The DOI matches its Crossref record; 74 of the 83 references it cites were found in a registry or an index; none of the 13 sources its claims cite is checked for support yet.
Deep Convolutional Neural Network for Inverse Problems in Imaging
Open accessAbstract
In this paper, we propose a novel deep convolutional neural network (CNN)-based algorithm for solving ill-posed inverse problems. Regularized iterative algorithms have emerged as the standard approach to ill-posed inverse problems in the past few decades. These methods produce excellent results, but can be challenging to deploy in practice due to factors including the high computational cost of the forward and adjoint operators and the difficulty of hyperparameter selection. The starting point of this paper is the observation that unrolled iterative methods have the form of a CNN (filtering followed by pointwise nonlinearity) when the normal operator (H*H, where H* is the adjoint of the forward imaging operator, H) of the forward model is a convolution. Based on this observation, we propose using direct inversion followed by a CNN to solve normal-convolutional inverse problems. The direct inversion encapsulates the physical model of the system, but leads to artifacts when the problem is ill posed; the CNN combines multiresolution decomposition and residual learning in order to learn to remove these artifacts while preserving image structure. We demonstrate the performance of the proposed network in sparse-view reconstruction (down to 50 views) on parallel beam X-ray computed tomography in synthetic phantoms as well as in real experimental sinograms. The proposed network outperforms total variation-regularized iterative reconstruction for the more realistic phantoms and requires less than a second to reconstruct a 512 × 512 image on the GPU.
Overview
AI snapshot
Written from the paper’s full text.
TL;DR
The paper proposes FBPConvNet, a deep CNN that takes a filtered back projection (FBP) reconstruction as input and learns to remove artifacts for inverse problems whose normal operator is a convolution, demonstrated on sparse-view X-ray CT. It reports competitive or better SNR than TV-regularized iterative reconstruction on realistic synthetic and real CT datasets, with sub-second GPU reconstruction.
Key findings
- FBPConvNet outperformed TV-regularized iterative reconstruction on the biomedical synthetic dataset: at 143 views, SNR was 36.15 dB vs. 31.92 dB for TV; at 50 views, 28.83 dB vs. 25.20 dB.
- On the real experimental CT dataset, FBPConvNet also had higher SNR than TV: 11.34 dB vs. 8.25 dB at 145 views, and 8.85 dB vs. 7.25 dB at 14× subsampling (table lists 51 views).
- On the ellipsoidal synthetic dataset, TV outperformed FBPConvNet: 29.48 dB vs. 28.96 dB at 143 views, and 27.69 dB vs. 23.84 dB at 50 views; the paper attributes this to TV being optimal for piecewise-constant phantoms.
- Reconstruction speed was about 200 ms for FBP plus 200–300 ms for the CNN on GPU for a 512 × 512 image, versus around 7 minutes for iterative reconstruction; the abstract states less than one second total.
Methods
- Design: FBPConvNet applies discretized FBP to measurements, then a CNN based on U-net with multilevel decomposition, multichannel filtering, and residual learning via a skip connection; zero-padding is used and the final layer reduces 64 channels to a single output image. Training pairs low-view FBP images with full-view FBP images as input and output.
- Data: three datasets—ellipsoid synthetic (500 images, sinograms 729 × 1,000), biomedical synthetic (500 real in-vivo CT images from the Mayo Clinic Low-dose Grand Challenge, sinograms 729 × 1,000), and experimental real CT (377 sinograms from the TOMCAT beamline, each 1,493 × 721, from rat brain z-slices). Synthetic data used 475 training and 25 test images; real data used 327 training and 25 test images with a 25-slice gap. Subsampling was ×7 and ×20 for synthetic (143 and 50 views) and ×5 and ×14 for real (145 and 52 views).
- Analysis: SNR was the quantitative metric; FBP, TV-regularized iterative reconstruction [13], and FBPConvNet were compared. Training used MatConvNet, a Titan Black GPU, about 15 hours for 101 iterations, learning rate decreasing from 0.01 to 0.001, batch size 1, momentum 0.99, gradient clipping at 10⁻², and flip-flop augmentation. TV parameters were selected by golden-section search to optimize performance on the training set.
Limitations
- Stated: the method lacks transfer between datasets; a network trained on ×7 subsampled sinograms retained artifacts when applied to ×20 subsampled sinograms, and different dimensions or subsampling factors require retraining.
- Stated: the work focuses on X-ray CT; adapting to MRI or diffraction tomography is expected to be non-trivial experimentally, requiring large training sets and a high-quality iterative reconstruction for comparison, and complex-valued data would need a CNN architecture that correctly handles complex values.
- Evident/acknowledged: TV outperformed FBPConvNet on the ellipsoidal piecewise-constant phantom, which the paper says is reasonable because TV regularization should be optimal there; the paper also notes TV parameters were tuned on the training set rather than learned from data, and the correct way to learn such parameters remains open.
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26 Own result · 14 Own interpretation · 12 No basis given · 6 Not checked yet · 4 Own method
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“requires less than a second to reconstruct a 512 × 512 image on the GPU.”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Rests on an assumption: High-view FBP reconstructions can serve as ground truth for training and evaluation.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In Figure 5, again, the sparse-view FBP contains line artifacts.”
plain- Asserted: passes — The authors put it forward as true.
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A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Both the TV and FBPConvNet methods significantly reduce these artifacts, giving visually indistinguishable results.”
plaindirectSingle case- Asserted: passes — The authors put it forward as true.
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- Rests on an assumption: The training and test sets for the synthetic datasets are disjoint.
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In the paper
“This yields a noticeable increase in performance compared to the same network without the skip connection.”
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- Rests on an assumption: High-view FBP reconstructions can serve as ground truth for training and evaluation.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The proposed network outperforms total variation-regularized iterative reconstruction for the more realistic phantoms”
plaindirect- Asserted: passes — The authors put it forward as true.
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- Rests on an assumption: High-view FBP reconstructions can serve as ground truth for training and evaluation.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The average SNR on the testing set for the TV method is higher than that of the the FBPConvNet.”
plain- Asserted: passes — The authors put it forward as true.
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- Rests on an assumption: The training and test sets for the synthetic datasets are disjoint.
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In the paper
“We demonstrate the performance of the proposed network in sparse-view reconstruction (down to 50 views) on parallel beam X-ray computed tomography in synthetic phantoms as well as in real experimental sinograms.”
boosted- Asserted: passes — The authors put it forward as true.
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- Rests on an assumption: A twenty times reduction in the number of views corresponds to a twenty times reduction in the radiation dose received by the patient.
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In the paper
“Especially, reconstructed images from the proposed network represented well complex textures which are important to diagnosis.”
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- Rests on an assumption: High-view FBP reconstructions can serve as ground truth for training and evaluation.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“A version of Theorem 1 also holds in the discrete case; we sketch the result here.”
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In the paper
“When the downsampling is increased to twenty times, Figure 4, the line artifacts in the sparse-view FBP reconstruction are even more pronounced.”
plaindirectSingle case- Asserted: passes — The authors put it forward as true.
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- Rests on an assumption: The training and test sets for the synthetic datasets are disjoint.
- It may state two separate claims as one. 20
- Probably not a substantial claim. 8
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“| avg. SNR (dB) | 50 views (x20) | 8.44 | 27.69 | 23.84 |”
plain- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
- Rests on an assumption: The training and test sets for the synthetic datasets are disjoint.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In the seven times downsampling case, Figure 3, the full-view FBP (ground truth) shows nearly artifact-free ellipsoids, while the sparseview FBP shows significant line artifacts (most visible in the background).”
plaindirectSingle case- Asserted: passes — The authors put it forward as true.
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In the paper
“| avg. SNR (dB) | 143 views (x7) | 16.09 | 29.48 | 28.96 |”
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“Both the TV and FBPConvNet reduce these artifacts, though the FBPConvNet retains some of the artifacts.”
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
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In the paper
“When the forward model is normal-convolutional and when W is a convolution, the algorithm consists of iteratively filtering by I -(1 /L ) W ∗ H ∗ HW , adding a bias, (1 /L ) W ∗ H ∗ y , and applying a point-wise nonlinearity, S θ .”
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In the paper
“| avg. SNR (dB) | 50 views (x20) | 13.52 | 25.2 | 28.83 |”
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In the paper
“In Fig. 7, we observe the same trend as for the biomedical dataset, where the TV method oversmooths and the FBPConvNet better preserves fine structures.”
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In the paper
“This is much faster than the iterative reconstruction, which, in our case, requires around 7 minutes even after the regularization parameters have been selected.”
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“Furthermore, after training, the computation time of the proposed network per one image is under a second.”
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In the paper
“The conventional iterative algorithm with TV regularization outperformed the FBPConvNet in the ellipsoidal dataset, while the reverse was true for the biomedical and experimental datasets.”
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In the paper
“| avg. SNR (dB) | 145 views (x5) | 5.38 | 8.25 | 11.34 |”
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In the paper
“| avg. SNR (dB) | 143 views (x7) 24.97 | | 31.92 36.15 | |”
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“Both TV and the proposed method remove streaking artifacts satisfactorily; however, the TV reconstruction shows the cartoon-like artifacts that are typical of TV reconstructions.”
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In the paper
“| avg. SNR (dB) | 51 views (x14) | 3.29 | 7.25 | 8.85 |”
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In the paper
“The computation time for the FBPConvNet was about 200 ms for the FBP and 200 ∼ 300 ms in GPU for the CNN for a 512 × 512 image.”
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In the paper
“For example, in the case of CT reconstruction, if the sinogram is used as input, the CNN must encode a change between polar and Cartesian coordinates, which is completely avoided when the FBP is used as input.”
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In the paper
“We use high-view FBP reconstructions for training, meaning that training is possible from real data (without oracle knowledge).”
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- Rests on an assumption: High-view FBP reconstructions can serve as ground truth for training and evaluation.
None of the sources cited for this claim have been checked yet.
In the paper
“We clip the computed gradients to a fixed range to prevent the divergence of the cost function [27], [46].”
What the sources say
- Not checked yetcites [27] [27] J. Kim, J. K. Lee, and K. M. Lee, 'Accurate image super-resolution using very deep convolutional networks,' in Proc. of IEEE Converence on Computer Vision and Pattern Recognition (CVPR) , 2016. - source
- Not checked yetcites [46] [46] R. Pascanu, T. Mikolov, and Y. Bengio, 'On the difficulty of training recurrent neural networks.' ICML (3) , vol. 28, pp. 1310-1318, 2013. - source
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“We use flip-flop data augmentation in both horizontal and vertical directions during the training phase to reduce overfitting [18].”
What the sources say
- Not checked yetcites [18] [18] A. Krizhevsky, I. Sutskever, and G. E. Hinton, 'Imagenet classification with deep convolutional neural networks,' in Advances in neural information processing systems , 2012, pp. 1097-1105. -
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In the paper
“Many other iterate methods for solving Eq. (5), including ADMM [40], FISTA [41], and SALSA [42], also rely on these basic building blocks.”
What the sources say
- Not matchedcites [40] [40] S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, 'Distributed optimization and statistical learning via the alternating direction method of multipliers,' Found. Trends in Mach. Learn. , vol. 3, no. 1, pp. 1-122, 2011. -
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not checked yetcites [41] [41] A. Beck and M. Teboulle, 'A fast iterative shrinkage-thresholding algorithm for linear inverse problems,' SIAM journal on imaging sciences , vol. 2, no. 1, pp. 183-202, 2009. - source
- Not checked yetcites [42] [42] M. V. Afonso, J. M. Bioucas-Dias, and M. A. Figueiredo, 'Fast image recovery using variable splitting and constrained optimization,' IEEE Transactions on Image Processing , vol. 19, no. 9, pp. 2345-2356, 2010. - source
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“In the case of the biomedical dataset, the test data is chosen from a different subject than the training set.”
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In the paper
“we instead simply select the parameters that optimize performance on the training set (via a golden-section search).”
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None of the sources cited for this claim have been checked yet.
In the paper
“The class trivially includes denoising and deconvolution, but also includes MRI [35], X-ray CT [13], [16], and diffraction tomography (DT).”
What the sources say
- Not matchedcites [35] [35] K. P. Pruessmann, M. Weiger, M. B. Scheidegger, P. Boesiger et al. , 'SENSE: sensitivity encoding for fast MRI,' Magn. Reson. Med. , vol. 42, no. 5, pp. 952-962, 1999. -
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not checked yetcites [13] [13] M. T. McCann, M. Nilchian, M. Stampanoni, and M. Unser, 'Fast 3D reconstruction method for differential phase contrast X-ray CT,' Opt. Express , vol. 24, no. 13, pp. 14 564-14 581, Jun 2016. - source
- Not checked yetcites [16] [16] E. Y. Sidky and X. Pan, 'Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,' Physics in medicine and biology , vol. 53, no. 17, p. 4777, 2008. - source
plaincorrelational- Asserted: passes — The authors put it forward as true.
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- Not matched
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In the paper
“In practice, inverse problems related with imaging are often ill-posed, which prohibits the use of direct inversion because measurement noise causes serve perturbations in the solution.”
What the sources say
- Not checked yetcites [2] [2] A. Chambolle, 'An algorithm for total variation minimization and applications,' J. Math. imaging and vision , vol. 20, no. 1-2, pp. 89-97, 2004. - source
- Not checked yetcites [9] [9] R. Tibshirani, 'Regression shrinkage and selection via the lasso,' Journal of the Royal Statistical Society. Series B (Methodological) , pp. 267-288, 1996. - source
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“This formulation does not admit a closed form solution, and, therefore, is typically solved iteratively.”
plaindirect- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This is critical for our application because the filters corresponding to H ∗ H (and its inverse) may have noncompact support, e.g. in CT. Thus, a CNN with a small, fixed filter size may not be able to effectively invert H ∗ H .”
hedgeddirect- Asserted: passes — The authors put it forward as true.
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In the paper
“The test data are obtained from the last z-slices with the gap of 25 slices left between testing and training data.”
plain- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This points to one advantage of the proposed method over iterative methods: the iterative methods must explicitly impose regularization, while the FBPConvNet effectively learns a regularizer from the data.”
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
The paper relies on this without support.
High-view FBP reconstructions can serve as ground truth for training and evaluation.
“We compute its FBP (standard high quality reconstruction) and take this as the ground truth.”
The paper relies on this without support.
A twenty times reduction in the number of views corresponds to a twenty times reduction in the radiation dose received by the patient.
“a twenty times reduction in the number of views corresponds to a twenty times reduction in the radiation dose received by the patient.”
The paper relies on this without support.
The training and test sets for the synthetic datasets are disjoint.
“In case of synthetic data, the total number of training images is 475. The number of test images is 25.”
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“Their study showed that U-net could effectively produce noiseless CT images from the subsampled sinogram.”
From Convolutional neural networks: an overview and application in radiology (2018) · cites this paper as [54]
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§ A. Data Preparation · p. 10
“For the real data, we subsampled by factors of 5 and 14 corresponding to 145 and 52 views.”
§ B. Biomedical Dataset · p. 15
“| avg. SNR (dB) | 51 views (x14) | 3.29 | 7.25 | 8.85 |”
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In the paper
§ C. Experimental Dataset · p. 15
“These trends also appears in twenty times downsampling case (x20) in Fig. 8.”
§ A. Data Preparation · p. 10
“For the real data, we subsampled by factors of 5 and 14 corresponding to 145 and 52 views.”
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Metadata
- Authors
- Jin, K. H., McCann, M. T., Froustey, E., & Unser, M.
- Affiliations
- École Polytechnique Fédérale de Lausanne (CH)
- Dassault Aviation (France) (FR)
- Venue
- IEEE Transactions on Image Processing
- Published
- 2017-09
- Volume
- 26
- Issue
- 9
- Pages
- 4509-4522
- ISSN
- 1057-7149 (ISSN-L)
- 1941-0042
- OpenAlex
- W2574952845
- Semantic Scholar
- 8a68ca7c5ba4d92de3435961f88188d0b769b920
- Cited by
- 2,529
- References
- 83
- Citations by year
- 2026: 184 · 2025: 280 · 2024: 291 · 2023: 318 · 2022: 317 · 2021: 383 · 2020: 326 · 2019: 277 · 2018: 128 · 2017: 23
- FWCI
- 228.23
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- IEEE Transactions on Image Processing (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- DORA PSI (Paul Scherrer Institute) (submittedVersion)
- Infoscience (Ecole Polytechnique Fédérale de Lausanne) (submittedVersion)
- Infoscience (Ecole Polytechnique Fédérale de Lausanne) (submittedVersion, cc-by-nc-nd)
- arXiv (Cornell University)
- Funders
- H2020 European Research Council (the European Research Council under Grant 692726 ()
- Centre dImagerie BioMedicale
- EPFL Fellows fellowship program co-funded by Marie Curie from the EuropeanUnions Horizon 2020 Framework Programme for Research and Innovation (665667)
- Topics
- Physical Sciences › Mathematics › Mathematical Physics › Numerical methods in inverse problems
- Health Sciences › Medicine › Radiology, Nuclear Medicine and Imaging › Medical Imaging Techniques and Applications
- Physical Sciences › Computer Science › Computer Vision and Pattern Recognition › Image and Signal Denoising Methods
- Keywords
- deep convolutional neural networks, ill-posed inverse problems, direct inversion, artifact removal, multiresolution decomposition, residual learning, sparse-view reconstruction, total variation regularization, image reconstruction
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
A language model drew the claims from the paper on 2026-10-09, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).
A second model labelled each claim on 2026-10-09 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
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