Ground-to-space image reconstruction
Conservative Flow-Matching forAstronomical Image Super-Resolution
Reconstruct finer structure from DESI observations while keeping source flux and measurement consistency in view.
Coming soon
Watch the FluxFlow introduction
From ground to space
Moving air blurs the light reaching a ground-based telescope. Hubble observes above this turbulent layer.
The challenge is to recover useful structure from the ground image without adding unsupported sources.
Meet the two observationsView the photograph ↗Hubble above Earth · NASA
The observations
DESI records the ground view. HST provides a matched, higher-resolution reference.
The input · from the ground
An i-band cutout is the measured input. Each 128 × 128 patch is paired with the same sky region in HST.
Observatory photograph from the paper's instrument figure; the imaging surveys combine observations from multiple telescopes.
The reference · from space
F814W images provide the higher-resolution target. Weight maps and source catalogues support reliable training and evaluation.
HST is an observational reference, not an artifact-free image of the physical sky. Photo: NASA, reproduced in the paper.
The research
Useful detail, supported by measured light
FluxFlow is a conditional flow-matching model for astronomical image super-resolution. It learns in pixel space from real paired DESI–HST observations, using uncertainty maps and source-region weights to guide training.
During inference, measurement-consistent flow sampling (MC-FS) uses a Wiener-regularized correction to bring generated detail toward agreement with the measured input. This addresses a central generative-model risk: source-like structures unsupported by the observation.
Experiments evaluate image structure, source flux and downstream measurements at ×2 and ×4. Reconstructed images remain estimates; measurement consistency reduces the risk of unsupported sources but does not establish that every faint source is real.
The dataset
Real DESI–HST pairs let us compare the input, reconstruction and reference at the same sky position.
Swipe through the same field — tap an image to inspect it
DESI input
128 × 128FluxFlow reconstruction
×2 · 256 × 256HST reference
×2 · 256 × 256DESI input
128 × 128FluxFlow reconstruction
×4 · 512 × 512HST reference
×4 · 512 × 512More than image pairs HST weight maps describe measurement reliability; source labels and catalogues support source-level evaluation.
Counting the data The paper describes approximately 19,500 pairs. The prepared split has 19,438 pairs per scale from COSMOS and UDS. The ×2 and ×4 versions share sample identities.
desi_sci.npyi-band ground observationhst_sci.npyHST observational referencehst_wht.npyInverse-variance weightshst_masks.npySource labelshst_sources.npzSource catalogueThe historical patch-level split contains overlapping sky regions between training and test samples. It supports comparison under that protocol; it should not be described as a spatially independent generalization test.
Start with the red boxesCompare the boxed spots in “w/o MCFS” with HST, then inspect the full method in “Ours”
The challenge
Generative detail can look convincing even when the observation does not support it.
The riskA model can turn a noise fluctuation into a plausible-looking star or galaxy, producing an unreliable source catalogue.
The checkBlur and downsample the reconstruction, then compare it with DESI. MC-FS uses this discrepancy to reduce unsupported structure.
The limitAgreement with the input reduces this risk. It cannot establish that every reconstructed faint source is real.
The method
Learn an update in pixel space, then check the reconstruction against the measured light.
View the whole model — switch to details to read its layers
A time-conditioned U-Net learns to move Gaussian noise toward an HST-style image, with DESI as the observation condition.
During training, square-root WHT weights and HST-derived source masks emphasize reliable pixels and scientifically useful structure.
At each sampling step, a Gaussian-PSF and downsampling model predicts the ground observation. A Wiener-regularized correction reduces the residual.
The architecture is the original paper figure. The three animations explain training and sampling; they are schematics rather than recorded intermediate states.
The evidence
Lower Flux-L1 indicates less source-flux error; higher SSIM indicates greater structural similarity.
DESI–HSTSource-level Flux-L1Paper Table 2
Flux-L1 · ×2
2.959
Lowest Flux-L1 in the reported comparison
×2 · lower is better
Source-level flux error
Structural similarity · SSIM0.714
FluxFlow achieves the lowest Flux-L1 in the reported comparison at both scales. The complete table retains all reported metrics and baselines.
Explore the results
Choose a field and method, then move the divider to inspect the same region in both reconstructions.
Read structure, then check the referenceInspect source shapes and diffraction patterns. The HST option shows the paired observation; displayed brightness is not a flux measurement.
Field A · ×2 · HAT vs FluxFlow
Drag to compare · Arrow keys also work
Field D · ×4 · HAT vs FluxFlow
Drag to compare · Arrow keys also work
Missing methods are disabled for that field. HST is an observational reference; no substitute images are used.
The fields
Drag to explore · Click for the original
HST observation
FluxFlow · Field A
HST observation
DESI observation
FluxFlow · Field B
HST observation
DESI observation
FluxFlow · Field C
Observed fields and reconstructed outputs from the original project. Original pixels, original sources.
The problem
Ground-based telescopes are limited by atmospheric seeing, which imposes a stochastic, spatially varying PSF that upsampling alone cannot undo. Space telescopes reach the diffraction limit but see almost none of the sky.
The 2.4 m HST aperture is diffraction-limited above the atmosphere; DESI is seeing-limited from the ground. The same point source, resolved two ways.
Both ends are the paper's own ablation — w/o MC-FS at 0 and the full model at 100. The values in between are a visual blend, not measurements.
The failure
Synthetic training pairs do not carry real atmospheric statistics, so a generator learns to please the eye. It produces crisp texture, and occasionally a source that was never in the sky.
Hover a method below to see how it renders the same sky — or drag the dial to move between the ablation and the full model.
Paper and citation
FluxFlow: Conservative Flow-Matching for Astronomical Image Super-Resolution
arXiv:2605.03749 · 2026
@misc{liu2026fluxflow,
title = {FluxFlow: Conservative Flow-Matching for Astronomical Image Super-Resolution},
author = {Liu, Shuhong and Ge, Xining and Xu, Quanfeng and Cui, Ziteng and Li, Liuzhuozheng and Chang, Gengjia and Liu, Jun and Gu, Ziying and Li, Dong and Chu, Xuangeng and Gu, Lin and Harada, Tatsuya},
year = {2026},
eprint = {2605.03749},
archivePrefix = {arXiv},
primaryClass = {cs.CV},
url = {https://arxiv.org/abs/2605.03749}
}Overview
A conservative flow learns the structure. A measurement-consistent correction keeps it grounded in the observation.
x⁰ ~ N(0, I)
Short, straight-path ODE
Wiener residual correction
Structure + integrated flux
THE SAMPLER
The sampler, step by step
Scroll to step through the sampler.
08AInside the sampler
Measurement-consistent flow sampling. The velocity network proposes; the forward model disposes. Every step is projected back onto the observation — which is what keeps hallucinated sources out of the estimate.
1Noisy state xtx⁰ ~ N(0, I)
2DESI observation ybicubic → HR grid
3Conditional U-Net vθpixel space · base_dim = 48 · FiLM time conditioning, 256-d
4Euler stepx̃ = x + vθ Δt
5Forward projectionA = DsHPSF, σPSF = 2
6Wiener correctionW(k) = H*(k)/(|H(k)|² + λSNR−1) · residual r = A(x̃) − y
7Corrected estimatex ← x̃ − ηi F−1(W F[Ds⊤r])The thumbnails are illustrative stand-ins, not the sampler’s actual intermediate states — step 1 starts from pure Gaussian noise, and the correction runs on residuals, not on Bicubic or HAT images.
01 · Weight the evidence
wp = √Wp · (1 + Mp)
hst_wht)Per-image loss normalised by the sum of pixel weights · Eq. 5, paper §5.1 · hst_wht.npy, hst_masks.npy
02 · Correct without retraining
W(k) = H*(k)|H(k)|2 + λSNR−1
Eq. (10) in the paper · removes sources the observation never had
03 · Settings as reported
ηi = η0 (1 − i/N) · optimum 1.92, rounded
04 · Ablation · weight term removed
Full model → w/o WHT · paper Table 3
05 · Algorithm 1 · MC-FS
Algorithm 1 MC-FS
inputs observation y, PSF h, steps N, base step eta_0
steps t_i = i / N, dt = 1 / N
precompute W(k) from Eq. (10)
x <- x0 ~ N(0, I)
for i = 0 .. N-1 do
xt <- x + v_theta(x, t_i, c) * dt Euler step
r <- A(xt) - y residual to forward model
eta_i <- eta_0 * (1 - i / N) linear decay
x <- xt - eta_i * F^-1(W * F[Ds^T r]) Wiener-corrected update
return xThe proposal is the network; the correction is the forward model.
git clone https://github.com/I2WM/FluxFlow
# inference settings as reported in the paper
# sampling steps N = 10 Euler steps
# base correction step eta_0 = 0.5 (eta_i = eta_0 (1 - i/N))
# Wiener regularisation lambda_SNR = 50
# forward PSF width sigma_PSF = 2 (data-driven optimum 1.92, rounded)
# correction mode adjoint --correction_mode adjoint swaps the Wiener
# inverse for a plain adjoint
# correction off --no_mcfm removes the correction entirely
#
# Check the repository README for the current command-line interface:
# https://github.com/I2WM/FluxFlowThe research / The people
FluxFlow: Conservative Flow-Matching for Astronomical Image Super-Resolution.
Every hyper‑parameter of MC‑FS is swept in the appendix, over the paper’s own ranges (T = 5–60 steps, η0 = 0.1–1.0). The curves are trend schematics only — hand‑drawn to match the reported shapes and optima of Figs. 18–21, not measured data. The vertical axis is a schematic 0–1 scale, and the curves must not be used for numerical comparison.
Default σPSF = 2 HR pixels at both scales.
Default λSNR = 50, the value used throughout Table 2.
Default η0 = 0.5, with per‑step decay ηi = η0(1 − i/N).
Default T = 10, all other MC‑FS hyper‑parameters at their defaults.
Photometric metrics are a proxy. The real question is whether a reconstruction makes the catalogue measurements better. Every number below is measured on the matched bright‑source set, under the same SEP segmentation footprint as the ground truth, with one common threshold Fthr = 0.5 nMgy at both scales.
Lower is better in every column. Magnitude and shear are dimensionless; stellar‑mass MAE is in Θ★‑scaled nanomaggies.
| Method | Magnitude residual Δm | Stellar‑mass MAE | Shear residual | |||
|---|---|---|---|---|---|---|
| mean | median | mean | median | mean | median | |
| Bicubic | 0.758 | 0.637 | 7.767 | 1.537 | 0.148 | 0.122 |
| SwinIR | 0.649 | 0.300 | 8.197 | 0.762 | 0.149 | 0.116 |
| HAT | 0.542 | 0.192 | 7.962 | 0.524 | 0.151 | 0.112 |
| FISR | 0.561 | 0.230 | 8.014 | 0.608 | 0.165 | 0.133 |
| cGAN | 0.548 | 0.215 | 7.996 | 0.586 | 0.159 | 0.131 |
| GD‑Net | 0.664 | 0.393 | 7.659 | 1.025 | 0.166 | 0.131 |
| FluxFlow | 0.516 | 0.187 | 7.632 | 0.508 | 0.161 | 0.126 |
| Method | Magnitude residual Δm | Stellar‑mass MAE | Shear residual | |||
|---|---|---|---|---|---|---|
| mean | median | mean | median | mean | median | |
| Bicubic | 1.067 | 0.863 | 16.90 | 3.648 | 0.144 | 0.115 |
| SwinIR | 0.969 | 0.461 | 16.51 | 2.43 | 0.118 | 0.090 |
| HAT | 0.886 | 0.397 | 16.19 | 2.207 | 0.108 | 0.081 |
| FISR | 0.876 | 0.417 | 16.25 | 2.246 | 0.124 | 0.097 |
| cGAN | 0.902 | 0.460 | 16.42 | 2.440 | 0.135 | 0.109 |
| GD‑Net | 1.509 | 1.308 | 17.58 | 4.630 | 0.170 | 0.139 |
| FluxFlow | 0.851 | 0.389 | 16.07 | 2.131 | 0.117 | 0.091 |
What improves. FluxFlow returns the lowest magnitude residual and the lowest stellar‑mass MAE — on both the mean and the median, at both scales. At ×4 the mean magnitude residual improves from 0.876 (FISR, best baseline) to 0.851, and the mean mass MAE from 16.19 (HAT, best baseline) to 16.07 Θ★‑scaled nanomaggies.
What does not. Shear is the flat metric. It is set by the second‑moment tensor of each source, and its orthogonal failure mode is catalog‑level miscentering rather than image quality, so the ranking compresses and the strongest baseline depends on scale and statistic: at ×4 HAT leads both shear mean (0.108) and median (0.081); at ×2 the shear mean is lowest for Bicubic (0.148) and the median for HAT (0.112). FluxFlow lands just behind the leader rather than ahead.
Why the mass median is the headline. The median stellar‑mass MAE is where bright, cleanly segmented sources live; the mean is dragged by a handful of blended or badly segmented objects, which is why GD‑Net posts a competitive mean (7.659) but a median of 1.025.
DESI–HST is a cross‑survey benchmark: ground‑based seeing‑limited spectra‑era imaging on one side, space‑based diffraction‑limited imaging on the other. Reddy et al. 2024 is also G→S in that table, but without flux calibration and at 2,880 pairs — this set is the only one there combining real data, absolute flux calibration and the ground‑to‑space direction, and the largest real G→S set in it.
The resolution gap is dominated by atmospheric turbulence, not by pixel scale: a 11–14× ratio in FWHM (a 2.4 m space aperture against DESI's time-varying, spatially extended i‑band seeing). That is what makes DESI→HST super‑resolution substantially harder than benchmarks that rely on pixel‑scale downsampling alone.
| Dataset | Size | Real | Flux | Type |
|---|---|---|---|---|
| Miao et al. 2024 | 2,000 | ✓ | G→G | |
| Shan et al. 2025 | 9,383 | ✓ | G→G | |
| Song et al. 2024 | 1,597 | ✓ | S→G | |
| Reddy et al. 2024 | 2,880 | ✓ | G→S | |
| Luo et al. 2025 | 17,000 | ✓ | G→G | |
| Li et al. 2022 | 14,604 | ✓ | G→G | |
| Wu et al. 2025 | 54,738 | ✓ | ✓ | S→S |
| Zhang et al. 2026 | 120,000 | ✓ | S→G | |
| FluxFlow (ours) | 19,500 | ✓ | ✓ | G→S |
G ground‑based imaging · S space‑based imaging · → the direction of the super‑resolution mapping.
The same architecture, with the terms that need flux calibration switched off — the inverse‑variance term, the source‑importance term and the Wiener‑deconvolved measurement consistency are not available on these sets, so an ablated configuration is evaluated instead.
The honest caveat. AstroSR is distributed as RGB composites without absolute flux calibration, so total brightness is no longer conserved across the SDSS→HSC mapping. The Flux‑L1 column on that benchmark carries limited physical meaning and is reported only for completeness. The margins over regression baselines are markedly smaller than on DESI–HST, because neither dataset exposes super‑resolution under atmospheric aberration — the dominant degradation mode this method is built for is simply absent.
A source whose morphology has no preimage under A still produces a non‑zero forward residual, so it is penalised wherever the correction enforces y = A(x). Hallucinations are therefore not undetectable in principle — they are trajectories that exit the consistency set C(y), and the whole sampler is built to push them back in.
Minimising the mean‑square error over the Fourier‑domain filter, setting the variation to zero and solving gives the optimal filter W(k) = H*(k) / (|H(k)|² + λSNR−1).
Naive adjoint back‑projection contracts high‑frequency modes at a rate of |H(k)|² per step, so the loss accumulates into PSF‑scale smoothing across integration steps. Replacing that multiplier with the Wiener kernel keeps the cumulative transfer function stable at high frequencies and preserves the resolution gain the flow provides.
A normalised 2‑D Gaussian PSF of width σh, probed with a single point source of amplitude α, restricted to the band the LR observation actually supports, iterated for N Euler steps — giving an exact per‑mode recovery factor rather than a fixed‑point approximation.
W(k) → H*(k)/|H(k)|² = 1/H(k)
The exact inverse filter. Maximally amplifies high frequencies and is unstable wherever H(k) ≈ 0.
W(k) → λSNR H*(k) → 0
The response stays proportional to the matched filter — equivalent, up to scaling, to the raw adjoint. This is exactly the variant the ablation calls w/o Wiener.
λSNR therefore interpolates continuously between the two extremes, and controls how aggressively the deconvolution restores the high‑frequency content the PSF suppressed.
At N = 10 the naive recovery is visibly broader than the true PSF — the FWHM is enlarged from 4.71 to 6.78 HR pixels, while the Wiener‑corrected profile comes in at 4.85 and tracks the true PSF with in‑line width. The widening is a low‑pass effect, not noise amplification.
The honest caveat, in the paper's own analysis. If the iteration were run to convergence instead of a finite N, the fixed‑point per‑pixel variance evaluates to roughly 0.21 σ²n at the DESI–HST setting (σh = 2, kmax = π/4). Modest in absolute terms, but it is a substantial amplification of the noise power inside the narrow LR‑supported band, and the prefactor grows exponentially as kmax approaches Nyquist. The practical N = 10 keeps the recovery short of that fixed point — so in this regime the dominant pathology is the finite‑N low‑pass effect of Eq. 27, not the asymptotic noise amplification of Eq. 30.
| Ablation | Photometry | Detection | ||||
|---|---|---|---|---|---|---|
| PSNR ↑ | SSIM ↑ | Flux‑L1 ↓ | Prec ↑ | Rec ↑ | F1 ↑ | |
| w/ VAE | 28.90 | 0.593 | 5.075 | 0.859 | 0.211 | 0.3402 |
| w/ Bridge | 30.77 | 0.663 | 3.206 | 0.702 | 0.393 | 0.5019 |
| w/o WHT | 30.17 | 0.685 | 2.814 | 0.688 | 0.483 | 0.5272 |
| w/o Src | 31.16 | 0.711 | 3.098 | 0.729 | 0.464 | 0.5382 |
| w/o MC‑FS | 30.78 | 0.698 | 3.025 | 0.602 | 0.553 | 0.5256 |
| w/o Wiener | 30.79 | 0.699 | 3.031 | 0.708 | 0.448 | 0.5066 |
| Ours | 31.23 | 0.714 | 2.959 | 0.738 | 0.458 | 0.5412 |
The one that matters most. Without MC‑FS the sampler produces compact hallucination sources, and the table shows the signature: the highest recall in the whole study (0.553) together with a precision collapse to 0.602. The spurious peaks float as false positives and drag F1 to 0.5256, below the full model's 0.5412.
The quiet trade‑off. w/o WHT posts a marginally better Flux‑L1 (2.814 vs 2.959) — and that is reported rather than hidden, because it comes from fitting drizzle artefacts that overlap the source apertures. Without the WHT term the velocity field treats cosmic‑ray trails and chip gaps as valid targets and reproduces them in the reconstruction; the drops in PSNR, F1 and precision are the real cost.
Source weighting. A uniform loss is dominated by background pixels and cannot preferentially preserve the faint outer galaxy profiles that carry morphological and PSF‑shape information. The two weights are complementary: WHT filters data quality, source weighting emphasises scientifically informative regions.
Latent space. Moving the flow into a frozen SD‑VAE feature space costs 2.33 dB of PSNR and 0.121 of SSIM, and collapses recall to 0.211; the bridge variant that replaces the Gaussian source with a noisy bicubic mean recovers most of the photometry but still trails the full model on precision and F1.
| Scale | Metric | Method | |||||||
|---|---|---|---|---|---|---|---|---|---|
| Bicubic | EDSR | SRGAN | RCAN | SwinIR | HAT | FISR | Ours | ||
| ×2 | PSNR ↑ | 28.71 | 33.50 | 32.25 | 33.89 | 34.05 | 34.38 | 34.40 | 34.66 |
| ×2 | SSIM ↑ | 0.6945 | 0.7729 | 0.7501 | 0.7808 | 0.7812 | 0.7836 | 0.7859 | 0.7918 |
| ×2 | Flux‑L1 ↓ | 125.4 | 48.78 | 51.35 | 43.44 | 42.41 | 41.27 | 41.05 | 41.84 |
| ×4 | PSNR ↑ | 26.21 | 30.73 | 30.33 | 32.13 | 32.24 | 32.67 | 32.51 | 32.44 |
| ×4 | SSIM ↑ | 0.5993 | 0.6549 | 0.6522 | 0.6838 | 0.6881 | 0.6936 | 0.6891 | 0.6867 |
| ×4 | Flux‑L1 ↓ | 239.2 | 67.47 | 74.72 | 60.52 | 55.35 | 48.49 | 50.94 | 57.20 |
| Scale | Metric | Method | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Bicubic | EDSR | SRGAN | RCAN | ENLCA | SwinIR | HAT | FISR | Ours | ||
| ×2 | PSNR ↑ | 17.75 | 23.71 | 23.46 | 23.72 | 23.70 | 23.74 | 23.81 | 23.84 | 23.87 |
| ×2 | SSIM ↑ | 0.171 | 0.350 | 0.348 | 0.351 | 0.350 | 0.352 | 0.352 | 0.354 | 0.352 |
| ×2 | Flux‑L1 ↓ | 1916 | 213.6 | 165.9 | 226.3 | 216.1 | 223.1 | 193.3 | 192.5 | 198.7 |
Read this column with care. AstroSR is distributed as RGB composites without absolute flux calibration, so total brightness is not conserved across the SDSS→HSC mapping: its Flux‑L1 column carries limited physical meaning and is reported only for completeness.
What it wins. On STAR at ×2 the model takes the highest PSNR (34.66) and the highest SSIM (0.7918), with a top‑tier Flux‑L1 (41.84, behind FISR's 41.05 and HAT's 41.27). At ×4 it stays competitive with the strongest regression baselines — HAT leads PSNR at 32.67 against 32.44. On AstroSR at ×2 it takes the highest PSNR (23.87) and a competitive SSIM (0.352 against FISR's 0.354).
Reading it. FluxFlow trains in 16 h at ×2 and 47 h at ×4 — 2.90% and 8.56% of the cost of the strongest Transformer regression baseline (HAT, 552 h / 549 h) respectively, and at ×4 about 7.7× faster than the heaviest generative baseline (AS‑Bridge, 360 h). Only the two cheapest baselines, cGAN and GD‑Net, train faster at ×2.
Every ConvBlock is Conv2d → LeakyReLU → FiLM → Conv2d → LeakyReLU. The FiLM layer applies a feature‑wise affine modulation conditioned on the time‑conditioning state τ, so the time signal is injected at every resolution rather than only at the input. The timestep t ∈ [0,1] is mapped to τ by a sinusoidal positional embedding followed by a two‑layer MLP of output dimension 256.
FluxFlow recovers the diffraction spikes of the central star and the morphology of the extended source, while avoiding the speckled background of AS‑Bridge.
On a bright central star surrounded by closely packed compact sources, FluxFlow reproduces the spike pattern and resolves the neighbouring sources without the merging or smearing seen in the baselines.
On a field with a saturated source in the upper‑left corner and several low‑SNR background galaxies, FluxFlow preserves faint background galaxies that Bicubic and GD‑Net erase, while reconstructing the saturated halo more cleanly than the speckled output of AS‑Bridge.
On a binary‑star field, FluxFlow reproduces the four‑pointed diffraction pattern on the lower companion together with surrounding faint sources — whereas regression baselines suppress the spike pattern and the generative baselines either over‑smooth or over‑sharpen the background.
The panels in the comparison gallery above are real cut‑outs sliced from the paper's own comparison grids — every one of the 42 tiles is a genuine render, not a reconstruction of one.