SDF: A self-supervised denoiser framework for X-Ray computed tomography

Emilien Valat; Andreas Hauptmann; Ozan Öktem

doi:10.3934/ammc.2025012

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September 2025, Volume 5, 86-100. Doi: 10.3934/ammc.2025012

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SDF: A self-supervised denoiser framework for X-Ray computed tomography

1.
Kungliga Tekniska Högskolan, Sweden
2.
Research Unit of Mathematical Sciences, University of Oulu, Finland
3.
Department of Computer Science, University College London, UK

^*Corresponding author: Emilien Valat
^*Corresponding author: Emilien Valat

Received: June 2025

Revised: September 2025

Published online: September 30, 2025

OÖ and EV acknowledge support from Swedish Energy Agency grant P2022-00286 and FORMAS grant 2022-00469. AH acknowledges support from DigitalFutures Scholar-in-Residence grant KTH-RPROJ-0146472 2 and from the Research Council of Finland project Nos. 353093, 359186, 338408.

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Abstract

We introduce the self-supervised denoiser framework (SDF), a training procedure for learned image denoisers in the context of X-ray computed tomography (CT). Computing a computed tomography CT image amounts to recovering a model parameter from indirect observations of the said model. A popular approach to solve this problem is to train a neural network in a supervised way to enhance an initial approximation of the model, learning the observation-to-image mapping, thus requiring image-observation pairs. In this work, we introduce a method that relies only on observations for pretraining and a few images for fine-tuning image denoisers for sparse-view and low-dose X-ray Computed Tomography (CT).

We propose to train an image denoiser in the sinogram space by defining the learning task of an image denoiser as the prediction of one sinogram subset from another. Our approach does not require ground-truth image data, leverages the abundant data modality in CT, the sinogram, and can drastically enhance the quality of images reconstructed from a fraction of the measurements. We demonstrate that SDF produces better image quality in terms of peak signal-to-noise ratio than other analytical and self-supervised frameworks in 2D fan-beam and 3D cone-beam CT settings. Moreover, we show that the enhancement provided by SDF carries over when fine-tuning the image denoiser on a few examples, making it a suitable pretraining technique when there is little high-quality image data. Our results are established on experimental datasets, making SDF a strong candidate for being the building block of foundational, image-enhancement models in CT.

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Mathematics Subject Classification: Primary: 94A08, 68U10, 65R32.

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Figure 1. The blue nodes represent sinogram-space data, and the orange nodes represent image-space data. $ \Gamma_{i,j}^{\theta} $, the composition of $ \operatorname{\mathcal{A}}^{\dagger}_i $ with $ \Lambda_{\theta} $ (denoted by $ \operatorname{\mathcal{E}}_i $) and $ A_j $ (denoted $ \operatorname{\mathcal{D}}_j $), can be seen as a sinogram autoencoder that has $ \mathcal{X} $ (the image manifold) as its latent space

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Figure 2. Slices of an object reconstructed from mode1 (A) and mode2 (B) sinograms of the 2DeteCT dataset, and the central slices of a volume reconstructed from Feldkamp-Davis-Kress (FDK) ran on orbit 1 (C), and least-squares (LS) ran on all orbits (D) of the Walnuts dataset

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Figure 3. Architecture of the image denoiser $ \Lambda_{\theta} $ used in all our experiments. $ C(n,m) $ denotes a convolutional layer with $ n $ input layers and $ m $ output layers, a kernel size of 5, and a padding of 2, with LeakyReLU activation. For 3D CT, $ n_{\text{filters}} = 8 $ and for 2D CT, $ n_{\text{filters}} = 32 $

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Figure 4. Qualitative comparison of a Mode2 image reconstructed with FBP 4a, LS 4b, SDF 4c, and N2I 4d

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Figure 5. Evolution of the PSNR (solid lines) and SSIM (dashed lines) against the angular sparsity

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Figure 6. Central slice of Walnut 39 reconstructed with LS on full sinogram and SDF on subsampled sinogram

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Table 1. Performance of self-supervised, analytic, and iterative methods, before and after fine tuning

		Before Fine-Tuning		After Fine-Tuning
Mode	Method	SSIM	PSNR	SSIM	PSNR
1	FBP	$ 0.04 \pm 0.01 $	$ 6.77 \pm 2.00 $	$ 0.21 \pm 0.02 $	$ 27.52 \pm 1.95 $
	LS	$ \mathbf{0.55 \pm 0.04} $	$ 26.65 \pm 1.94 $	$ 0.32 \pm 0.03 $	$ \mathbf{29.19 \pm 1.90} $
	SDF	$ 0.26 \pm 0.03 $	$ \mathbf{26.96 \pm 1.94} $	$ \mathbf{0.34 \pm 0.04} $	$ 28.34 \pm 1.95 $
	N2I	$ 0.23 \pm 0.03 $	$ 23.05 \pm 1.96 $	$ 0.28 \pm 0.03 $	$ 27.61 \pm 1.94 $
	EI	$ 0.09 \pm 0.01 $	$ 14.23 \pm 2.00 $	$ 0.20 \pm 0.03 $	$ 25.13 \pm 1.98 $
2	FBP	$ 0.14 \pm 0.01 $	$ 21.99 \pm 1.95 $	$ \mathbf{0.46 \pm 0.05} $	$ 33.32 \pm 1.93 $
	LS	$ \mathbf{0.81 \pm 0.04} $	$ \mathbf{31.03 \pm 1.88} $	$ 0.27 \pm 0.03 $	$ 30.59 \pm 1.89 $
	SDF	$ 0.24 \pm 0.02 $	$ 30.59 \pm 1.91 $	$ 0.31 \pm 0.02 $	$ 33.97 \pm 1.94 $
	N2I	$ 0.41 \pm 0.04 $	$ 28.66 \pm 1.93 $	$ 0.39 \pm 0.03 $	$ 32.59 \pm 1.92 $
	EI	$ 0.29 \pm 0.03 $	$ 29.25 \pm 1.93 $	$ 0.44 \pm 0.04 $	$ \mathbf{34.14 \pm 1.94} $

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Table 2. Results of experiments on angular and orbital subsets of the Walnuts dataset

Method	PSNR	SSIM
FDK	$ 28.65 \pm 2.34 $	$ 0.58 \pm 0.08 $
SDF	$ 36.76 \pm 2.39 $	$ 0.98 \pm 0.08 $
CNN	$ 25.31 \pm 1.99 $	$ 0.03 \pm 0.01 $

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References

[1]	W. van Aarle, W. J. Palenstijn, J. Cant, E. Janssens, F. Bleichrodt, A. Dabravolski, J. D. Beenhouwer, K. J. Batenburg and J. Sijbers, Fast and flexible X-ray tomography using the ASTRA toolbox, Opt. Express, 24 (2016), 25129-25147. doi: 10.1364/OE.24.025129.
[2]	J. Adler, S. Lunz, O. Verdier, C.-B. Schönlieb and O. Öktem, Task adapted reconstruction for inverse problems, Inverse Problems, 38 (2022), 075006, 21 pp. doi: 10.1088/1361-6420/ac28ec.
[3]	J. Adler and O. Oktem, Learned primal-dual reconstruction, IEEE Transactions on Medical Imaging, 37 (2018), 1322-1332. doi: 10.1109/TMI.2018.2799231.
[4]	S. Arridge, P. Maass, O. Öktem and C.-B. Schönlieb, Solving inverse problems using data-driven models, Acta Numerica, 28 (2019), 1-174. doi: 10.1017/S0962492919000059.
[5]	I. Ayad, N. Larue and M. K. Nguyen, QN-Mixer: A quasi-Newton MLP-mixer model for sparse-view CT reconstruction, 2024, URL https://arXiv.org/abs/2402.17951, eprint: arXiv: 2402.17951.
[6]	J. Batson and L. Royer, Noise2Self: Blind denoising by self-supervision, 2019, URL https://arXiv.org/abs/1901.11365, eprint: arXiv: 1901.11365.
[7]	D. Chen, J. Tachella and M. E. Davies, Equivariant imaging: Learning beyond the range space, 2021, URL http://arXiv.org/abs/2103.14756, arXiv: 2103.14756 [cs].
[8]	D. Chen, J. Tachella and M. E. Davies, Robust equivariant imaging: A fully unsupervised framework for learning to image from noisy and partial measurements, 2022, URL https://arXiv.org/abs/2111.12855, eprint: arXiv: 2111.12855.
[9]	H. Der Sarkissian, F. Lucka, M. van Eijnatten, G. Colacicco, S. B. Coban and K. J. Batenburg, A cone-beam X-ray computed tomography data collection designed for machine learning, Scientific Data, 6 (2019), 215. doi: 10.1038/s41597-019-0235-y.
[10]	J. Devlin, M.-W. Chang, K. Lee and K. Toutanova, BERT: Pre-training of deep bidirectional transformers for language understanding, 2019, URL https://arXiv.org/abs/1810.04805, eprint: arXiv: 1810.04805.
[11]	M. van Eijnatten, L. Rundo, K. J. Batenburg, F. Lucka, E. Beddowes, C. Caldas, F. A. Gallagher, E. Sala, C.-B. Schönlieb and R. Woitek, 3D deformable registration of longitudinal abdominopelvic CT images using unsupervised deep learning, Computer Methods and Programs in Biomedicine, 208 (2021), 106261. doi: 10.1016/j.cmpb.2021.106261.
[12]	M. Genzel, I. Gühring, J. Macdonald and M. März, Near-exact recovery for tomographic inverse problems via deep learning, in International Conference on Machine Learning, PMLR, 2022, 7368-7381.
[13]	N. Gruber, J. Schwab, E. Gizewski and M. Haltmeier, Sparse2inverse: Self-supervised inversion of sparse-view ct data, 2024, URL https://arXiv.org/abs/2402.16921.
[14]	I. E. Hamamci, S. Er, F. Almas, A. G. Simsek, S. N. Esirgun, I. Dogan, M. F. Dasdelen, O. F. Durugol, B. Wittmann, T. Amiranashvili, E. Simsar, M. Simsar, E. B. Erdemir, A. Alanbay, A. Sekuboyina, B. Lafci, C. Bluethgen, M. K. Ozdemir and B. Menze, Developing generalist foundation models from a multimodal dataset for 3D computed tomography, 2024, URL https://arXiv.org/abs/2403.17834, eprint: arXiv: 2403.17834.
[15]	L. He, W. Du, P. Liao, F. Fan, H. Chen, H. Yang and Y. Zhang, Solving zero-shot sparse-view CT reconstruction with variational score solver, IEEE Transactions on Medical Imaging, 44 (2025), 3586-3599. doi: 10.1109/TMI.2024.3475516.
[16]	A. A. Hendriksen, D. M. Pelt and K. J. Batenburg, Noise2Inverse: Self-supervised deep convolutional denoising for tomography, IEEE Trans. Computational Imaging, 6 (2020), 1320-1335, URL http://dblp.uni-trier.de/db/journals/tci/tci6.html#HendriksenPB20. doi: 10.1109/TCI.2020.3019647.
[17]	A. A. Hendriksen, D. Schut, W. J. Palenstijn, N. Viganò, J. Kim, D. Pelt, T. van Leeuwen and K. J. Batenburg, Tomosipo: Fast, flexible, and convenient 3D tomography for complex scanning geometries in python, Optics Express, 29 (2021), 40494-40513. doi: 10.1364/OE.439909.
[18]	M. Jaderberg, K. Simonyan, A. Zisserman and K. Kavukcuoglu, Spatial transformer networks, 2016, URL https://arXiv.org/abs/1506.02025, eprint: arXiv: 1506.02025.
[19]	A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, Society for Industrial and Applied Mathematics, 2001, URL https://epubs.siam.org/doi/abs/10.1137/1.9780898719277. doi: 10.1137/1.9780898719277.
[20]	U. S. Kamilov, C. A. Bouman, G. T. Buzzard and B. Wohlberg, Plug-and-play methods for integrating physical and learned models in computational imaging: Theory, algorithms, and applications, IEEE Signal Processing Magazine, 40 (2023), 85-97. doi: 10.1109/MSP.2022.3199595.
[21]	M. B. Kiss, A. Biguri, Z. Shumaylov, F. Sherry, K. J. Batenburg, C.-B. Schönlieb and F. Lucka, Benchmarking learned algorithms for computed tomography image reconstruction tasks, 2024, URL https://arXiv.org/abs/2412.08350, eprint: arXiv: 2412.08350.
[22]	M. B. Kiss, S. B. Coban, K. J. Batenburg, T. van Leeuwen and F. Lucka, 2DeteCT – A large 2D expandable, trainable, experimental computed tomography dataset for machine learning, Scientific Data, 10 (2023), 576, URL http://arXiv.org/abs/2306.05907, arXiv: 2306.05907 [cs, eess]. doi: 10.1038/s41597-023-02484-6.
[23]	O. Kosomaa, S. Laine, T. Karras, M. Aittala and J. Lehtinen, Simulator-based self-supervision for learned 3D tomography reconstruction, 2023, URL https://arXiv.org/abs/2212.07431, eprint: arXiv: 2212.07431.
[24]	J. Leuschner, M. Schmidt, P. S. Ganguly, V. Andriiashen, S. B. Coban, A. Denker, D. Bauer, A. Hadjifaradji, K. J. Batenburg, P. Maass and M. van Eijnatten, Quantitative comparison of deep learning-based image reconstruction methods for low-dose and sparse-angle CT applications, Journal of Imaging, 7 (2021), 44. doi: 10.3390/jimaging7030044.
[25]	T. R. Moen, B. Chen, D. R. Holmes, X. Duan, Z. Yu, L. Yu, S. Leng, J. G. Fletcher and C. H. McCollough, Low-dose CT image and projection dataset, Medical Physics, 48 (2021), 902-911. doi: 10.1002/mp.14594.
[26]	A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, A. Desmaison, A. Köpf, E. Yang, Z. DeVito, M. Raison, A. Tejani, S. Chilamkurthy, B. Steiner, L. Fang, J. Bai and S. Chintala, PyTorch: An imperative style, high-performance deep learning library, 2019, URL https://arXiv.org/abs/1912.01703, eprint: arXiv: 1912.01703.
[27]	J. Rudzusika, B. Bajić, T. Koehler and O. Öktem, 3D helical CT reconstruction with a memory efficient learned primal-dual architecture, IEEE Transactions on Computational Imaging, 10 (2024), 1414-1424. doi: 10.1109/TCI.2024.3463485.
[28]	J. Tachella, D. Chen and M. Davies, Unsupervised learning from incomplete measurements for inverse problems, 2022, URL https://arXiv.org/abs/2201.12151, eprint: arXiv: 2201.12151.
[29]	M. O. Unal, M. Ertas and I. Yildirim, Proj2Proj: Self-supervised low-dose CT reconstruction, PeerJ. Computer Science, 10 (2024), e1849. doi: 10.7717/peerj-cs.1849.
[30]	E. Valat, A. Biguri, L. E. Sanchez, C. McCague, O. Öktem and C.-B. Schönlieb, Empirical evidence of the task-adapted reconstruction framework for joint CT reconstruction and segmentation, Applied Mathematics for Modern Challenges, 2 (2024), 287-300. doi: 10.3934/ammc.2024015.
[31]	B. Yaman, S. A. H. Hosseini, S. Moeller, J. Ellermann, K. Uğurbil and M. Akçakaya, Self‐supervised learning of physics‐guided reconstruction neural networks without fully sampled reference data, Magnetic Resonance in Medicine, 84 (2020), 3172-3191, Publisher: Wiley. doi: 10.1002/mrm.28378.
[32]	K. Zhang, W. Zuo, Y. Chen, D. Meng and L. Zhang, Beyond a Gaussian denoiser: Residual learning of deep CNN for image denoising, IEEE Transactions on Image Processing, 26 (2017), 3142-3155. doi: 10.1109/TIP.2017.2662206.