressive sensing in medical imaging. Appl. Opt.
**2015**, 54, C23–C44. [Google Scholar] [CrossRef][Green Version] - Tian, Z.; Jia, X.; Yuan, K.; Pan, T.; Jiang, S.B. Low Dose CT Reconstruction via Edge-preserving Total Variation Regularization. Phys. Med. Biol.
**2011**, 56, 5949–5967. [Google Scholar] [CrossRef] - Jensen, T.L.; Jørgensen, J.H.; Hansen, P.C.; Jensen, S.H. Implementation of an optimal first-order method for strongly convex total variation regularization. BIT Numer. Math.
**2012**, 52, 329–356. [Google Scholar] [CrossRef][Green Version] - Sidky, E.; Chartrand, R.; Boone, J.; Pan, X. Constrained T p V-minimization for enhanced exploitation of gradient sparsity: Application to CT image reconstruction. IEEE J. Transl. Eng. Health Med.
**2013**, 2, 1–15. [Google Scholar] [CrossRef] [PubMed] - Liu, L. Model-based Iterative Reconstruction: A Promising Algorithm for Today’s Computed Tomography Imaging. J. Med. Imaging Radiat. Sci.
**2014**, 45, 131–136. [Google Scholar] [CrossRef] [PubMed][Green Version] - Loli Piccolomini, E.; Morotti, E. A Model-Based Optimization Framework for Iterative Digital Breast Tomosynthesis Image Reconstruction. J. Imaging
**2021**, 7, 36. [Google Scholar] [CrossRef] - Rantala, M.; Vanska, S.; Jarvenpaa, S.; Kalke, M.; Lassas, M.; Moberg, J.; Siltanen, S. Wavelet-based reconstruction for limited-angle X-ray tomography. IEEE Trans. Med. Imaging
**2006**, 25, 210–217. [Google Scholar] [CrossRef] [PubMed] - Purisha, Z.; Rimpeläinen, J.; Bubba, T.; Siltanen, S. Controlled wavelet domain sparsity for x-ray tomography. Meas. Sci. Technol.
**2017**, 29, 014002. [Google Scholar] [CrossRef][Green Version] - Zhang, H.M.; Dong, B. A review on deep learning in medical image reconstruction. J. Oper. Res. Soc. China
**2020**, 8, 311–340. [Google Scholar] [CrossRef][Green Version] - Ahishakiye, E.; Van Gijzen, M.B.; Tumwiine, J.; Wario, R.; Obungoloch, J. A survey on deep learning in medical image reconstruction. Intell. Med.
**2021**. [Google Scholar] [CrossRef] - Zhang, H.; Li, L.; Qiao, K.; Wang, L.; Yan, B.; Li, L.; Hu, G. Image prediction for limited-angle tomography via deep learning with convolutional neural network. arXiv
**2016**, arXiv:1607.08707. [Google Scholar] - Ronneberger, O.; Fischer, P.; Brox, T. U-net: Convolutional networks for biomedical image segmentation. In Proceedings of the International Conference on Medical Image Computing and Computer-Assisted Intervention, Munich, Germany, 5–9 October 2015; pp. 234–241. [Google Scholar]
- Li, H.; Mueller, K. Low-dose CT streak artifacts removal using deep residual neural network. In Proceedings of the Fully 3D Conference, Xi’an, China, 18–23 June 2017; pp. 191–194. [Google Scholar]
- Han, Y.; Ye, J.C. Framing U-Net via deep convolutional framelets: Application to sparse-view CT. IEEE Trans. Med. Imaging
**2018**, 37, 1418–1429. [Google Scholar] [CrossRef] [PubMed][Green Version] - Zhang, T.; Gao, H.; Xing, Y.; Chen, Z.; Zhang, L. DualRes-UNet: Limited Angle Artifact Reduction for Computed Tomography. In Proceedings of the 2019 IEEE Nuclear Science Symposium and Medical Imaging Conference (NSS/MIC), Manchester, UK, 26 October–2 November 2019; pp. 1–3. [Google Scholar]
- Han, Y.; Ye, J.C. Deep residual learning approach for sparse-view CT reconstruction. Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine. In Proceedings of the Fully 3D Conference Organization, Xi’an, China, 18–23 June 2017. [Google Scholar]
- Schnurr, A.K.; Chung, K.; Russ, T.; Schad, L.R.; Zöllner, F.G. Simulation-based deep artifact correction with convolutional neural networks for limited angle artifacts. Z. Med. Phys.
**2019**, 29, 150–161. [Google Scholar] [CrossRef] [PubMed] - Han, Y.S.; Yoo, J.; Ye, J.C. Deep residual learning for compressed sensing CT reconstruction via persistent homology analysis. arXiv
**2016**, arXiv:1611.06391. [Google Scholar] - Huang, Y.; Würfl, T.; Breininger, K.; Liu, L.; Lauritsch, G.; Maier, A. Some investigations on robustness of deep learning in limited angle tomography. In Proceedings of the International Conference on Medical Image Computing and Computer-Assisted Intervention, Granada, Spain, 16–20 September 2018; pp. 145–153. [Google Scholar]
- Liu, C.; Huang, Y.; Maier, J.; Klein, L.; Kachelrieß, M.; Maier, A. Robustness Investigation on Deep Learning CT Reconstruction for Real-Time Dose Optimization. arXiv
**2020**, arXiv:2012.03579. [Google Scholar] - Schwartz, R.; Dodge, J.; Smith, N.A.; Etzioni, O. Green ai. Commun. ACM
**2020**, 63, 54–63. [Google Scholar] [CrossRef] - Strubell, E.; Ganesh, A.; McCallum, A. Energy and policy considerations for deep learning in NLP. arXiv
**2019**, arXiv:1906.02243. [Google Scholar] - Strubell, E.; Ganesh, A.; McCallum, A. Energy and Policy Considerations for Modern Deep Learning Research. In Proceedings of the AAAI Conference on Artificial Intelligence, New York, NY, USA, 2–17 February 2020; Volume 34, pp. 13693–13696. [Google Scholar] [CrossRef]
- Asperti, A.; Evangelista, D.; Piccolomini, E.L. A Survey on Variational Autoencoders from a Green AI Perspective. SN Comput. Sci.
**2021**, 2, 1–23. [Google Scholar] [CrossRef] - Goodfellow, I.; Bengio, Y.; Courville, A. Deep Learning; MIT Press: Cambridge, MA, USA, 2016. [Google Scholar]
- Kang, E.; Min, J.; Ye, J.C. A deep convolutional neural network using directional wavelets for low-dose X-ray CT reconstruction. Med. Phys.
**2017**, 44, e360–e375. [Google Scholar] [CrossRef][Green Version] - Ye, J.C.; Han, Y.; Cha, E. Deep convolutional framelets: A general deep learning framework for inverse problems. SIAM J. Imaging Sci.
**2018**, 11, 991–1048. [Google Scholar] [CrossRef] - Bubba, T.A.; Kutyniok, G.; Lassas, M.; Maerz, M.; Samek, W.; Siltanen, S.; Srinivasan, V. Learning the invisible: A hybrid deep learning-shearlet framework for limited angle computed tomography. Inverse Probl.
**2019**, 35, 064002. [Google Scholar] [CrossRef][Green Version] - Heinrich, M.; Stille, M.; Buzug, T. Residual U-Net Convolutional Neural Network Architecture for Low-Dose CT Denoising. Curr. Dir. Biomed. Eng.
**2018**, 4, 297–300. [Google Scholar] [CrossRef] - Wang, J.; Zeng, L.; Wang, C.; Guo, Y. ADMM-based deep reconstruction for limited-angle CT. Phys. Med. Biol.
**2019**, 64, 115011. [Google Scholar] [CrossRef] [PubMed] - Chen, H.; Zhang, Y.; Zhang, W.; Liao, P.; Li, K.; Zhou, J.; Wang, G. Low-dose CT via convolutional neural network. Biomed. Opt. Express
**2017**, 8, 679–694. [Google Scholar] [CrossRef] [PubMed] - Le, H.; Borji, A. What are the Receptive, Effective Receptive, and Projective Fields of Neurons in Convolutional Neural Networks? arXiv
**2017**, arXiv:1705.07049. [Google Scholar] - Araujo, A.; Norris, W.; Sim, J. Computing Receptive Fields of Convolutional Neural Networks. Distill
**2019**, 4, e21. [Google Scholar] [CrossRef] - McCollough, C. TU-FG-207A-04: Overview of the Low Dose CT Grand Challenge. Med. Phys.
**2016**, 43, 3759–3760. [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][Green Version] - 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] - Zhang, L.; Zhang, L.; Mou, X.; Zhang, D. FSIM: A Feature Similarity Index for Image Quality Assessment. IEEE Trans. Image Process.
**2011**, 20, 2378–2386. [Google Scholar] [CrossRef] [PubMed][Green Version] - Segars, W.P.; Sturgeon, G.; Mendonca, S.; Grimes, J.; Tsui, B.M. 4D XCAT phantom for multimodality imaging research. Med. Phys.
**2010**, 37, 4902–4915. [Google Scholar] [CrossRef] [PubMed] - Russ, T.; Goerttler, S.; Schnurr, A.K.; Bauer, D.F.; Hatamikia, S.; Schad, L.R.; Zöllner, F.G.; Chung, K. Synthesis of CT images from digital body phantoms using CycleGAN. Int. J. Comput. Assist. Radiol. Surg.
**2019**, 14, 1741–1750. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**The three different CT geometric protocols. In the traditional setting (

**a**), the X-ray source and the detector walk a full circle trajectory with a very small angular step, which is enlarged in sparse-view CT (

**b**), whereas in limited-angle CT (

**c**), the source-detector rotation is restricted to a C-shape path.

**Figure 2.**Graphical draft of the considered two-step workflow for tomographic reconstruction from sparse-view data.

**Figure 3.**On the right: graphical representation of the ResUNet architecture; On the left: details on the maximum receptive fields for each of the five levels of the network encoder (RF percentage respect to the input 512 × 512 image and size of RF).

**Figure 4.**On the

**left**: graphical representation of the 3L-SSNet architecture; on the

**right**: details on the receptive fields for each of the three layers of the network (RF percentage respect to the input 512 × 512 image and size of RF). The name of the three layers follows the notation in [34].

**Figure 5.**Ground truth image (

**a**) and the two considered zooms-in (

**b**,

**c**), which are depicted by the red squares on the full image (

**a**).

**Figure 6.**Full-range geometry reconstructions. The results obtained with FPB (

**left column**), ResUNet (

**central column**) and 3l-SSNet (

**right column**). Below each image, the values of its RE and SSIM metrics.

**Figure 7.**Half-range geometry reconstructions. The results obtained with FPB (

**left column**), ResUNet (

**central column**) and 3L-SSNet (

**right column**). Below each image, the values of its RE and SSIM metrics.

**Figure 8.**Crops of the reconstructions of the test patient with unseen noise and half-range geometry. ResUNet in (

**a**,

**c**), 3L-SSNet in (

**b**,

**d**).

**Figure 9.**XCAT phantom test image with half-range geometry. From the left to right: (

**first column**): Ground Truth image (

**a**) and the considered zooms-in (

**e**,

**i**). Reconstructions from half-range geometry with FBP (

**second column**), ResUNet (

**third column**) and 3L-SSNet (

**fourth column**).

**Table 1.**A comparison of the cost of the considered networks. The training time is expressed in sec/epoch in the third column.

Parameters | FLOPs | Training Time | |
---|---|---|---|

ResUNet | $34.5\times {10}^{6}$ | $406\times {10}^{9}$ | 209 |

3L-SSNet | $85\times {10}^{3}$ | $44\times {10}^{9}$ | 53 |

**Table 2.**The average of the full-reference metrics on the test set in the case of full-range geometry.

RE | PSNR | SSIM | FSIM | |
---|---|---|---|---|

FBP | 0.9966 | 86.42 (33.89) | 0.2924 | 0.5456 |

ResUNet | 0.0942 | 106.99 (41.95) | 0.9262 | 0.9709 |

3L-SSNet | 0.0840 | 107.92 (42.32) | 0.9480 | 0.9627 |

**Table 3.**The average of the full-reference metrics on the test set in the case of half-range geometry.

RE | PSNR | SSIM | FSIM | |
---|---|---|---|---|

FBP | 0.9932 | 86.45 (33.90) | 0.2962 | 0.6819 |

ResUNet | 0.1016 | 106.38 (41.71) | 0.9324 | 0.9478 |

3L-SSNet | 0.1309 | 104.34 (40.91) | 0.9021 | 0.9474 |

**Table 4.**Full-reference metrics on the test image with unseen noise in full-range and half-range cases.

FBP | ResUNet | 3L-SSNet | ||||
---|---|---|---|---|---|---|

RE | SSIM | RE | SSIM | RE | SSIM | |

Full-range | 0.9966 | 0.2526 | 0.0966 | 0.9172 | 0.0896 | 0.9295 |

Half-range | 0.9932 | 0.2567 | 0.0986 | 0.9212 | 0.1162 | 0.8866 |

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**MDPI and ACS Style**

Morotti, E.; Evangelista, D.; Loli Piccolomini, E. A Green Prospective for Learned Post-Processing in Sparse-View Tomographic Reconstruction. *J. Imaging* **2021**, *7*, 139.
https://doi.org/10.3390/jimaging7080139

**AMA Style**

Morotti E, Evangelista D, Loli Piccolomini E. A Green Prospective for Learned Post-Processing in Sparse-View Tomographic Reconstruction. *Journal of Imaging*. 2021; 7(8):139.
https://doi.org/10.3390/jimaging7080139

**Chicago/Turabian Style**

Morotti, Elena, Davide Evangelista, and Elena Loli Piccolomini. 2021. "A Green Prospective for Learned Post-Processing in Sparse-View Tomographic Reconstruction" *Journal of Imaging* 7, no. 8: 139.
https://doi.org/10.3390/jimaging7080139