Flux-Invariant Super Resolution for Astronomical Imaging

• Flux-Invariant Super Resolution (FISR) is a method that maintains the total integrated flux in astronomical images, ensuring measurement consistency across training and inference.
• It integrates PSF blurring, flux-consistent downsampling, and specialized network modules like FGG and FGC to optimize photometric fidelity using metrics such as Flux Error and Flux-L1.
• FISR delivers significant improvements in photometric accuracy over conventional SR methods, enhancing stellar photometry, population analysis, and weak lensing studies.

Flux-Invariant Super Resolution (FISR) designates a class of super-resolution (SR) methods for astronomical imaging in which the total integrated astronomical flux—i.e., the sum of detected photon counts across all pixels—is stringently preserved throughout both training and inference. This property is of primary astrophysical importance, as many downstream science tasks (e.g., photometry, stellar population estimation, mass mapping) depend critically on reliable flux reconstruction. Recent advances such as the STAR benchmark and FISR model (Wu et al., 22 Jul 2025), as well as closely related frameworks like FluxFlow (Liu et al., 5 May 2026), provide rigorous and measurement-consistent solutions to flux invariance in the SR context.

1. Flux-Preserving Data Generation

High-resolution source images are drawn from calibrated Hubble Space Telescope (HST) I-band (F814W) mosaics, covering wide star fields with both radiometric and astrometric fidelity (Wu et al., 22 Jul 2025). The data generation pipeline comprises:

• PSF Blurring: Each HST image is convolved with either a Gaussian or Airy point-spread function (PSF) to emulate instrumental and atmospheric degradation. For the Gaussian PSF, the kernel is $$\mathrm{PSF}_{\mathcal{G}}(x,y) = \exp\left[-(x^2 + y^2)/(2\sigma^2)\right]$$; for the Airy disk, $$\mathrm{PSF}_{\mathcal{A}}(r)\propto [J_1(kr)/(kr)]^2$$.
• Flux-Consistent Downsampling: The low-resolution (LR) pixel flux $F_\mathrm{LR}(i)$ is computed by accumulating HR pixel fluxes $f_\mathrm{HR}(j)$ weighted by the precise overlap in sky coordinates:

$$F_\mathrm{LR}(i) = \sum_{j\in S_i} w_{i,j} f_\mathrm{HR}(j),\qquad w_{i,j} = A_{i,j}/A_\mathrm{HR}(j)$$

where $A_{i,j}$ is the overlapped area between LR pixel $i$ and HR pixel $j$’s sky patch. This exactly conserves total flux, replicating physical photon integration on the sky rather than merely interpolating intensities.

• Patch Extraction: Overlapping sub-images (e.g., $128\times128$ px) are extracted, discarding those with excessive NaN regions to ensure data quality.

This data protocol ensures that synthetic low-resolution observations retain the true integrated flux of their high-resolution counterparts, thereby serving as a valid foundation for developing and testing flux-invariant SR models.

2. Flux Error Metric

Photometric fidelity, as opposed to mere pixel-level accuracy, is quantified by the Flux Error (FE) metric (Wu et al., 22 Jul 2025):

1. Detect $N$ objects in the ground-truth HR image via algorithms such as Starfinder.
2. For each object $$\mathrm{PSF}_{\mathcal{A}}(r)\propto [J_1(kr)/(kr)]^2$$0, perform aperture photometry to extract its true flux $$\mathrm{PSF}_{\mathcal{A}}(r)\propto [J_1(kr)/(kr)]^2$$1 from $$\mathrm{PSF}_{\mathcal{A}}(r)\propto [J_1(kr)/(kr)]^2$$2 and measured flux $$\mathrm{PSF}_{\mathcal{A}}(r)\propto [J_1(kr)/(kr)]^2$$3 from the SR result $$\mathrm{PSF}_{\mathcal{A}}(r)\propto [J_1(kr)/(kr)]^2$$4 (reusing object positions).
3. Compute:

$$\mathrm{PSF}_{\mathcal{A}}(r)\propto [J_1(kr)/(kr)]^2$$5

Lower FE values indicate improved preservation of integrated source fluxes—crucial for scientific analysis—compared to metrics such as PSNR or SSIM, which do not reflect photometric correctness.

A related metric, Flux-L1, is used in other works (Liu et al., 5 May 2026): $$\mathrm{PSF}_{\mathcal{A}}(r)\propto [J_1(kr)/(kr)]^2$$6 where $$\mathrm{PSF}_{\mathcal{A}}(r)\propto [J_1(kr)/(kr)]^2$$7 is the mask for the $$\mathrm{PSF}_{\mathcal{A}}(r)\propto [J_1(kr)/(kr)]^2$$8th source.

3. FISR Model Architecture and Optimization

The Flux-Invariant Super Resolution (FISR) model (Wu et al., 22 Jul 2025) is an encoder–decoder network incorporating explicit flux guidance:

• Encoder: Processes the LR input through convolutional and transformer blocks, extracting hierarchical feature maps $$\mathrm{PSF}_{\mathcal{A}}(r)\propto [J_1(kr)/(kr)]^2$$9.
• Flux Guidance Generation (FGG) Module: Detects object locations and photometric flux, then encodes these as multi-scale Gaussian "flux maps." A convolutive block-pyramid transforms flux maps into guidance tensors $F_\mathrm{LR}(i)$0 reflecting local photometric importance.
• Flux Guidance Controller (FGC): At each scale, combines $F_\mathrm{LR}(i)$1, $F_\mathrm{LR}(i)$2, and a learnable prompt $F_\mathrm{LR}(i)$3 through:
  • Guidance Controller Module (GCM): Channel-wise feature re-weighting via

  $F_\mathrm{LR}(i)$4

  where $F_\mathrm{LR}(i)$5 is derived from pooled $F_\mathrm{LR}(i)$6 and $F_\mathrm{LR}(i)$7. - Prompt Interaction Module (PIM): Merges GCM output with $F_\mathrm{LR}(i)$8 via a local transformer, yielding enriched features $F_\mathrm{LR}(i)$9.

• Decoder: Progressive upsampling with PixelShuffle reconstructs the HR output.

Losses are defined as: $f_\mathrm{HR}(j)$0 where $f_\mathrm{HR}(j)$1 is $f_\mathrm{HR}(j)$2 or $f_\mathrm{HR}(j)$3 error, and $f_\mathrm{HR}(j)$4 is a flux-weighted pixelwise loss: $f_\mathrm{HR}(j)$5 $f_\mathrm{HR}(j)$6 is the flux guidance map and $f_\mathrm{HR}(j)$7 controls the trade-off between global and photometric accuracy.

4. Quantitative Performance and Empirical Properties

On the STAR benchmark (54,738 HST pairs), FISR outperforms established SR methods such as EDSR, RCAN, SwinIR, and HAT:

Upsample	Model	PSNR (dB)	SSIM	FE
×2	FISR	37.88	0.8311	0.5739
×2	HAT	37.25	0.8295	0.7636

At ×2 upsampling, FISR achieves a 24.8% reduction in FE relative to HAT. At ×4, FE improvement is smaller but FISR still attains the best photometric results among baselines. Qualitatively, FISR restores sharper stellar point-spread cores, accurately preserves faint objects, and suppresses unphysical flux leakage into backgrounds. Distributional divergence metrics (KL, JS) between patches further confirm that FISR reconstructs source photon statistics with high fidelity (Wu et al., 22 Jul 2025).

5. Impact on Astronomical Imaging

Ensuring flux invariance in super-resolved images is critical for astrophysical data utility:

• Stellar Photometry: Enables reliable detection and measurement of faint stars, transients, and variable sources (incl. supernovae and Cepheid variables).
• Stellar Population Analysis: Accurate fluxes are essential for constructing color–magnitude diagrams and inferring masses, ages, and chemical abundances.
• Weak Gravitational Lensing: Quantitative shape- and flux-preserving reconstructions are fundamental for mass-mapping and cosmological inference.

By tightly coupling super-resolution to physical flux conservation, FISR bridges the gap between algorithmic image enhancement and scientifically trustworthy astrophysical measurement, thus supporting deeper, more reliable analyses without increased telescope time (Wu et al., 22 Jul 2025).

6. Comparison with Alternative Flux-Invariant Approaches

Flux invariance is approached both architecturally and algorithmically in other methods such as FluxFlow (Liu et al., 5 May 2026):

• Continuous Pixel-Space Flow: Enforces divergence-free evolution in pixel flux via ODEs that maintain $f_\mathrm{HR}(j)$8 at each time step, with test-time measurement-consistent correction.
• Source-Weighted Training: Assigns higher loss weight to source pixels (e.g., via $f_\mathrm{HR}(j)$9) based on segmentation masks from SExtractor, focusing network attention on photometric fidelity of scientifically important regions.
• Wiener-Regularized Back-Projection: At inference, adjusts generated images to exactly preserve total measured flux by a Wiener-filtered correction in Fourier space; area-downsampling and normalized PSF convolution guarantee that net flux is strictly conserved.

These design patterns offer an alternative but complementary route to the direct encoder–decoder plus flux-guidance architecture of FISR.

7. Scientific and Methodological Implications

Flux-invariant approaches, exemplified by FISR and related models, mark a significant methodological shift in astronomical SR: from visually plausible enhancement to measurement-consistent synthesis. These models underpin reliable science extraction from large sky surveys and archival data, free from artifacts that would otherwise compromise derived source properties. A plausible implication is that flux invariance, once a secondary consideration, is becoming a primary design constraint for imaging pipelines targeting next-generation astrophysical surveys.