Convergent data-driven regularizations for CT reconstruction. (English) Zbl 07883806
Summary: The reconstruction of images from their corresponding noisy Radon transform is a typical example of an ill-posed linear inverse problem as arising in the application of computerized tomography (CT). As the (naïve) solution does not depend on the measured data continuously, regularization is needed to reestablish a continuous dependence. In this work, we investigate simple, but yet still provably convergent approaches to learning linear regularization methods from data. More specifically, we analyze two approaches: one generic linear regularization that learns how to manipulate the singular values of the linear operator in an extension of our previous work, and one tailored approach in the Fourier domain that is specific to CT-reconstruction. We prove that such approaches become convergent regularization methods as well as the fact that the reconstructions they provide are typically much smoother than the training data they were trained on. Finally, we compare the spectral as well as the Fourier-based approaches for CT-reconstruction numerically, discuss their advantages and disadvantages and investigate the effect of discretization errors at different resolutions.
MSC:
| 47A52 | Linear operators and ill-posed problems, regularization |
| 65J22 | Numerical solution to inverse problems in abstract spaces |
| 68T05 | Learning and adaptive systems in artificial intelligence |
References:
| [1] | Adler, J.; Öktem, O., Learned primal-dual reconstruction, IEEE Trans. Med. Imaging, 37, 6, 1322-1332, 2018 · doi:10.1109/TMI.2018.2799231 |
| [2] | Aharon, M.; Elad, M.; Bruckstein, A., K-SVD: an algorithm for designing overcomplete dictionaries for sparse representation, IEEE Trans. Signal Process., 54, 11, 4311-4322, 2006 · Zbl 1375.94040 · doi:10.1109/TSP.2006.881199 |
| [3] | Alberti, GS; De Vito, E.; Lassas, M.; Ratti, L.; Santacesaria, M.; Ranzato, M.; Beygelzimer, A.; Dauphin, Y.; Liang, PS; Vaughan, JW, Learning the optimal Tikhonov regularizer for inverse problems, Advances in Neural Information Processing Systems, 25205-25216, 2021, New York: Curran Associates Inc, New York |
| [4] | Ambrosio, L.; Gigli, N., A user’s guide to optimal transport, In: Modelling and Optimisation of Flows on Networks, 1-155, 2013, Berlin: Springer, Berlin · doi:10.1007/978-3-642-32160-3_1 |
| [5] | Amos, B., Xu, L., Kolter, J.Z.: Input convex neural networks. In: ICML, pp. 146-155. PMLR (2017) |
| [6] | Arridge, S.; Maass, P.; Öktem, O.; Schönlieb, C-B, Solving inverse problems using data-driven models, Acta Numer., 28, 1-174, 2019 · Zbl 1429.65116 · doi:10.1017/S0962492919000059 |
| [7] | Aspri, A.; Banert, S.; Öktem, O.; Scherzer, O., A data-driven iteratively regularized Landweber iteration, Numer. Funct. Anal. Optim., 41, 10, 1190-1227, 2020 · Zbl 07241785 · doi:10.1080/01630563.2020.1740734 |
| [8] | Baguer, DO; Leuschner, J.; Schmidt, M., Computed tomography reconstruction using deep image prior and learned reconstruction methods, Inverse Prob., 36, 9, 2020 · Zbl 1460.68093 · doi:10.1088/1361-6420/aba415 |
| [9] | Bai, S., Kolter, J.Z., Koltun, V. Deep equilibrium models. In: Wallach, H., Larochelle, H., Beygelzimer, A., d’Alché-Buc, F., Fox, E., Garnett, R. (eds.) Advances in Neural Information Processing Systems, vol. 32, Curran Associates, Inc., New York (2019) |
| [10] | Barutcu, S.; Aslan, S.; Katsaggelos, AK; Gürsoy, D., Limited-angle computed tomography with deep image and physics priors, Sci. Rep., 11, 1, 17740, 2021 · doi:10.1038/s41598-021-97226-2 |
| [11] | Bauermeister, H., Burger, M., Moeller, M.: Learning spectral regularizations for linear inverse problems. In: NeurIPS 2020 Workshop on Deep Learning and Inverse Problems (2020) |
| [12] | Benning, M.; Burger, M., Modern regularization methods for inverse problems, Acta Numer., 27, 1-111, 2018 · Zbl 1431.65080 · doi:10.1017/S0962492918000016 |
| [13] | Bora, A., Jalal, A., Price, E., Dimakis, A.G.: Compressed sensing using generative models. In: ICML, pp. 537-546. PMLR (2017) |
| [14] | Chen, Y.; Ranftl, R.; Pock, T., Insights into analysis operator learning: from patch-based sparse models to higher order MRFs, IEEE Trans. Image Process., 23, 3, 1060-1072, 2014 · Zbl 1374.94065 · doi:10.1109/TIP.2014.2299065 |
| [15] | Chen, H.; Zhang, Y.; Chen, Y.; Zhang, J.; Zhang, W.; Sun, H.; Lyu, Y.; Liao, P.; Zhou, J.; Wang, G., LEARN: learned experts’ assessment-based reconstruction network for sparse-data CT, IEEE Trans. Med. Imaging, 37, 6, 1333-1347, 2018 · doi:10.1109/TMI.2018.2805692 |
| [16] | Chen, H.; Zhang, Y.; Kalra, MK; Lin, F.; Chen, Y.; Liao, P.; Zhou, J.; Wang, G., Low-dose CT with a residual encoder-decoder convolutional neural network, IEEE Trans. Med. Imaging, 36, 12, 2524-2535, 2017 · doi:10.1109/TMI.2017.2715284 |
| [17] | Chung, J.; Chung, M.; O’Leary, DP, Designing optimal spectral filters for inverse problems, SIAM J. Sci. Comput., 33, 6, 3132-3152, 2011 · Zbl 1269.65040 · doi:10.1137/100812938 |
| [18] | Engl, HW; Hanke, M.; Neubauer, A., Regularization of Inverse Problems, 1996, Dordrecht: Kluwer, Dordrecht · Zbl 0859.65054 · doi:10.1007/978-94-009-1740-8 |
| [19] | He, J.; Wang, Y.; Ma, J., Radon inversion via deep learning, IEEE Trans. Med. Imaging, 39, 6, 2076-2087, 2020 · doi:10.1109/TMI.2020.2964266 |
| [20] | Jin, KH; McCann, MT; Froustey, E.; Unser, M., Deep convolutional neural network for inverse problems in imaging, IEEE Trans. Image Process., 26, 9, 4509-4522, 2017 · Zbl 1409.94275 · doi:10.1109/TIP.2017.2713099 |
| [21] | Kobler, E., Effland, A., Kunisch, K., Pock, T.: Total deep variation for linear inverse problems. In: CVPR, pp. 7546-7555 (2020) |
| [22] | Latorre, F., Ektekhari, A., Cevher, V. Fast and provable ADMM for learning with generative priors. In: Wallach, H., Larochelle, H., Beygelzimer, A., d’Alché-Buc, F., Fox, E., Garnett, R. (eds.) Advances in Neural Information Processing Systems, vol. 32, Curran Associates, Inc., New York (2019) |
| [23] | Leuschner, J.; Schmidt, M.; Baguer, DO; Maass, P., LoDoPaB-CT, a benchmark dataset for low-dose computed tomography reconstruction, Sci. Data, 8, 109, 2021 · doi:10.1038/s41597-021-00893-z |
| [24] | Leuschner, J.; Schmidt, M.; Ganguly, PS; Andriiashen, V.; Coban, SB; Denker, A.; Bauer, D.; Hadjifaradji, A.; Batenburg, KJ; Maass, P.; van Eijnatten, M., Quantitative comparison of deep learning-based image reconstruction methods for low-dose and sparse-angle CT applications, J. Imaging, 7, 3, 44, 2021 · doi:10.3390/jimaging7030044 |
| [25] | Li, H.; Schwab, J.; Antholzer, S.; Haltmeier, M., NETT: solving inverse problems with deep neural networks, Inverse Prob., 36, 6, 2020 · Zbl 1456.65038 · doi:10.1088/1361-6420/ab6d57 |
| [26] | 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, 38, 10, 2469-2481, 2019 · doi:10.1109/TMI.2019.2910760 |
| [27] | Mairal, J., Ponce, J., Sapiro, G., Zisserman, A., Bach, F.: Supervised dictionary learning. In: NeurIPS (2008) |
| [28] | Meinhardt, T., Moeller, M., Hazirbas, C., Cremers, D.: Learning proximal operators: using denoising networks for regularizing inverse imaging problems. In: ICCV, pp. 1781-1790 (2017) |
| [29] | Moeller, M., Mollenhoff, T., Cremers, D.: Controlling neural networks via energy dissipation. In: ICCV, pp. 3256-3265 (2019) |
| [30] | Riccio, D., Ehrhardt, M.J., Benning, M.: Regularization of inverse problems: deep equilibrium models versus bilevel learning. arXiv:2206.13193 (2022) |
| [31] | Rick Chang, J., Li, C.-L., Poczos, B., Vijaya Kumar, B., Sankaranarayanan, A.C.: One network to solve them all—solving linear inverse problems using deep projection models. In: ICCV, pp. 5888-5897 (2017) |
| [32] | Romano, Y.; Elad, M.; Milanfar, P., The little engine that could: regularization by denoising (RED), SIAM J. Imaging Sci., 10, 4, 1804-1844, 2017 · Zbl 1401.62101 · doi:10.1137/16M1102884 |
| [33] | Ronneberger, O.; Fischer, P.; Brox, T.; Navab, N.; Hornegger, J.; Wells, WM; Frangi, AF, U-net: convolutional networks for biomedical image segmentation, Medical Image Computing and Computer-Assisted Intervention—MICCAI 2015, 234-241, 2015, Cham: Springer, Cham |
| [34] | Roth, S.; Black, MJ, Fields of experts, Int. J. Comput. Vis., 82, 2, 205-229, 2009 · Zbl 1477.68419 · doi:10.1007/s11263-008-0197-6 |
| [35] | Servieres, M.C.J., Normand, N., Subirats, P., Guedon, J.: Some links between continuous and discrete Radon transform. In: Fitzpatrick, J.M., Sonka, M. (eds.) Medical Imaging 2004: Image Processing, vol. 5370, pp. 1961-1971. SPIE, WA, United States. International Society for Optics and Photonics (2004). doi:10.1117/12.533472 |
| [36] | Ulyanov, D., Vedaldi, A., Lempitsky, V.: Deep image prior. In: CVPR, pp. 9446-9454 (2018) |
| [37] | Wang, G.; Ye, JC; De Man, B., Deep learning for tomographic image reconstruction, Nat. Mach. Intell., 2, 12, 737-748, 2020 · doi:10.1038/s42256-020-00273-z |
| [38] | Xiang, J.; Dong, Y.; Yang, Y., FISTA-Net: learning a fast iterative shrinkage thresholding network for inverse problems in imaging, IEEE Trans. Med. Imaging, 40, 5, 1329-1339, 2021 · doi:10.1109/TMI.2021.3054167 |
| [39] | Xu, M.; Hu, D.; Luo, F.; Liu, F.; Wang, S.; Wu, W., Limited-angle x-ray CT reconstruction using image gradient \(\ell_0\)-norm with dictionary learning, IEEE Trans. Radiat. Plasma Med. Sci., 5, 1, 78-87, 2021 · doi:10.1109/TRPMS.2020.2991887 |
| [40] | Zhang, M.; Gu, S.; Shi, Y., The use of deep learning methods in low-dose computed tomography image reconstruction: a systematic review, Complex Intell. Syst., 8, 5545-5561, 2022 · doi:10.1007/s40747-022-00724-7 |
| [41] | Zhu, B.; Liu, JZ; Cauley, SF; Rosen, BR; Rosen, MS, Image reconstruction by domain-transform manifold learning, Nature, 555, 7697, 487-492, 2018 · doi:10.1038/nature25988 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
