# Low-Cost Probabilistic 3D Denoising with Applications for Ultra-Low-Radiation Computed Tomography

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Patient-Specific Generation of Synthetic CT Images, Radiation Exposure Estimation, and LAR Computations

#### 2.2. Probabilistic Mumford–Shah Model Formulation

#### 2.3. Relation of Probabilistic Mumford–Shah and rSPA Algorithm to Regularized Mumford–Shah Framework (MS) and Rudin–Osher–Fatemi (ROF) Total Variation Model

#### 2.4. Practical Implementation

#### 2.4.1. Synthetic CT Image Generation Model

`imnoise()`with the parameter value ‘speckle’. The parameter $\sigma $ is, in both cases, selected according to the description below. The MATLAB code implementing this CT image generation workflow is available at https://www.dropbox.com/sh/rr0no9vdo8osx44/AAAHQxXJnxT8P0LPs7wTRBv7a?dl=0 (accessed on 18 March 2022). Generation of the nonparametric empirical CT noise was implemented in the function

`create_CT_image_noise()`available at https://www.dropbox.com/s/xbwwrk9y2napgpy/create_CT_image_noise.m?dl=0 (accessed on 18 March 2022).

#### 2.4.2. Common CT Image Denoising and Image Quality Assessment Methods

`imgaussfilt3()`, 3D local median filtering with the MATLAB function

`medfilt3()`and bilateral filtering with the MATLAB function

`imbilatfilt()`) [14,16,17,18], spectral denoising methods (the 3D wavelets denoising with the MATLAB function

`wavedec3()`) [15,19,20,21,22] and a deep learning denoising method based on pre-trained feed-forward denoising convolutional neural networks (DnCNNs, with the MATLAB functions

`denoiseImage()`and

`denoisingNetwork()`) [13,25,26,27].

## 3. Results and Discussion

#### 3.1. Application and Comparison of the PMS Model with Standard Methods

#### 3.2. Implementation Details

`immse()`; 3D Peak Signal-to-Noise Ratio (PSNR) was obtained as an average over the 2D PSNR image error measures [67] implemented in the MATLAB function

`psnr()`; 3D Multi-Scale Structural Similarity Index Measure (3D MS-SSIM) [68] was obtained with the 3D image volume measure MATLAB function

`multissim3()`.

`quantile()`.

## 4. Conclusions

**3DMS-SSIM**around 0.9) of a 3D image with ${10}^{7}$ voxels in the ultra-low-radiation regime (SNR = 0.5) in only 3 min on a MacBook Pro laptop with 4 cores. None of the other denoising methods considered were able to come close to this performance.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Communicating Radiation Risks in Paediatric Imaging: Information to Support Health Care Discussions about Benefit and Risk; World Health Organization: Geneva, Switzerland, 2016; 88p.
- Ai, T.; Yang, Z.; Hou, H.; Zhan, C.; Chen, C.; Lv, W.; Tao, Q.; Sun, Z.; Xia, L. Correlation of Chest CT and RT-PCR Testing in Coronavirus Disease 2019 (COVID-19) in China: A Report of 1014 Cases. Radiology
**2020**, 296, E32–E40. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bernheim, A.; Mei, X.; Huang, M.; Yang, Y.; Fayad, Z.A.; Zhang, N.; Diao, K.; Lin, B.; Zhu, X.; Li, K.; et al. Chest CT Findings in Coronavirus Disease-19 (COVID-19): Relationship to Duration of Infection. Radiology
**2020**, 295, 200463. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Brenner, D.J.; Doll, R.; Goodhead, D.T.; Hall, E.J.; Land, C.E.; Little, J.B.; Lubin, J.H.; Preston, D.L.; Preston, R.J.; Puskin, J.S.; et al. Cancer risks attributable to low doses of ionizing radiation: Assessing what we really know. Proc. Natl. Acad. Sci. USA
**2003**, 100, 13761–13766. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Radiation, committee and research, board and studies, division and council, national. In Health Risks from Exposure to Low Levels of Ionizing Radiation: BEIR VII Phase 2; The National Academies Press: Washington, DC, USA, 2006; pp. 1–406. [CrossRef]
- Brenner, D.J.; Hall, E.J. Computed Tomography—An Increasing Source of Radiation Exposure. N. Engl. J. Med.
**2007**, 357, 2277–2284. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Gillespie, D.T. A diffusional bimolecular propensity function. J. Chem. Phys.
**2009**, 131, 164109. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Siegel, J.A.; Greenspan, B.S.; Maurer, A.H.; Taylor, A.T.; Phillips, W.T.; Van Nostrand, D.; Sacks, B.; Silberstein, E.B. The BEIR VII Estimates of Low-Dose Radiation Health Risks Are Based on Faulty Assumptions and Data Analyses: A Call for Reassessment. J. Nucl. Med.
**2018**, 59, 1017–1019. [Google Scholar] [CrossRef] [Green Version] - de González, A.B.; Mahesh, M.; Kim, K.P.; Bhargavan, M.; Lewis, R.; Mettler, F.; Land, C. Projected Cancer Risks From Computed Tomographic Scans Performed in the United States in 2007. Arch. Intern. Med.
**2009**, 169, 2071–2077. [Google Scholar] [CrossRef] - Miglioretti, D.L.; Johnson, E.; Williams, A.; Greenlee, R.T.; Weinmann, S.; Solberg, L.I.; Feigelson, H.S.; Roblin, D.; Flynn, M.J.; Vanneman, N.; et al. The Use of Computed Tomography in Pediatrics and the Associated Radiation Exposure and Estimated Cancer Risk. JAMA Pediatr.
**2013**, 167, 700–707. [Google Scholar] [CrossRef] - Duncan, J.R.; Lieber, M.R.; Adachi, N.; Wahl, R.L. Radiation Dose Does Matter: Mechanistic Insights into DNA Damage and Repair Support the Linear No-Threshold Model of Low-Dose Radiation Health Risks. J. Nucl. Med.
**2018**, 59, 1014–1016. [Google Scholar] [CrossRef] [Green Version] - Huang, R.; Liu, X.; He, L.; Zhou, P.K. Radiation Exposure Associated With Computed Tomography in Childhood and the Subsequent Risk of Cancer: A Meta-Analysis of Cohort Studies. Dose-Response Publ. Int. Hormesis Soc.
**2020**, 18, 1559325820923828. [Google Scholar] [CrossRef] - Choy, G.; Khalilzadeh, O.; Michalski, M.; Do, S.; Samir, A.E.; Pianykh, O.S.; Geis, J.R.; Pandharipande, P.V.; Brink, J.A.; Dreyer, K.J. Current Applications and Future Impact of Machine Learning in Radiology. Radiology
**2018**, 288, 318–328. [Google Scholar] [CrossRef] [PubMed] - Koziol, P.; Raczkowska, M.K.; Skibinska, J.; Urbaniak-Wasik, S.; Paluszkiewicz, C.; Kwiatek, W.; Wrobel, T.P. Comparison of spectral and spatial denoising techniques in the context of High Definition FT-IR imaging hyperspectral data. Sci. Rep.
**2018**, 8, 14351. [Google Scholar] [CrossRef] [PubMed] - Roels, J.; Vernaillen, F.; Kremer, A.; Gonçalves, A.; Aelterman, J.; Luong, H.Q.; Goossens, B.; Philips, W.; Lippens, S.; Saeys, Y. An interactive ImageJ plugin for semi-automated image denoising in electron microscopy. Nat. Commun.
**2020**, 11, 771. [Google Scholar] [CrossRef] [PubMed] - Wirjadi, O.; Breuel, T. Approximate separable 3D anisotropic Gauss filter. In Proceedings of the IEEE International Conference on Image Processing 2005, Genova, Italy, 14 September 2005; Volume 2, pp. 11–149. [Google Scholar] [CrossRef]
- Tomasi, C.; Manduchi, R. Bilateral Filtering for Gray and Color Images. In Proceedings of the Sixth International Conference on Computer Vision, Bombay, India, 7 January 1998; IEEE Computer Society: Washington, DC, USA, 1998; p. 839. [Google Scholar]
- Harms, J.; Wang, T.; Petrongolo, M.; Niu, T.; Zhu, L. Noise suppression for dual-energy CT via penalized weighted least-square optimization with similarity-based regularization. Med. Phys.
**2016**, 43, 2676–2686. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Donoho, D.L. De-noising by soft-thresholding. IEEE Trans. Inf. Theory
**1995**, 41, 613–627. [Google Scholar] [CrossRef] [Green Version] - Arias-Castro, E.; Donoho, D.L. Does median filtering truly preserve edges better than linear filtering? Ann. Stat.
**2009**, 37, 1172–1206. [Google Scholar] [CrossRef] [Green Version] - Wang, Y.; Che, X.; Ma, S. Nonlinear filtering based on 3D wavelet transform for MRI denoising. EURASIP J. Adv. Signal Process.
**2012**, 2012, 40. [Google Scholar] [CrossRef] [Green Version] - Tang, S.; Tang, X. Statistical CT noise reduction with multiscale decomposition and penalized weighted least squares in the projection domain. Med. Phys.
**2012**, 39, 5498–5512. [Google Scholar] [CrossRef] - Yang, Q.; Yan, P.; Zhang, Y.; Yu, H.; Shi, Y.; Mou, X.; Kalra, M.K.; Zhang, Y.; Sun, L.; Wang, G. Low-Dose 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] - Konefal, A.; Tang, C.; Li, J.; Wang, L.; Li, Z.; Jiang, L.; Cai, A.; Zhang, W.; Liang, N.; Li, L.; et al. Unpaired Low-Dose CT Denoising Network Based on Cycle-Consistent Generative Adversarial Network with Prior Image Information. Comput. Math. Methods Med.
**2019**, 2019, 8639825. [Google Scholar] [CrossRef] - Zhang, K.; Zuo, W.; Chen, Y.; Meng, D.; Zhang, L. Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising. Trans. Img. Proc.
**2017**, 26, 3142–3155. [Google Scholar] [CrossRef] [PubMed] [Green Version] - 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] - Topal, E.; Löffler, M.; Zschech, E. Deep Learning-based Inaccuracy Compensation in Reconstruction of High Resolution XCT Data. Sci. Rep.
**2020**, 10, 7682. [Google Scholar] [CrossRef] [PubMed] - Tian, C.; Fei, L.; Zheng, W.; Xu, Y.; Zuo, W.; Lin, C.W. Deep learning on image denoising: An overview. Neural Netw.
**2020**, 131, 251–275. [Google Scholar] [CrossRef] - Bhadra, S.; Kelkar, V.A.; Brooks, F.J.; Anastasio, M.A. On Hallucinations in Tomographic Image Reconstruction. IEEE Trans. Med. Imaging
**2021**, 40, 3249–3260. [Google Scholar] [CrossRef] - Kaur, P.; Singh, G.; Kaur, P. A review of denoising medical images using machine learning approaches. Curr. Med. Imaging
**2018**, 14, 675–685. [Google Scholar] [CrossRef] - Litjens, G.; Kooi, T.; Bejnordi, B.E.; Setio, A.A.A.; Ciompi, F.; Ghafoorian, M.; Van Der Laak, J.A.; Van Ginneken, B.; Sánchez, C.I. A survey on deep learning in medical image analysis. Med. Image Anal.
**2017**, 42, 60–88. [Google Scholar] [CrossRef] [Green Version] - Lundervold, A.S.; Lundervold, A. An overview of deep learning in medical imaging focusing on MRI. Z. Med. Phys.
**2019**, 29, 102–127. [Google Scholar] [CrossRef] - Razzak, M.I.; Naz, S.; Zaib, A. Deep learning for medical image processing: Overview, challenges and the future. Classif. Bioapps
**2018**, 26, 323–350. [Google Scholar] - Liu, S.; Deng, W. Very deep convolutional neural network based image classification using small training sample size. In Proceedings of the 2015 3rd IAPR Asian Conference on Pattern Recognition (ACPR), Kuala Lumpur, Malaysia, 3–6 November 2015; IEEE: Piscataway, NJ, USA, 2015; pp. 730–734. [Google Scholar]
- Gaonkar, B.; Hovda, D.; Martin, N.; Macyszyn, L. Deep learning in the small sample size setting: Cascaded feed forward neural networks for medical image segmentation. In Proceedings of the Medical Imaging 2016: Computer-Aided Diagnosis, San Diego, CA, USA, 27 February–3 March 2016; International Society for Optics and Photonics: Bellingham, WA, USA, 2016; Volume 9785, p. 97852I. [Google Scholar]
- Zhao, W. Research on the deep learning of the small sample data based on transfer learning. AIP Conf. Proc.
**2017**, 1864, 020018. [Google Scholar] - Keshari, R.; Ghosh, S.; Chhabra, S.; Vatsa, M.; Singh, R. Unravelling small sample size problems in the deep learning world. In Proceedings of the 2020 IEEE Sixth International Conference on Multimedia Big Data (BigMM), New Delhi, India, 24–26 September 2020; IEEE: Piscataway, NJ, USA, 2020; pp. 134–143. [Google Scholar]
- D’souza, R.N.; Huang, P.Y.; Yeh, F.C. Structural analysis and optimization of convolutional neural networks with a small sample size. Sci. Rep.
**2020**, 10, 834. [Google Scholar] [CrossRef] [PubMed] - Dietterich, T. Overfitting and undercomputing in machine learning. ACM Comput. Surv.
**1995**, 27, 326–327. [Google Scholar] [CrossRef] - Zhang, C.; Vinyals, O.; Munos, R.; Bengio, S. A study on overfitting in deep reinforcement learning. arXiv
**2018**, arXiv:1804.06893. [Google Scholar] - Rice, L.; Wong, E.; Kolter, Z. Overfitting in adversarially robust deep learning. In Proceedings of the International Conference on Machine Learning, Virtual, 13–18 July 2020; pp. 8093–8104. [Google Scholar]
- Hosseini, M.; Powell, M.; Collins, J.; Callahan-Flintoft, C.; Jones, W.; Bowman, H.; Wyble, B. I tried a bunch of things: The dangers of unexpected overfitting in classification of brain data. Neurosci. Biobehav. Rev.
**2020**, 119, 456–467. [Google Scholar] [CrossRef] [PubMed] - Gerber, S.; Pospisil, L.; Sys, S.; Hewel, C.; Torkamani, A.; Horenko, I. Co-inference of data mislabeling reveals improved models in genomics and breast cancer diagnostics. Front. Artif. Intell.
**2022**, 4, 739432. [Google Scholar] [CrossRef] - Srivastava, N.; Hinton, G.; Krizhevsky, A.; Sutskever, I.; Salakhutdinov, R. Dropout: A simple way to prevent neural networks from overfitting. J. Mach. Learn. Res.
**2014**, 15, 1929–1958. [Google Scholar] - Ying, X. An overview of overfitting and its solutions. J. Phys. Conf. Ser.
**2019**, 1168, 022022. [Google Scholar] [CrossRef] - Pan, S.J.; Yang, Q. A survey on transfer learning. IEEE Trans. Knowl. Data Eng.
**2009**, 22, 1345–1359. [Google Scholar] [CrossRef] - Weiss, K.; Khoshgoftaar, T.M.; Wang, D. A survey of transfer learning. J. Big Data
**2016**, 3, 9. [Google Scholar] [CrossRef] [Green Version] - Jang, Y.; Lee, H.; Hwang, S.J.; Shin, J. Learning what and where to transfer. In Proceedings of the International Conference on Machine Learning, Long Beach, CA, USA, 9–15 June 2019; pp. 3030–3039. [Google Scholar]
- Raghu, M.; Zhang, C.; Kleinberg, J.; Bengio, S. Transfusion: Understanding transfer learning for medical imaging. Adv. Neural Inf. Process. Syst.
**2019**, 32, 3347–3357. [Google Scholar] - Alzubaidi, L.; Fadhel, M.A.; Al-Shamma, O.; Zhang, J.; Santamaría, J.; Duan, Y.; Oleiwi, S.R. Towards a better understanding of transfer learning for medical imaging: A case study. Appl. Sci.
**2020**, 10, 4523. [Google Scholar] [CrossRef] - Alzubaidi, L.; Al-Amidie, M.; Al-Asadi, A.; Humaidi, A.J.; Al-Shamma, O.; Fadhel, M.A.; Zhang, J.; Santamaría, J.; Duan, Y. Novel Transfer Learning Approach for Medical Imaging with Limited Labeled Data. Cancers
**2021**, 13, 1590. [Google Scholar] [CrossRef] [PubMed] - Tartaglione, E.; Barbano, C.A.; Berzovini, C.; Calandri, M.; Grangetto, M. Unveiling covid-19 from chest X-ray with deep learning: A hurdles race with small data. Int. J. Environ. Res. Public Health
**2020**, 17, 6933. [Google Scholar] [CrossRef] [PubMed] - Tsymbal, A. The problem of concept drift: Definitions and related work. Comput. Sci. Dep. Trinity Coll. Dublin
**2004**, 106, 58. [Google Scholar] - Žliobaitė, I. Learning under concept drift: An overview. arXiv
**2010**, arXiv:1010.4784. [Google Scholar] - Gama, J.; Žliobaitė, I.; Bifet, A.; Pechenizkiy, M.; Bouchachia, A. A survey on concept drift adaptation. ACM Comput. Surv.
**2014**, 46, 1–37. [Google Scholar] [CrossRef] - Alippi, C. Learning in Non-stationary Environments. In Proceedings of the ECTA 2014—Proceedings of the International Conference on Evolutionary Computation Theory and Applications, part of IJCCI 2014, Rome, Italy, 22–24 October 2014; Rosa, A.C., Guervós, J.J.M., Filipe, J., Eds.; SciTePress: Setúbal, Portugal, 2014; p. IS-11. [Google Scholar]
- Souza, V.M.A.; dos Reis, D.M.; Maletzke, A.; Batista, G.E.A.P.A. Challenges in benchmarking stream learning algorithms with real-world data. Data Min. Knowl. Discov.
**2020**, 34, 1805–1858. [Google Scholar] [CrossRef] - Horenko, I. On a Scalable Entropic Breaching of the Overfitting Barrier for Small Data Problems in Machine Learning. Neural Comput.
**2020**, 32, 1563–1579. [Google Scholar] [CrossRef] - Hochreiter, S.; Schmidhuber, J. Long Short-term Memory. Neural Comput.
**1997**, 9, 1735–1780. [Google Scholar] [CrossRef] - Horenko, I. Finite Element Approach to Clustering of Multidimensional Time Series. SIAM J. Sci. Comput.
**2010**, 32, 62–83. [Google Scholar] [CrossRef] - Metzner, P.; Putzig, L.; Horenko, I. Analysis of persistent nonstationary time series and applications. Commun. Appl. Math. Comput. Sci.
**2012**, 7, 175–229. [Google Scholar] [CrossRef] [Green Version] - Gerber, S.; Horenko, I. Improving clustering by imposing network information. Sci. Adv.
**2015**, 1, e1500163. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Pospisil, L.; Gagliardini, P.; Sawyer, W.; Horenko, I. On a scalable nonparametric denoising of time series signals. Commun. Appl. Math. Comput. Sci.
**2018**, 13, 107–138. [Google Scholar] [CrossRef] - Rodrigues, D.R.; Everschor-Sitte, K.; Gerber, S.; Horenko, I. A deeper look into natural sciences with physics-based and data-driven measures. Iscience
**2021**, 24, 102171. [Google Scholar] [CrossRef] - Gerber, S.; Horenko, I. Towards a direct and scalable identification of reduced models for categorical processes. Proc. Natl. Acad. Sci. USA
**2017**, 114, 4863–4868. [Google Scholar] [CrossRef] [Green Version] - Wackerly, D.; Mendenhall, W., III; Scheaffer, R.L. Mathematical Statistics with Applications, 6th ed.; Duxbury Advanced Series; Cengage Learning: Boston, MA, USA, 2002. [Google Scholar]
- Huynh-Thu, Q.; Ghanbari, M. The accuracy of PSNR in predicting video quality for different video scenes and frame rates. Telecommun. Syst.
**2012**, 49, 35–48. [Google Scholar] [CrossRef] - Wang, Z.; Bovik, A.C.; Sheikh, H.R.; Simoncelli, E.P. Image Quality Assessment: From Error Visibility to Structural Similarity. Trans. Img. Proc.
**2004**, 13, 600–612. [Google Scholar] [CrossRef] [Green Version] - Hallgrímsson, B.; Hall, B. Variation: A Central Concept in Biology; Elsevier Science: Amsterdam, The Netherlands, 2011. [Google Scholar]
- Meyer, M.; Ronald, J.; Vernuccio, F.; Nelson, R.C.; Ramirez-Giraldo, J.C.; Solomon, J.; Patel, B.N.; Samei, E.; Marin, D. Reproducibility of CT Radiomic Features within the Same Patient: Influence of Radiation Dose and CT Reconstruction Settings. Radiology
**2019**, 293, 583–591. [Google Scholar] [CrossRef] - De Man, B.; Basu, S.; Chandra, N.; Dunham, B.; Edic, P.; Iatrou, M.; McOlash, S.; Sainath, P.; Shaughnessy, C.; Tower, B.; et al. CatSim: A new computer assisted tomography simulation environment. In Proceedings of the Medical Imaging 2007: Physics of Medical Imaging, San Diego, CA, USA, 17–22 February 2007; International Society for Optics and Photonics: Bellingham, WA, USA, 2007; Volume 6510, p. 65102. [Google Scholar]
- Yu, L.; Shiung, M.; Jondal, D.; McCollough, C.H. Development and validation of a practical lower-dose-simulation tool for optimizing computed tomography scan protocols. J. Comput. Assist. Tomogr.
**2012**, 36, 477–487. [Google Scholar] [CrossRef] - 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. Low-dose 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] [Green Version] - Moen, T.R.; Chen, B.; Holmes, D.R., III; Duan, X.; Yu, Z.; Yu, L.; Leng, S.; Fletcher, J.G.; McCollough, C.H. Low-dose CT image and projection dataset. Med. Phys.
**2021**, 48, 902–911. [Google Scholar] [CrossRef] [PubMed] - Li, X.; Samei, E. Comparison of patient size-based methods for estimating quantum noise in CT images of the lung. Med. Phys.
**2009**, 36, 541–546. [Google Scholar] [CrossRef] [PubMed] - Solomon, J.; Lyu, P.; Marin, D.; Samei, E. Noise and spatial resolution properties of a commercially available deep? Learning based CT reconstruction algorithm. Med. Phys.
**2020**, 47, 3961–3971. [Google Scholar] [CrossRef] [PubMed] - Samei, E.; Kinahan, P.; Nishikawa, R.M.M.; Maidment, A. Virtual Clinical Trials: Why and What (Special Section Guest Editorial). J. Med. Imaging
**2020**, 7, 042801. [Google Scholar] [CrossRef] [PubMed] - Anam, C.; Haryanto, F.; Widita, R.; Arif, I.; Dougherty, G.; McLean, D. Volume computed tomography dose index (CTDIvol) and size-specific dose estimate (SSDE) for tube current modulation (TCM) in CT scanning. Int. J. Radiat. Res.
**2018**, 16, 289–297. [Google Scholar] - Karimi, D.; Deman, P.; Ward, R.; Ford, N. A sinogram denoising algorithm for low-dose computed tomography. BMC Med. Imaging
**2016**, 16, 11. [Google Scholar] [CrossRef] [Green Version] - Koyuncu, H.; Ceylan, R. Elimination of white Gaussian noise in arterial phase CT images to bring adrenal tumours into the forefront. Comput. Med. Imaging Graph.
**2018**, 65, 46–57. [Google Scholar] [CrossRef] - Sheppard, J.P.; Nguyen, T.; Alkhalid, Y.; Beckett, J.S.; Salamon, N.; Yang, I. Risk of Brain Tumor Induction from Pediatric Head CT Procedures: A Systematic Literature Review. Brain Tumor Res. Treat.
**2018**, 6, 1–7. [Google Scholar] [CrossRef] [Green Version] - Bezdek, J.C.; Ehrlich, R.; Full, W. FCM: The Fuzzy c-MEANS Clustering Algorithm. Comput. Geosci.
**1984**, 10, 191–203. [Google Scholar] [CrossRef] - Höppner, F.; Klawonn, F.; Kruse, R.; Runkler, T. Fuzzy Cluster Analysis: Methods for Classification, Data Analysis and Image Recognition; John Wiley & Sons: Hoboken, NJ, USA, 1999. [Google Scholar]
- Jain, A.K. Data clustering: 50 years beyond K-means. Pattern Recognit. Lett.
**2010**, 31, 651–666. [Google Scholar] [CrossRef] - Mumford, D.; Shah, J. Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math.
**1989**, 42, 577–685. [Google Scholar] [CrossRef] [Green Version] - Gerber, S.; Pospisil, L.; Navandar, M.; Horenko, I. Low-cost scalable discretization, prediction, and feature selection for complex systems. Sci. Adv.
**2020**, 6, eaaw0961. [Google Scholar] [CrossRef] [PubMed] [Green Version] - de Wiljes, J.; Majda, A.; Horenko, I. An Adaptive Markov Chain Monte Carlo Approach to Time Series Clustering of Processes with Regime Transition Behavior. SIAM Multiscale Model. Simul.
**2013**, 11, 415–441. [Google Scholar] [CrossRef] - Rudin, L.I.; Osher, S.; Fatemi, E. Nonlinear total variation based noise removal algorithms. Phys. D Nonlinear Phenom.
**1992**, 60, 259–268. [Google Scholar] [CrossRef] - Chambolle, A. Finite-differences discretizations of the mumford-shah functional. ESAIM Math. Model. Numer. Anal.
**1999**, 33, 261–288. [Google Scholar] [CrossRef] [Green Version] - Lysaker, O.M.; Lundervold, A.; Tai, X. Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time. IEEE Trans. Image Process.
**2003**, 12, 1579–1590. [Google Scholar] [CrossRef] - Chan, T.F.; Shen, J. Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods; SIAM: Philadelphia, PA, USA, 2005. [Google Scholar]
- Pock, T.; Cremers, D.; Bischof, H.; Chambolle, A. An algorithm for minimizing the Mumford-Shah functional. In Proceedings of the 2009 IEEE 12th International Conference on Computer Vision, Kyoto, Japan, 29 September–2 October 2009; pp. 1133–1140. [Google Scholar] [CrossRef] [Green Version]
- Hohm, K.; Storath, M.; Weinmann, A. An algorithmic framework for Mumford–Shah regularization of inverse problems in imaging. Inverse Probl.
**2015**, 31, 115011. [Google Scholar] [CrossRef] [Green Version] - Paragios, N.; Duncan, J.; Ayache, N. Handbook of Biomedical Imaging: Methodologies and Clinical Research; Springer: Berlin/Heidelberg, Germany, 2015; p. 590. [Google Scholar] [CrossRef]
- Barzilai, J.; Borwein, J.M. Two point step size gradient methods. IMA J. Numer. Anal.
**1988**, 8, 141–148. [Google Scholar] - Birgin, E.G.; Martínez, J.M.; Raydan, M.M. Nonmonotone spectral projected gradient methods on convex sets. SIAM J. Optim.
**2000**, 10, 1196–1211. [Google Scholar] - Boyd, L.; Vandenberghe, L. Convex Optimization, 1st ed.; Cambridge University Press: New York, NY, USA, 2004. [Google Scholar]
- Chen, Y.; Ye, X. Projection onto a simplex. arXiv
**2011**, Unpublished manuscript. arXiv:101.6081. [Google Scholar] - Grippo, L.; Lampariello, F.; Lucidi, S. nonmonotone line search technique for Newton’s method. SIAM J. Numer. Anal.
**1986**, 23, 707–716. [Google Scholar]

**Figure 1.**Graphical representation of our proposed pipeline workflow for automated generation and risk assessment of CT images. (

**A**): Initial reference data can be either a set of real CT-data generated using high-dose radiation or artificially simulated data. (

**B**) Exemplary high-quality and low-quantum noise image of lung vessels. (

**C**) exemplary low-dose CT images with high quantum noise. (

**D**) Workflow from image generation to subsequent benchmarking of ML/DL-denoising methods. Starting with high-quality data or artificially generated reference data, respectively, a spectrum of image noise $\sigma $ is added for a multitude of combinations from patient-specific and CT control variables, as suggested in Equation (1). The noisy images are then denoised using various state-of-the-art methods and the processed images are compared to the original reference data.

**Figure 2.**Graphical overview of the Probabilistic Mumford–Shah (PMS) framework. (

**A**) Summary of the parameters and variables. (

**B**) Core rSPA algorithm idea: 3D-denoising with the regularized Scalable Probabilistic Approximation algorithm (rSPA). Given the (noisy) CT voxel data V, rSPA minimizes the function $L(C,\Gamma )$ and seeks for the optimal segmentation of V in terms of the K spatially-persistent latent features characterized by the latent feature probabilities in K rows of the matrix $\Gamma $, as well as by the latent colors as K columns of the latent color matrix C. Persistency of the feature segmentation is imposed by the second term of the right-hand side of the function $L(C,\Gamma )$, which penalizes the differences in feature probability values in the spatially-neighboring points. (

**C**) Denoising idea: latent feature probabilities are persistent (slowly-changing) 3D functions. (

**D**) Graphical representation of the overlapping domain decomposition used in the parallel DD-rSPA algorithm.

**Figure 3.**Radiation exposure, quantum noise, and denoising performance of CNNs and rSPA in low-radiation and ultra-low-radiation thorax CT regimes. (

**A**) Reference data of a thorax CT voxel fragment (approx. 5 cm${}^{3}$) of a 19-year-old female with the BMI 27.5, acquired with the Somatum Emotion 16 2007 (Siemens Aktiengesellschaft, Berlin, Germany) at 130 kV tube voltage. (

**B**) Simulated decrease in the radiation exposure ${\mathrm{CTDI}}_{\mathrm{vol}}$ from 15.6 mGy (reference frame) to 3.3 mGy (for low-radiation simulations) and 0.5 mGy (ultra-low-radiation) results in a significant increase of quantum noise. (

**C**) Reconstructed images using CNNs. (

**D**) Reconstructed images using rSPA. (

**E**) 3D segmentation of the original reference frame. (

**F**) 3D segmentation based on the images denoised using CNNs. (

**G**) 3D-segmentation of the images denoised by rSPA.

**Figure 4.**Comparing denoising performance on synthetic CT images of noisy circles, with DL from Figure 3 additionally trained to recognize circles for non-Gaussian noise model: (

**A**) medium noise scenario, corresponding to low-radiation regime with around 3.3 mGy; (

**B**) high noise scenario, corresponding to ultra-low-radiation regime with 0.5 mGy.

**Figure 5.**Comparing denoising quality, cost and parallelizability: (

**A**–

**C**) comparison of PMS rSPA algorithm to the regularized Mumford–Shah denoising tool introduced in [93] and to the additionally trained DL denoising algorithm from Figure 3 and Figure 4; (

**D**,

**E**) computational cost scaling and performance for DL (without taking into account time for additional training), sequential rSPA, parallel DD-rSPA and DD-rSPA followed by DL. Each point of each method’s curve and surface is obtained from statistical averaging of the respective values obtained by analyzing 10 randomly-generated images with these particular combinations of image size and noise level.

**Figure 6.**Comparing CT image denoising performances for three CT noise models: (

**A**) additive Gaussian noise model (CT noise variance is independent of the feature color); (

**B**) multiplicative non-Gaussian noise model (CT noise variance changes with the amplitude of the underlying color signal); (

**C**) empirical noise obtained from the thorax CT patient data. In (

**A**,

**B**), generation of synthetic images was performed for a patient with a water-equivalent diameter of 30 cm, which is subject to a Thorax CT with a typical tube voltage of 120 kV in the range of tube currents between 5–180 mA and a set of artificial anatomic features from Figure 2A (with a feature contrast of 200 HU). In (

**C**), real patient data were used. Comparison is performed with three primary image quality criteria: with mean squared error (left panels); with peak signal-to-noise ratio (middle panels); and with the 3D multiscale structural similarity index (right panels).

**Figure 7.**Comparing denoising methods with the average Multiscale Structural Similarity Index (3D MS-SSIM): (

**A**) varying the true underlying feature contrast and LAR for a synthetic 30-year-old female patient with a water-equiv. cross-section of 27 cm; (

**B**) varying the true underlying feature contrast and LAR for a synthetic 1-year-old female infant patient with a water-equiv. cross-section of 12.7 cm; (

**C**) denoising performance comparison when varying the patient size and the effective absorbed radiation dose density, with the 200 Hounsfield Units (HU) feature contrast differences.

**Figure 8.**Comparing denoising methods with the average Multiscale Structural Similarity Index (3D MS-SSIM) for simulated thorax CT: (

**A**) varying the absorbed radiation dose for a synthetic 30-year-old female patient with a water-equiv. cross-section of 27 cm; (

**B**) varying the absorbed radiation dose for a synthetic 1-year-old female infant patient with a water-equiv. cross-section of 12.7 cm. Noiseless thorax CT image used as reference in this performance comparison is available at https://www.dropbox.com/s/29x0xivg8l80q10/female_lung_thorax_CT_image_section_v2.mat?dl=0 (accessed on 18 March 2022). Dotted lines show 95% nonparametric confidence intervals (c.i.) obtained for every value of ${\mathrm{CTDI}}_{\mathrm{vol}}$ from 100 different independently-generated noisy synthetic CT images, using the MATLAB function quantile().

**Table 1.**Deterioration of CT image quality (decrease in 3D MS-SSIM index, baseline = 100%) caused by a reduction of lifetime attributable risk (LAR) for different methods. The CT scans pertain to the infant patient, with a fixed feature contrast of 200 HU.

Image Quality Loss (3D MS-SSIM, in %) | |||||
---|---|---|---|---|---|

Reduction ofLAR (in %) | Raw Image | 3D GaussianFiltering | DL CNNDenoising | 3D WaveletFiltering | rSPA |

6 | 1.16 | 0.77 | 0.54 | 0.79 | 0.01 |

16 | 1.26 | 0.84 | 0.59 | 0.86 | 0.01 |

23 | 1.34 | 0.90 | 0.63 | 0.92 | 0.01 |

29 | 1.44 | 0.96 | 0.67 | 0.99 | 0.01 |

36 | 1.56 | 1.04 | 0.73 | 1.07 | 0.01 |

42 | 1.70 | 1.13 | 0.79 | 1.16 | 0.01 |

49 | 1.88 | 1.25 | 0.88 | 1.29 | 0.01 |

56 | 2.12 | 1.41 | 0.99 | 1.45 | 0.02 |

63 | 2.45 | 1.63 | 1.14 | 1.67 | 0.02 |

70 | 2.92 | 1.95 | 1.36 | 2.00 | 0.02 |

77 | 3.69 | 2.46 | 1.72 | 2.52 | 0.03 |

83 | 5.11 | 3.40 | 2.38 | 3.49 | 0.04 |

90 | 8.72 | 5.81 | 4.07 | 5.97 | 0.06 |

97 | 25.96 | 18.20 | 13.37 | 18.16 | 0.17 |

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## Share and Cite

**MDPI and ACS Style**

Horenko, I.; Pospíšil, L.; Vecchi, E.; Albrecht, S.; Gerber, A.; Rehbock, B.; Stroh, A.; Gerber, S.
Low-Cost Probabilistic 3D Denoising with Applications for Ultra-Low-Radiation Computed Tomography. *J. Imaging* **2022**, *8*, 156.
https://doi.org/10.3390/jimaging8060156

**AMA Style**

Horenko I, Pospíšil L, Vecchi E, Albrecht S, Gerber A, Rehbock B, Stroh A, Gerber S.
Low-Cost Probabilistic 3D Denoising with Applications for Ultra-Low-Radiation Computed Tomography. *Journal of Imaging*. 2022; 8(6):156.
https://doi.org/10.3390/jimaging8060156

**Chicago/Turabian Style**

Horenko, Illia, Lukáš Pospíšil, Edoardo Vecchi, Steffen Albrecht, Alexander Gerber, Beate Rehbock, Albrecht Stroh, and Susanne Gerber.
2022. "Low-Cost Probabilistic 3D Denoising with Applications for Ultra-Low-Radiation Computed Tomography" *Journal of Imaging* 8, no. 6: 156.
https://doi.org/10.3390/jimaging8060156