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REVIEW 4 major objections 6 minor 47 references

Solar image denoising with convolutional neural networks

T0 review · 4 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A convolutional neural network trained only on paired noisy solar spectropolarimetric images, with no clean targets and no noise model, denoises the data about as well as a network trained with clean targets.

desk verdict Solid first application of Noise2Noise to solar Stokes polarimetry; abstract overclaims generality and real-data validation is qualitative, but the synthetic proof-of-concept and honest limitations make it worthy of review. read the letter →

arxiv 1908.02815 v1 pith:D2BVCIVG submitted 2019-08-07 astro-ph.SR eess.IV

C. J. Díaz Baso , J. de la Cruz Rodríguez , S. Danilovic This is my paper · ORCID
classification astro-ph.SReess.IV
keywords convolutionalneuralnetworksimagedenoisingspectropolarimetrysolarchromosphereStokesparametersNoise2NoiseU-networktemporalredundancy
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that a convolutional neural network can denoise solar spectropolarimetric images using only pairs of independent noisy observations of the same scene, with no clean images and no explicit noise model. On synthetic data, the noisy-pair network leaves residuals of about $6\times10^{-4}I_c$, the same level as a clean-target network and roughly what averaging 25 frames would achieve. On real CRISP/SST observations of the Ca ii 8542 Å line, the procedure removes noise and fixed-pattern artifacts while yielding spectrally coherent Stokes profiles. If the claim holds, weak chromospheric magnetic-field polarization signals, normally buried near the detection limit, become recoverable from existing time series without clean calibrations.

What carries the argument

The central mechanism is Noise2Noise training: the network minimizes the mean squared difference between its output and a second, independent noisy image of the same underlying scene rather than a clean image, so the expected loss is minimized by the clean scene when the noise has zero mean. The architecture is a U-network with encoder-decoder convolutional blocks and skip connections; it predicts each pixel from its spatial neighbors, using spatial coherence as the signal. Independent pairs come from temporal redundancy in time series, with data augmentation (rotations, sign flips, reversed wavelength order) used to balance any solar evolution between frames. The paper also pairs this with a Bayesian uncertainty estimate using dropout and a heteroscedastic loss to return per-pixel error bars.

What would settle it

Use a synthetic dataset with known ground truth where each paired frame carries the same scene plus independent noise, but inject a small, known, slowly varying shift in one frame. If the trained network output attenuates or removes that injected change, the zero-mean assumption has been violated and the method cannot separate real temporal evolution from noise. A realistic version applies the network to two real frames separated by a known flow and compares the output in the moving region with a third, temporally averaged reference.

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Extended reading notes

Core claim

A U-network convolutional encoder-decoder trained with the Noise2Noise objective recovers weak Stokes signals under complex, signal-correlated corruption. The paper demonstrates this in two settings: on synthetic profiles from an MHD simulation, where training on noisy input/noisy target pairs leaves a residual standard deviation of about $6\times10^{-4}I_c$, matching the clean-target baseline; and on real full-Stokes observations from the Swedish 1-meter Solar Telescope, where the same architecture suppresses photon noise and post-processing artifacts. The recovered Stokes Q and V profiles are spectrally coherent even though the network is trained monochromatically, and the network can be extended to take wavelength cubes as input for slightly smoother profiles. The conclusion is that clean targets are unnecessary for this denoising task: temporal redundancy supplies the independent noise realizations the loss function needs.

Load-bearing premise

The load-bearing premise is that each training pair consists of two independent noise realizations of the same underlying scene with zero-mean noise; if the solar scene changes between frames or post-processing leaves signal-correlated, non-zero-mean residuals, the network will learn to suppress real changes or true signals.

Editorial extensions

If this is right

  • Weak chromospheric polarization signals can be recovered from existing observations with no clean reference data, improving the empirical basis for chromospheric magnetic-field studies.
  • Because the pairing relies only on repeated exposures of the same scene, the method transfers across spectral lines and wavelength samplings, including cases where PCA degrades because wavelength sampling is scarce.
  • On real data, the network removes not just random noise but also fixed-pattern post-processing artifacts, such as vertical stripes, that are hard to model analytically.
  • Adding the spectral dimension as input yields smoother Stokes profiles, with average differences around $1\times10^{-4}I_c$ relative to the spatial-only version.
  • The Bayesian extension produces uncertainty maps of order $6\times10^{-4}I_c$, giving downstream magnetic-field inversions a quantitative error estimate.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Any instrument that repeatedly images the same target, not only Fabry-Perot filtergraphs, could use the same pairing trick; the practical limit is how stationary the scene remains between paired exposures.
  • The cleanest stress test is to inject a small, known temporal evolution into one frame of a synthetic pair: if the network suppresses or attenuates it, the zero-mean-noise condition has failed in exactly the regime where real chromospheric dynamics vary.
  • For real data, cross-checking the denoised output against temporally averaged profiles at the same pixels, as the paper does qualitatively, is a cheap way to detect whether the network has learned to erase genuine evolution rather than noise.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper applies the Noise2Noise paradigm to denoising solar spectropolarimetric images. The authors train a U-Net-like convolutional network on pairs of independent noisy realizations of the same scene, without any clean targets or explicit noise model, and compare it to a network trained on noisy-clean pairs. In a synthetic test based on MURaM MHD simulations, both training strategies produce similar residuals (about 6e-4 Ic against an injected noise level of 3e-3 Ic). The same architecture is then trained on CRISP/SST observations of Ca II 8542 Å using temporally separated frames as noisy pairs, and the denoised maps and spectral profiles are compared qualitatively with the originals and with time-averaged profiles. The paper also presents an uncertainty-estimation extension using heteroscedastic aleatoric uncertainty and MC-dropout, and explores a spatial-plus-spectral variant. The central claim is that the method recovers weak chromospheric polarization signals without needing clean data or a noise model.

Significance. If the central claim holds, the method is genuinely useful for chromospheric spectropolarimetry, where polarization signals are often at the detection limit and where post-processing such as MOMFBD produces correlated, non-Gaussian noise that is difficult to model. A notable strength is that the synthetic proof of concept is independently grounded: the noisy-noisy network is benchmarked against external clean synthetic images, so the equivalence of the two training strategies is demonstrated rather than assumed. The authors also make the code and trained weights publicly available, which supports reproducibility. The limitations of the work are openly discussed, including the possibility that the uncertainty estimates are too low and that the spectral variant may introduce unphysical correlations. However, the real-data validation is qualitative and rests on an unverified assumption about the independence of temporal frames, and the uncertainty calibration is partly circular. These issues need to be addressed before the method can be recommended for routine use on real observations.

major comments (4)
  1. [Section 3.2.2, Fig. 8] The real-data validation does not establish that the network preserves real signal, including temporal evolution, rather than merely smoothing it. The comparison of reconstructed profiles to time-averaged profiles is exactly what a smoothing or evolution-suppressing network would produce, so agreement with the average cannot discriminate between noise suppression and signal suppression. The authors should provide a quantitative test on real data, for example by injecting synthetic (known) signals into real frames and measuring recovery, or by comparing the network output against an independent higher-S/N observation, to support the claim that temporal evolution is preserved.
  2. [Section 3.1.1, Appendix A] The uncertainty calibration is circular as presented. The manuscript states that the MC-dropout rate is 'chosen to produces the expected error', and the dropout rate is the free parameter controlling the epistemic uncertainty. Therefore the agreement between the uncertainty estimate and the residual error shown in Fig. A.1 is not an independent confirmation that the network's uncertainties are well calibrated; it is a consequence of tuning that parameter. The authors should either fix the dropout rate a priori on a validation set independent of the residual-error comparison, or present the calibration as a demonstration of the tuning procedure rather than as evidence that the uncertainty estimates are trustworthy.
  3. [Section 3.1, Fig. 4] The synthetic proof of concept covers one MHD snapshot, one noise level (3e-3 Ic), and reports a single residual statistic (6e-4 Ic) with no repeated runs, multiple noise levels, or error bars. In addition, Fig. 4 shows that in the critically sampled case the network suppresses the highest spatial frequencies, so the recovery is not complete. The claim that the method 'succeeds in correcting up to a level of 6e-4 Ic' should be framed as a single realization rather than a general performance estimate, and the power-spectrum suppression should be discussed as a known limitation of the method.
  4. [Sections 2.1 and 3.2] The load-bearing assumption for the real-data application is that two time-separated frames are independent zero-mean noise realizations of the same underlying signal, as stated in Section 2.1 ('the noise is the main difference between the two images'). For the CRISP data, this is mitigated only by selecting periods that are 'stable over time' and by data augmentation, but no quantitative test on real data is provided to show that the network is not suppressing real temporal evolution or signal-correlated MOMFBD residuals. Given that the entire real-data claim depends on this assumption, the authors should provide an explicit diagnostic, such as measuring the residual statistics against a known injected signal on real frames, or comparing network outputs with an independent data product.
minor comments (6)
  1. [Abstract] The statement that the method 'can recover weak signals equally well no matter on what spectral line or spectral sampling is used' overreaches the presented evidence; the experiments cover one spectral line pair (Fe I 6302/6301 Å at 50 mÅ sampling) and one real line (Ca II 8542 Å). Please soften the claim to reflect the tested configurations.
  2. [Section 3.2] In the sentence about optical flow, 'wrap one frame to the other' should be 'warp one frame to the other'.
  3. [Section 3.2.3] The phrase 'around 1 .10−4Ic' should read 'around 1x10^-4 Ic'.
  4. [Appendix B, Table B.2] The 'Variance Output' row is unclear: '−Abs(x)−12' does not specify the operation unambiguously. Please state that the final variance is computed as exp of the network output or otherwise clarify the parameterization.
  5. [Figures 3 and 6] The labels in the figure captions such as 'Difference (3e-03)/(6e-04)' are ambiguous. Please clarify that the difference panels are scaled to the stated levels or indicate the color scale explicitly.
  6. [Section 2.2] The claim that the chosen topology is 'the perfect balance between accuracy and speed of execution' is not quantified; the comparison with other architectures in Section 3.2.3 is qualitative. Please provide quantitative comparisons (e.g., network size, runtime, and residuals) or remove the word 'perfect'.

Circularity Check

1 steps flagged · score 4.0 of 10

Uncertainty calibration is partly by construction, but the central Noise2Noise denoising claim is independently benchmarked against clean synthetic images.

  1. fitted input called prediction [Section 3.1.1 and Appendix A (Eq. 1, Fig. A.1)]
    "The free parameter, dropout rate, is chosen to produces the expected error, which is the average difference between the result of different neural networks. [...] In the real case, the value of σθ will be higher than the noise, so the deviation from the known noise level will capture the error of the network, a quantity that perfectly matches with the residuals calculated between the output of the network and the original image (see Fig. A.1 in the Appendix)."

    The heteroscedastic loss (Eq. 1) trains the variance head by minimizing (fθ(x)-y)^2/(2σθ^2)+log(σθ^2)/2; for a fixed prediction the optimal σθ^2 is exactly the squared residual between the network output and the target. The MC-dropout rate is then explicitly chosen to make the uncertainty spread match the expected error. Hence the agreement between the estimated uncertainty and the residual error displayed in Fig. A.1 is imposed by the loss and by the hyperparameter choice, rather than being an independent prediction. This does not affect the main denoising benchmark, which uses external clean synthetic images as ground truth.

full rationale

The central claim—that a network trained on noisy-noisy pairs denoises as well as one trained on noisy-clean pairs—is validated against known clean MHD-simulation images (Section 3.1), so it is not circular: the residuals are computed against externally provided ground truth and the noisy-noisy result matches the noisy-clean result. The real-data demonstration relies on the Noise2Noise assumption that temporally adjacent frames differ mainly by independent zero-mean noise; this is a substantive physical assumption and a possible correctness risk, but it is not a construction-equivalent prediction. I found no load-bearing self-citation: the references to the authors' prior work concern network architecture and inversion/synthesis codes, not the validity of the denoising result. The only step that reduces by construction is the uncertainty calibration: the variance is fit to squared residuals via Eq. (1) and the dropout rate is chosen to match the expected error, so the agreement in Fig. A.1 is calibration rather than an independent check. The paper itself concedes that the uncertainty estimate 'may be lower than the real one' and is restricted to Gaussian aleatoric noise. Because this circularity is confined to the secondary uncertainty claim and does not touch the main denoising benchmark, the overall score is 4 rather than higher.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central claim adds no new physical entities or free physical parameters. It rests on the Noise2Noise zero-mean independent noise assumption, on temporal stationarity of the observed solar scene, and on spatial coherence of Stokes images. The uncertainty estimate is partly calibrated by tuning the dropout rate.

free parameters (5)
  • Dropout rate for MC-dropout uncertainty = 0.01
    Section 3.1.1: chosen 'to produce the expected error', i.e., tuned so the Monte Carlo dropout uncertainty matches the residual between network output and clean image, making the uncertainty calibration partly fitted.
  • Number of filters in first convolutional layer = 32
    Section 2.2: selected as 'a good balance between accuracy and speed'; affects capacity and the reported performance.
  • Training patch size = 52 x 52 pixels
    Section 3: described as 'chosen arbitrarily so it is large enough to capture the scale of the noise'; affects border effects and training.
  • Learning rate = 1e-4
    Section 2.2: kept constant during training; a standard optimizer choice, not fitted to the target result.
  • Number of training epochs = 20
    Section 3.1: found sufficient for convergence, selected by monitoring the loss.
assumptions (4)
  • domain assumption Noise2Noise convergence: minimizing L2 loss against independent noisy targets yields the clean signal when the noise is zero-mean
    Section 2.1 relies on Lehtinen et al. (2018); requires independent noise realizations and zero-mean conditional noise, which is not established for MOMFBD-processed solar data.
  • domain assumption Temporal stationarity: frames of the same solar region taken close in time share the same underlying signal, differing mainly in noise
    Section 3.2: 'the noise is the main difference between the two images'; if the scene evolves, the network may remove real signal.
  • domain assumption Spatial coherence: solar Stokes images contain enough spatial redundancy for a 2D CNN to predict a clean pixel from its neighbors, per wavelength and per Stokes parameter
    Core premise introduced in Section 2; validated only indirectly through the synthetic test and qualitative real-data results.
  • domain assumption Time-averaged real profiles are a proxy for the clean signal
    Section 3.2.2 compares reconstructed profiles to the average profile over the time series as validation, assuming the temporal average approximates the true signal.

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Cite this review

Pith. "Pith review of Solar image denoising with convolutional neural networks." pith.science (2026). https://pith.science/paper/D2BVCIVG

@misc{pith2026190802815,
  author       = {Pith},
  title        = {Pith review of: Solar image denoising with convolutional neural networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D2BVCIVG}},
  note         = {Machine review of arXiv:1908.02815}
}
read the original abstract

The topology and dynamics of the solar chromosphere are greatly affected by the presence of magnetic fields. The magnetic field can be inferred by analyzing polarimetric observations of spectral lines. Polarimetric signals induced by chromospheric magnetic fields are, however, particularly weak, and in most cases very close to the detection limit of current instrumentation. Because of this, there are only few observational studies that have successfully reconstructed the three components of the magnetic field vector in the chromosphere. Traditionally, the signal-to-noise ratio of observations has been improved by performing time-averages or spatial averages, but in both cases, some information is lost. More advanced techniques, like principal-component-analysis, have also been employed to take advantage of the sparsity of the observations in the spectral direction. In the present study, we propose to use the spatial coherence of the observations to reduce the noise using deep-learning techniques. We design a neural network that is capable of recovering weak signals under a complex noise corruption (including instrumental artifacts and non-linear post-processing). The training of the network is carried out without a priori knowledge of the clean signals, or an explicit statistical characterization of the noise or other corruption. We only use the same observations as our generative model. The performance of this method is demonstrated on both, synthetic experiments and real data. We show examples of the improvement in typical signals obtained in current telescopes such as the Swedish 1-meter Solar Telescope. The presented method can recover weak signals equally well no matter on what spectral line or spectral sampling is used. It is especially suitable for cases when the wavelength sampling is scarce.

Figures

Figures reproduced from arXiv: 1908.02815 by the authors.

Figure 1
Figure 1. Sketch of different training processes: the typical supervised training process where the clean target (answer of the problem) is used (upper panel) and, when different uncorrelated corrupted images are used as the input and output of the network (lower panel). Lehtinen et al. (2018) demonstrated that in case of data cor￾rupted with zero-mean noise, the neural network can be trained with a large number of pairs (xi … view at source ↗
Figure 2
Figure 2. Architecture of the convolutional network used for denoising, made of many Conv+ELU gray blocks and the concatenation (arrows) of previous feature maps. In an autoencoder, the information is com￾pressed (encoder) and recovered later (decoder). This specific topology is commonly known as U-network1 (Ronneberger et al. 2015) because it includes skip connections between mirrored layers in the encoder and decoder part. … view at source ↗
Figure 3
Figure 3. Result of the reconstructions of synthetic data with an original noise level of 3×10−3 Ic , using clean targets (upper panel) and using noisy targets (lower panel). The left column shows the original image (and affected by noise in the upper half), the central panel the image inferred by the neural network and the right panel the difference between the previous panels of the same row. 3. Results This section shows h… view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Comparison of the power spectrum calculated for the simulation and the reconstruction with the neural network trained with the original or a degraded resolution. The red curve depicts the critically sampled case, whereas the black curve depicts an oversampled case. In …
Figure 5
Figure 5. Figure 5: Stokes Q and V profiles extracted from 4 different points of the map: the synthetic profiles (in green), with noise (in gray) and the output of the neural network (in red), are shown with their respective color bands indicating twice the uncertainty. The reference wave…
Figure 6
Figure 6. Figure 6: Results for different wavelengths and Stokes parameters after apply the neural network to real observations. For Stokes Q at −0.17 Å (top panel) and an example of Stokes V (lower panel) at −0.765 Å. van Noort et al. 2005). The MOMFBD code applies a Fourier fil￾ter to t…
Figure 8
Figure 8. Figure 8: Stokes Q profiles extracted from 4 different points of the map: the original (in gray), the reconstructed by the neural network (in red) and the average profile in those pixels of the entire time series (in brown). The reference wavelength is λ0 = 8542.1Å. signals, at …
Figure 7
Figure 7. Figure 7: Spatial Fourier power spectra of the original image, the output of the neural network and the difference between them, which is our esti￾mate of the noise. We have marked the diffraction limit of the telescope as a reference. 3.2.2. Stokes profiles Once all wavelength …
Figure 9
Figure 9. Figure 9: Output of the network trained with synthetic data and Gaussian noise independent at each pixel (upper half) and with CRISP data (lower half). length or whether the image displays linear or circular polar￾ization. This allows for a more simple and diverse training be￾ca…
Figure 10
Figure 10. Figure 10: Stokes Q and V profiles extracted from 4 different points of the map: the observed profiles (in gray) and the output of the neural network (in red) are shown with their respective color bands indicating twice the uncertainty calculated by the neural network. wavelengt…
Figure 11
Figure 11. Figure 11: Same Stokes profiles of [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]
Figure 12
Figure 12. Figure 12: Monochromatic images at the core of Stokes Q after apply the network trained with monochromatic images (left panel) and with spectral profiles (right panel). The signals has been clipped to ±8 × 10−3 Ic for both panels. 3.3. Other instrumentation This technique could …

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Reference graph

Works this paper leans on

47 extracted references · 27 canonical work pages

  1. [1]

    & de la Cruz Rodríguez, J

    Asensio Ramos, A. & de la Cruz Rodríguez, J. 2015, A&A, 577, A140 Asensio Ramos, A., de la Cruz Rodríguez, J., Martínez González, M. J., & Socas-

    work page 2015

  2. [2]

    Real-time multiframe blind deconvolution of solar images

    Navarro, H. 2017a, A&A, 599, A133 Asensio Ramos, A., de la Cruz Rodriguez, J., & Pastor Yabar, A. 2018, ArXiv e-prints, arXiv:1806.07150 Asensio Ramos, A. & Díaz Baso, C. 2019, arXiv e-prints, arXiv:1904.03714 Asensio Ramos, A., Martínez González, M. J., & Rubiño-Martín, J. A. 2007, A&A, 476, 959 Asensio Ramos, A., Requerey, I. S., & Vitas, N. 2017b, A&A,...

    work page Pith review arXiv 2018

  3. [3]

    2019, HelioML /HelioML: HelioML 0.2.0 (2019-02-22)

    Bobra, M., Holdgraf, C., Mason, J., et al. 2019, HelioML /HelioML: HelioML 0.2.0 (2019-02-22)

    work page 2019

  4. [4]

    A., Kopf, M., Ilyin, I., & Strassmeier, K

    Carroll, T. A., Kopf, M., Ilyin, I., & Strassmeier, K. G. 2007, Astronomische Nachrichten, 328, 1043

    work page 2007

  5. [5]

    Casini, R. & Li, W. 2018, arXiv e-prints, arXiv:1811.03211

    work page Pith review arXiv 2018

  6. [6]

    Cheung, M. C. M., Rempel, M., Chintzoglou, G., et al. 2019, Nature Astronomy, 3, 160

    work page 2019

  7. [7]

    2013, Memorie della Societa As- tronomica Italiana, 84, 379 de la Cruz Rodríguez, J., Leenaarts, J., & Asensio Ramos, A

    Collados, M., Bettonvil, F., Cavaller, L., et al. 2013, Memorie della Societa As- tronomica Italiana, 84, 379 de la Cruz Rodríguez, J., Leenaarts, J., & Asensio Ramos, A. 2016, ApJ, 830, L30 de la Cruz Rodríguez, J., Leenaarts, J., Danilovic, S., & Uitenbroek, H. 2019, A&A, 623, A74 de la Cruz Rodríguez, J., Löfdahl, M. G., Sütterlin, P., Hillberg, T., & ...

    work page 2013

  8. [8]

    Model-blind Video Denoising Via Frame-to-frame Training

    Ehret, T., Davy, A., Facciolo, G., Morel, J.-M., & Arias, P. 2018, arXiv e-prints, arXiv:1811.12766

    work page Pith review arXiv 2018

Show all 47 references
  1. [9]

    & Ghahramani, Z

    Gal, Y . & Ghahramani, Z. 2015a, arXiv e-prints, arXiv:1506.02157

    arXiv

  2. [10]

    & Ghahramani, Z

    Gal, Y . & Ghahramani, Z. 2015b, arXiv e-prints, arXiv:1506.02142

    arXiv

  3. [11]

    2017, arXiv e-prints, arXiv:1705.07832

    Gal, Y ., Hron, J., & Kendall, A. 2017, arXiv e-prints, arXiv:1705.07832

    2017 arXiv

  4. [12]

    M., Snik, F., Keller, C

    Harrington, D. M., Snik, F., Keller, C. U., Sueoka, S. R., & van Harten, G. 2017, Journal of Astronomical Telescopes, Instruments, and Systems, 3, 048001

    2017

  5. [13]

    2015, arXiv e-prints, arXiv:1502.01852

    He, K., Zhang, X., Ren, S., & Sun, J. 2015, arXiv e-prints, arXiv:1502.01852

    2015 arXiv

  6. [14]

    E., Srivastava, N., Krizhevsky, A., Sutskever, I., & Salakhutdinov, R

    Hinton, G. E., Srivastava, N., Krizhevsky, A., Sutskever, I., & Salakhutdinov, R. R. 2012, arXiv e-prints, arXiv:1207.0580

    2012 arXiv

  7. [15]

    Huang, G., Liu, Z., van der Maaten, L., & Weinberger, K. Q. 2016, arXiv e- prints, arXiv:1608.06993

    2016 arXiv

  8. [16]

    2018, ApJ, 856, 7

    Huang, X., Wang, H., Xu, L., et al. 2018, ApJ, 856, 7

    2018

  9. [17]

    Illarionov, E. A. & Tlatov, A. G. 2018, MNRAS, 481, 5014

    2018

  10. [18]

    & Seung, H

    Jain, V . & Seung, H. S. 2008, in Proceedings of the 21st International Confer- ence on Neural Information Processing Systems, NIPS’08 (USA: Curran As- sociates Inc.), 769–776

    2008

  11. [19]

    C., Macek, J

    Kemp, J. C., Macek, J. H., & Nehring, F. W. 1984, ApJ, 278, 863

    1984

  12. [20]

    & Gal, Y

    Kendall, A. & Gal, Y . 2017, arXiv e-prints, arXiv:1703.04977

    2017 arXiv

  13. [21]

    Kingma, D. P. & Ba, J. 2014, ArXiv e-prints [arXiv:1412.6980]

    2014 arXiv

  14. [22]

    & Bengio, Y

    LeCun, Y . & Bengio, Y . 1998, in The Handbook of Brain Theory and Neural Networks, ed. M. A. Arbib (Cambridge, MA, USA: MIT Press), 255–258

    1998

  15. [23]

    2016, arXiv e-prints, arXiv:1609.04802

    Ledig, C., Theis, L., Huszar, F., et al. 2016, arXiv e-prints, arXiv:1609.04802

    2016 arXiv

  16. [24]

    2018, ArXiv e-prints, arXiv:1803.04189

    Lehtinen, J., Munkberg, J., Hasselgren, J., et al. 2018, ArXiv e-prints, arXiv:1803.04189

    2018 arXiv

  17. [25]

    Leung, H. W. & Bovy, J. 2019, MNRAS, 483, 3255

    2019

  18. [26]

    Lites, B. W. 1991, in Solar Polarimetry, ed. L. J. November, 166–172 Loève, M. 1963, Probability Theory, Graduate texts in mathematics (Van Nos- trand Reinhold Company) Löfdahl, M. G. 2002, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, V ol. 4...

    1991

  19. [27]

    T., Bovy, J., Leung, H

    Mackereth, J. T., Bovy, J., Leung, H. W., et al. 2019, arXiv e-prints, arXiv:1901.04502

    2019 arXiv

  20. [28]

    Maddox, W., Garipov, T., Izmailov, P., Vetrov, D., & Wilson, A. G. 2019, arXiv e-prints, arXiv:1902.02476

    2019 arXiv

  21. [29]

    2018, arXiv e-prints, arXiv:1807.11888 Martínez González, M

    Mansar, Y . 2018, arXiv e-prints, arXiv:1807.11888 Martínez González, M. J., Asensio Ramos, A., Carroll, T. A., et al. 2008, A&A, 486, 637

    2018 arXiv

  22. [30]

    & Hinton, G

    Nair, V . & Hinton, G. E. 2010, in Proceedings of the 27th International Con- ference on Machine Learning (ICML-10), June 21-24, 2010, Haifa, Israel, 807–814

    2010

  23. [31]

    2018, ApJ, 858, 113

    Nishizuka, N., Sugiura, K., Kubo, Y ., Den, M., & Ishii, M. 2018, ApJ, 858, 113

    2018

  24. [32]

    Osborne, C. M. J., Armstrong, J. A., & Fletcher, L. 2019, ApJ, 873, 128

    2019

  25. [33]

    Pelt, D. M. & Sethian, J. A. 2018, Proceedings of the National Academy of Sciences, 115, 254

    2018

  26. [34]

    Rojo, P. M. & Harrington, J. 2006, ApJ, 649, 553

    2006

  27. [35]

    2015, arXiv e-prints, arXiv:1505.04597

    Ronneberger, O., Fischer, P., & Brox, T. 2015, arXiv e-prints, arXiv:1505.04597

    2015 arXiv

  28. [36]

    Scharmer, G. B. 2006, A&A, 447, 1111

    2006

  29. [37]

    B., Bjelksjo, K., Korhonen, T

    Scharmer, G. B., Bjelksjo, K., Korhonen, T. K., Lindberg, B., & Petterson, B. 2003, in Proc. SPIE, V ol. 4853, Innovative Telescopes and Instrumentation for Solar Astrophysics, ed. S. L. Keil & S. V . Avakyan, 341–350

    2003

  30. [38]

    B., Narayan, G., Hillberg, T., et al

    Scharmer, G. B., Narayan, G., Hillberg, T., et al. 2008, ApJ, 689, L69

    2008

  31. [39]

    & Lopez-Paz, D

    Tagasovska, N. & Lopez-Paz, D. 2018, arXiv e-prints, arXiv:1811.00908

    2018 arXiv

  32. [40]

    R., Beruko ff, S., et al

    Tritschler, A., Rimmele, T. R., Beruko ff, S., et al. 2016, Astronomische Nachrichten, 337, 1064 Trujillo Bueno, J. 2010, Astrophysics and Space Science Proceedings, 19, 118

    2016

  33. [41]

    2017, arXiv e-prints, arXiv:1711.10925 van Noort, M., Rouppe van der V oort, L., & Löfdahl, M

    Ulyanov, D., Vedaldi, A., & Lempitsky, V . 2017, arXiv e-prints, arXiv:1711.10925 van Noort, M., Rouppe van der V oort, L., & Löfdahl, M. G. 2005, Sol. Phys., 228, 191

    2017 arXiv

  34. [42]

    2008, in Proceedings of the 25th International Conference on Machine Learning, ICML ’08 (New

    Vincent, P., Larochelle, H., Bengio, Y ., & Manzagol, P.-A. 2008, in Proceedings of the 25th International Conference on Machine Learning, ICML ’08 (New

    2008

  35. [43]

    2005, A&A, 429, 335

    York, NY , USA: ACM), 1096–1103 Vögler, A., Shelyag, S., Schüssler, M., et al. 2005, A&A, 429, 335

    2005

  36. [44]

    K., & Solanki, S

    Wiegelmann, T., Thalmann, J. K., & Solanki, S. K. 2014, Astronomy and Astro- physics Review, 22, 78

    2014

  37. [45]

    2018, ArXiv e-prints, arXiv:1801.09710

    Xie, Y ., Franz, E., Chu, M., & Thuerey, N. 2018, ArXiv e-prints, arXiv:1801.09710

    2018 arXiv

  38. [46]

    2017, IEEE Transactions on Image Processing, 26, 3142

    Zhang, K., Zuo, W., Chen, Y ., Meng, D., & Zhang, L. 2017, IEEE Transactions on Image Processing, 26, 3142

    2017

  39. [47]

    2018, IEEE Transactions on Image Processing, 27, 4608 Article number, page 11 of 13 A&A proofs: manuscript no

    Zhang, K., Zuo, W., & Zhang, L. 2018, IEEE Transactions on Image Processing, 27, 4608 Article number, page 11 of 13 A&A proofs: manuscript no. aanda Appendix A: Uncertainty calibration At this point, we show the spatial distribution of the uncertainty estimated by the network ...

    2018

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.