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Super-Resolution of Solar Magnetograms via Adaptive Stratified Ensemble Learning with Uncertainty Estimation

Sina Norouzi Kandalan, Haodi Jiang, Jason T. L. Wang, Qin Li

arXiv:2609.27131v1cs.CVastro-ph.SR

TL;DR

MDI-to-HMI super-resolution could extend higher-resolution magnetogram coverage backward, while variation in image complexity challenges a single model trained across the full pixel distribution. The paper uses complexity-routed specialist networks with adaptive loss and reports the best performance among closely related methods on three metrics.

  • Problem

    MDI-to-HMI super-resolution could extend HMI-quality magnetogram coverage backward, but single-network performance may be constrained by heterogeneous image complexity.

    Source: §I · p. 1

  • Method

    The SSE combines three complexity-specialist networks, image-statistic-based routing, weighted sampling, adaptive metric-gated loss, D4 test-time augmentation, and uncertainty maps.

    Source: §I · p. 2, §VII · p. 7

  • Results

    37.66 dB PSNR, 0.9443 SSIM, and 0.9439 CC; SSE achieved the best performance among SolarCNN, RRDBNet, and the baseline.

    Source: §VII · p. 7

  • Takeaways & Limitations

    Epistemic uncertainty maps corresponded spatially to error-prone, structured regions, suggesting potential use as confidence estimates when the HMI reference is unavailable.

    Source: §VII · p. 7

    Evaluation is limited by the short MDI–HMI overlap period; future work will expand it with larger synthetic datasets or out-of-distribution validation.

    Source: §VII · p. 7

Abstract

from arXiv · show

Single-image super-resolution of Sun's photospheric magnetograms enables consistent analysis across heterogeneous space-based instruments and supports long-term studies of solar magnetic field evolution. We address the super-resolution task from SOHO/MDI (low-resolution) to SDO/HMI (high-resolution) line-of-sight (LOS) magnetograms using a modified RRDBNet architecture initialized by ESRGAN pretrained weights. Through systematic per-image diagnostic analysis, we identify image complexity as the dominant predictor of reconstruction errors. To exploit this finding, we introduce an adaptive stratified specialist ensemble (SSE) of three specialist networks with uncertainty estimation, where each specialist network is trained by images from three different complexity strata using a weighted random sampling strategy. During inference, a lightweight router based on input image statistics assigns each test image to the appropriate specialist network. Our experimental results demonstrate the good performance of the proposed ensemble and its superiority over closely related methods.

I. INTRODUCTION

MDI-to-HMI super-resolution could extend higher-resolution magnetogram coverage backward, but magnetogram complexity challenges a single model trained across the full pixel distribution. The paper proposes adapted RRDBNet and a complexity-aware specialist ensemble with adaptive loss and uncertainty estimates.

  • MDI-to-HMI super-resolution could extend HMI-quality magnetogram coverage backward by more than a decade.Source: §I · p. 1
  • Magnetogram inputs are single-channel signed-magnitude images, and their content varies from quiet-Sun regions to complex active regions, challenging direct natural-image model application.Source: §I · p. 1
  • The modified RRDBNet uses learned input/output adapters and a residual refinement head to transfer ESRGAN-pretrained RGB weights to single-channel magnetogram super-resolution.Source: §I · p. 1
  • Per-image diagnostics identify image-content complexity as the dominant reconstruction-error driver, motivating the Stratified Specialist Ensemble with complexity-aware sampling and test-time augmentation.Source: §I · pp. 1–2
  • The framework adds metric-gated adaptive loss and reconstruction-error and epistemic uncertainty maps that identify complex, error-prone regions.Source: §I · p. 2
  • The proposed SSE framework outperforms closely related methods in three evaluation metrics.Source: §I · p. 2

II. RELATED WORK

The related-work section situates magnetogram super-resolution within natural-image models, solar-specific methods, and stratified or adaptive learning. It also describes the paired SOHO/MDI and SDO/HMI data used for the task.

A. Single Image Super-Resolution

B. Solar Magnetogram Super-Resolution

C. Stratified Training and Adaptive Losses

A. Data Acquisition and Pairing

B. Data Normalization

The data are normalized and split into training and held-out test sets, then characterized by image-level variability. D4 augmentation expands the limited paired training set through geometric transforms.

B. Data Normalization

  • LR patches are 64 × 64 pixels and HR patches 256 × 256, with values scaled by 1987.34 and clamped to [−1, 1].Source: B. Data Normalization · p. 2

C. Train-Test Split

  • The last 76 pairs from April 2011 form a temporal-tail test set, while the remaining 1,493 pairs are used for training and 10% of training data for validation.Source: C. Train-Test Split · p. 2

D. Image-Content Statistics

E. D4 Augmentation

  • Applying the eight D4 rotations-and-flips to each paired image increases the effective training set from 1,493 to 11,944 samples without additional data collection.Source: E. D4 Augmentation · p. 3

A. Architecture Overview

The baseline adapts an ESRGAN-pretrained RRDBNet for single-channel magnetograms, adding structural and pixel-level refinement while retaining a conservative bicubic correction. Its design avoids adversarial training because hallucinated texture would constitute fabricated physical measurements.

A. Architecture Overview

  • The baseline uses the 23-block ESRGAN RRDBNet generator without its adversarial component to avoid hallucinating unobserved magnetic-field texture.The authors regard fabricated pixel values as scientifically unacceptable for magnetograms, which encode physical measurements.Source: A. Architecture Overview · p. 3

B. Layer-by-Layer Specification

  • A learned 1→3 input adapter and symmetric 3→1 output adapter connect single-channel magnetograms to the pretrained RGB backbone and return a single-channel output.A bicubic ×4 skip scaled by 0.10 supplies a conservative bias correction, and the output is clamped to [−1, 1].Source: Figure 3 · p. 3, A. Architecture Overview · p. 3, B. Layer-by-Layer Specification · pp. 3–4
  • The SSIM-enhancer applies a small residual correction in the three-channel input feature space before the RRDB trunk.Its 0.20 residual scale is intended to adjust local contrast and structure conservatively.Source: B. Layer-by-Layer Specification · pp. 3–4
  • The pretrained RRDB trunk performs ×4 upsampling, while a full-resolution PSNR refiner adds a learned residual correction to reduce pixel-level errors.The refiner targets regression-toward-the-mean smoothing that degrades MSE and PSNR.Source: B. Layer-by-Layer Specification · p. 4
  • The baseline objective combines pixel MSE with a hinged SSIM penalty and a variance-matching term.The stated weights are λSSIM = 0.10 and λvar = 0.05.Source: B. Layer-by-Layer Specification · p. 4

C. Training Schedule

Training runs for 100 epochs in a warmup phase with the RRDB trunk frozen, followed by fine-tuning with the trunk unfrozen. The staged schedule lets the lightweight interface modules adapt before the pretrained backbone is updated.

C. Training Schedule

  • The 100-epoch schedule freezes the RRDB trunk for epochs 1–25, then unfreezes it for epochs 26–100.The trunk contains approximately 17.9 × 10^6 of the network's 18.0 × 10^6 parameters.Source: C. Training Schedule · p. 4
  • During warmup, only the lightweight adapters and refinement heads are trained to establish a domain-adapted interface to the pretrained RGB feature space.The authors state this provides a better-conditioned starting point for subsequent backbone fine-tuning.Source: C. Training Schedule · p. 4
  • Fine-tuning uses a 2 × 10−6 learning rate for the backbone and 3 × 10−6 for the adapter and refiner heads.The lower backbone rate is intended to preserve pretrained representations while allowing domain adaptation.Source: C. Training Schedule · p. 4
  • Mixed precision, gradient clipping at norm 2.0, and EMA weight averaging with decay 0.999 are used throughout training.Source: C. Training Schedule · p. 4

V. STRATIFIED SPECIALIST ENSEMBLE

The SSE trains three specialists with different sampling emphases over low-, mid-, and high-complexity image buckets. Its adaptive loss adjusts metric contributions according to validation-metric progress toward prescribed thresholds.

A. Weighted Random Sampling

  • Three specialists sample from all low-, mid-, and high-complexity buckets using distinct weight vectors: [1.00, 0.30, 0.05], [0.30, 1.00, 0.30], and [0.05, 0.30, 1.00].The vectors correspond respectively to the low, mid, and high specialists.Source: A. Weighted Random Sampling · p. 4

B. Adaptive Metric-Gated Curriculum Loss

  • The metric-gated curriculum loss addresses the baseline loss's fixed metric emphasis by adaptively weighting metric losses as validation quality changes.It uses running PSNR, SSIM, and correlation metrics and their prescribed thresholds to define soft saturation gates.Source: B. Adaptive Metric-Gated Curriculum Loss · pp. 4–5
  • The adaptive loss uses PSNR, SSIM, and correlation terms with base weights 1.0, 0.5, and 0.5, and thresholds 37.5 dB, 0.94, and 0.94, respectively.The corresponding gate sharpness values are 2.0, 400, and 400.Source: B. Adaptive Metric-Gated Curriculum Loss · p. 5
  • Below threshold, a metric's gate remains open and its loss contributes strongly; as the metric approaches or exceeds threshold, the gate closes and reduces its effective weight.Gates are recomputed each epoch from validation metrics and held fixed during that epoch, so they do not carry backpropagation gradients.Source: B. Adaptive Metric-Gated Curriculum Loss · p. 5

C. Test-Time Augmentation

At test time, the framework averages predictions from eight D4-transformed inputs to reduce orientation sensitivity and prediction variance. It routes each image to one of three specialists using the standard deviation of its LR pixels.

C. Test-Time Augmentation

  • Eight D4-transformed copies of each LR input are processed, inverse-transformed, and averaged to reduce orientation sensitivity and prediction variance.The authors report a consistent, measurable test-set improvement over single-pass inference.Source: C. Test-Time Augmentation · p. 5

D. Training and Inference Algorithms

  • Inference assigns each image to the low, mid, or high specialist according to LR-pixel standard deviation, using thresholds 0.0732 and 0.1037.These cutoffs are the 33rd and 67th percentiles of training-set LR pixel standard deviations.Source: D. Training and Inference Algorithms · p. 5
  • Each routed specialist produces an enhanced MDI image at a resolution close to the corresponding HMI magnetogram.The routed prediction uses ×8 D4 test-time augmentation averaging.Source: D. Training and Inference Algorithms · p. 5

E. Uncertainty Estimation

The paper estimates reconstruction reliability with four uncertainty maps, divided into ground-truth-dependent reconstruction-error maps and deployment-available epistemic maps. The reconstruction-error maps capture pixel-level and structural disagreement with HMI images.

E. Uncertainty Estimation

1) Reconstruction-Error Maps:

  • U1 measures the absolute pixel residual between the super-resolved MDI image and corresponding high-resolution HMI image.It directly localizes pixel-level reconstruction error.Source: 1) Reconstruction-Error Maps · p. 5
  • U2 increases where local structural similarity with the corresponding HMI image is lower, highlighting structural disagreement.The local similarity uses Gaussian-weighted moments over an 11 × 11 window.Source: 1) Reconstruction-Error Maps · pp. 5–6

2) Epistemic Uncertainty Maps:

The two epistemic maps estimate uncertainty from predictions alone, without a reference image. They capture disagreement across test-time orientations and across the three specialist networks.

2) Epistemic Uncertainty Maps:

  • U3 is the pixel-wise standard deviation across eight inverse-transformed D4 test-time predictions, with disagreement indicating low model confidence.The eight predictions are averaged for the final output; the standard deviation retains their orientation-dependent variation.Source: 2) Epistemic Uncertainty Maps · p. 6
  • U4 is the pixel-wise standard deviation across the three specialists’ outputs for the same LR test image.Agreement gives zero uncertainty at a pixel, while greater disagreement gives a larger U4 value.Source: 2) Epistemic Uncertainty Maps · p. 6

A. Performance Evaluation and Comparison

On the 76-image test set, SSE outperforms the compared methods across PSNR, SSIM, and CC. Per-complexity comparisons and ablations report gains across all three complexity strata and contributions from the framework’s components.

A. Performance Evaluation and Comparison

B. Ablation Tests

  • Ablations report PSNR of 37.26 dB for SSEw, 37.31 dB for SSEwt, and 37.54 dB for SSEwa; the authors attribute contributions to all three components.SSEw uses weighted random sampling; SSEwt adds TTA, while SSEwa adds adaptive loss without TTA.Source: B. Ablation Tests · pp. 6–7, Table III · p. 7

C. A Case Study

In a high-complexity case study, SSE sharpened magnetic structures and improved agreement with HMI; the paper reports that it also outperformed closely related methods overall and across complexity strata.

C. A Case Study

  • For the high-complexity test image, CC increased from 0.9393 for MDI to 0.9531 for SSE, and the scatter became more concentrated around the diagonal.The image was routed to the high specialist network; its SSE output appeared sharper and visually closer to HMI.Source: C. A Case Study · p. 7, Figure 4 · p. 8
  • The reconstruction-error maps highlighted regions rich in small-scale structures, while epistemic maps emphasized sunspot umbrae and penumbrae.The epistemic maps spatially corresponded with error-prone and highly structured regions, suggesting potential use as confidence estimates without HMI ground truth.Source: C. A Case Study · p. 7, §VII · p. 7
  • In the zoomed case-study region, SSE achieved the highest PSNR (32.94 dB), SSIM (0.8926), and CC (0.9568) among the compared methods.The comparison includes SolarCNN, RRDBNet, the baseline network, and HMI ground truth.Source: C. A Case Study · p. 7

VII. CONCLUSION

  • SSE combines three complexity-stratified specialists, weighted random sampling, eightfold D4 inference augmentation, and an adaptive metric-gated curriculum loss.Per-bucket evaluation showed SSE consistently outperformed the baseline across low-, mid-, and high-complexity strata.Source: §VII · p. 7
  • SSE achieved the best reported overall results among closely related methods: 37.66 dB PSNR, 0.9443 SSIM, and 0.9439 CC.Ablations also showed contributions from weighted random sampling, adaptive loss, and test-time augmentation.Source: §VII · p. 7
  • The evaluation is limited to the MDI–HMI overlap period; expanding it and performing out-of-sample or out-of-distribution validation remain future work.Source: §VII · p. 7

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