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description: How SYNAPSE-SR works: a Sentinel-2 camera model, a Tikhonov physics baseline, a Mamba state-space network constrained to the null space, trust maps and calibrated uncertainty.

How it works

x_hat = x_base + P_N(delta)
Component Role
A Sentinel-2 forward operator: per-band point-spread function on a 0.5 m grid, sampled at source-pixel centres
x_base deterministic baseline: bicubic anchor corrected by Tikhonov regularisation. One regularisation weight is shared by all four bands (so every band is sharpened equally) and set by the Morozov discrepancy principle (discrepancy, in units of the L2A sensor noise: 0.5 for Pro, a tight fit; 4 for Flash, which absorbs forward-model error such as point-spread function and registration). Flash then restores each 10 m pixel's measured mean reflectance with a smooth bicubic correction (restore_mean), counted in x_base
delta structural correction predicted by the network
P_N projector onto the null space of A, I - A^T (A A^T)^+ A: removes every component the sensor could have observed

Because A P_N = 0, re-observing x_hat gives exactly what re-observing x_base gives, whatever the network predicts. The network can add detail below the sensor's resolving power. It cannot contradict the measurement.

Network

SynapseProX5, 14.4 M parameters:

  • State-space backbone: 6 residual groups of 8 visual state-space blocks, with four-direction scanning so that no orientation is preferred, followed by channel attention.
  • Two stems: an RGBN spatial stem, plus a gated stem for the six 20 m bands. The gate starts closed, so the 20 m context is used only as far as training finds it useful.
  • Frequency mixer: splits features into a local low band and its high-frequency complement, mixes each, and re-fuses them.
  • Direct x5 head: PixelShuffle x5 with ICNR initialisation, so all 25 sub-pixel phases start identical and no 10 m cell pattern is imprinted.
  • Two outputs: delta (2 m, RGBN) and a confidence map computed from detached features, so the confidence loss cannot steer the reconstruction.

Geometry

x5 is odd. Output pixel 5i + r spans [10i + 2r, 10i + 2r + 2] m inside source pixel i, and r = 2 is centred on the source-pixel centre. The output grid therefore shares the input origin exactly.

Training (summary)

The network was trained on Sentinel-2 L2A paired with 0.6 m aerial reference imagery. The reference was registered to Sentinel-2, integrated onto the 2 m grid, and gain-matched per band to Sentinel-2 through the forward operator. Every loss term acts only on the null-space component P_N(x_hat - target). The model is therefore never trained to reproduce cross-sensor radiometric differences that the Sentinel-2 measurement already determines. The backbone was initialised from public CC0 weights; see Acknowledgements.