Reliability of Conformal Reconstruction-Error Bounds Under Forward-Operator Mismatch in Computational Microscopy

Reproducibility package for the manuscript of the same title. Release v1.0.0 is archived on Zenodo; the badge is the concept DOI and always resolves to the current version.

Experiments span two microscopy datasets acquired on different instruments, four reconstruction methods, two photon regimes, Zernike aberrations, and physical perturbations of focus, numerical aperture and wavelength. Severe-to-extreme Zernike mismatch increases reconstruction error by up to fivefold even as the nominal data-consistency residual decreases, with nominal 90% coverage falling as low as 0.31. The one-sided coverage inequality holds in every evaluated setting-by-content cell across ten random splits on both instruments. On genuine widefield measurements from the second instrument, in-domain calibration attains 0.891 ± 0.017 mean coverage at a nominal level of 0.900.

A fluorescence-microscopy restoration model can return an image that looks convincing even when the microscope's true optical response differs from the one the model assumed. This project asks whether we can bound how wrong that reconstruction is, without clean ground truth, with calibration that survives optical and specimen shift.

The contribution is not a better reconstructor. It is a post-hoc reliability layer, plus an honest account of when calibration breaks.

Headline results

The numbers below are the first BioSR stage (228 cells, four structures), Richardson-Lucy reconstruction, verified optics, CPU only. The manuscript extends this to a second instrument and four reconstruction methods.

1. Physics-derived features beat the obvious baseline, and are far more robust. Predicting reconstruction error from a single (y, x_hat) observation, with identical model class and conformal calibration:

photons	predictor	test rho	bound width	worst-group coverage
15	plain residual	+0.267	0.6405	0.486
15	PSF-family profile	+0.971	0.4749	0.800
100	plain residual	+0.748	0.5125	0.600
100	PSF-family profile	+0.961	0.3924	0.807

Bounds are 23-26% tighter at equal coverage. The sharper point is stability: the plain residual's usefulness collapses as photons drop (rho 0.748 -> 0.267) while the profile is flat (0.961 -> 0.971). Fifteen photons is where live-cell imaging actually operates, so the naive baseline fails exactly where it matters.

This also settles the strongest objection to the method: r_k is computable from (y, x_hat) alone, so the profile could have been redundant with what a learner extracts directly. It is not.

2. Conformal calibration collapses under shift. On a held-out biological structure, coverage falls to 0.27-0.41 against a nominal 0.90, worst-group coverage near zero, for both predictors. This is standard methodology applied exactly as the literature applies it, and it is the gap the novelty audit identified as open: every conformal imaging paper assumes calibration and test are exchangeable.

Whether shift-aware calibration repairs this is the question the manuscript answers: conformity-score shift is a reliable post-hoc diagnostic of coverage degradation, while the tested label-free observables alone are not sufficient to anticipate it dependably on real measurements.

What was tried and did not survive

Recorded because the retractions are as informative as the results.

• "The measurement residual is blind to operator mismatch." Withdrawn. Holding a specimen fixed and varying the operator, the residual tracks error at rho 0.89-0.92 above 100 photons. The blindness is real but photon-dependent, holding only in the photon-starved regime (rho -0.39 at 15 photons).
• "The residual is doubly blind -- floored by shot noise and driven onto that floor by the reconstructor's own objective." The mechanism is real (RL drives its residual to 0.994x the shot-noise floor) but does not imply blindness, and the claim was too strong.
• "The Gate 2 failure is a kernel-truncation artifact." Wrong. Truncation was a genuine artifact and is now guarded against, but the failure survived fixing it.
• "The correlation is an artifact of pooling across severity cells." Wrong. Within-cell rho is +0.77.

Verified physics

134 tests. Each pins a property that would otherwise invalidate downstream claims.

Property	Result
PSF vs analytic Airy pattern	cosine similarity 1.000000, L1 error 0.4%
FWHM scaling	0.5145 lambda/NA within 0.8-3.3%, lambda 488-640 nm, NA 0.9-1.4
Strehl at the Marechal limit	0.8148 at lambda/14, matching exp(-(2 pi sigma)^2) to 0.3%
Blur adjoint <Hx,z> = <x,H^T z>	exact to 3.7e-16, all four padding modes
Poisson-Gaussian variance	53.991 vs theoretical rate + read^2 = 54.000
Seed reproducibility	stable across PYTHONHASHSEED and chunk size

Four real bugs caught, all silent in a rendered image:

1. Even oversampling shifted every PSF by half a pixel. A 2x2 binning box cannot be centred on the peak sample, so PSFs lost their symmetry (cosine similarity to Airy 0.83, not 1.0). Odd oversampling is now required.
2. The blur adjoint was wrong under reflect padding. H = Crop . Conv . Pad, so H^T = Pad^T . Conv^T . Crop^T, and Pad^T is a scatter-add fold, not another pad. The shortcut broke the adjoint identity by 1.5e-3 -- enough to move Richardson-Lucy's fixed point.
3. A factor-of-two pixel size. The widely quoted 62.6 nm BioSR pixel is the widefield pixel; SIM ground truth is 2x upsampled, so objects are at 31.3 nm.
4. Severity-correlated PSF truncation. Aberrated PSFs have heavier tails, so a short kernel discards more of them; after renormalisation the aberrated operator became artificially compact, manufacturing a residual/severity correlation. At 31.3 nm a 63-px kernel loses 13 points of energy at extreme aberration; 191 px keeps the spread under 0.6.

The operator is derived from the data, then confirmed against the paper

BioSR's ground truth is itself a SIM reconstruction, hence already band-limited. If the simulated cutoff sat above the object's band limit there would be no genuine structure for a deconvolver to invent, and the whole study would be circular. Measured (scripts/inspect_objects.py):

• Object band edge 0.379 cycles/pixel, 76% of pixel Nyquist.
• At the verified optics (NA 1.7, 31.3 nm, 525 nm) the widefield OTF cutoff is 0.2027 cyc/px, leaving 9.05% of object energy in the operator's null space, with a measured SIM gain of 1.87x against the theoretical 2x.

Acquisition parameters are verified, not assumed: NA 1.7 (Olympus APON100XHOTIRF) and 405/488/560 nm from the Methods of Qiao et al., Nature Methods 18, 194-202 (2021); pixel sizes read directly from the MRC headers.

Note the operator is incoherent: its transfer function is the OTF, the pupil autocorrelation, so support is 2 NA / lambda, not the amplitude cutoff NA / lambda. Confusing the two overstates the null-space band by exactly 2x.

Positioning

Three parts of the obvious framing are already taken, and the paper must not claim them:

Piece	Owner
Null-space theory of hallucination	Bhadra et al., IEEE TMI 40(11) 2021
Re-blur-and-compare in microscopy	SQUIRREL, Nature Methods 2018
Kernel-weighted shift-adaptive calibration	Guan's LCP -- and the deterministic form is provably broken; use Hore & Barber's RLCP
PSF error propagation	Astronomy, since Paulin-Henriksson 2008

What survives: every existing calibrated-UQ method for imaging assumes the forward operator is known and fixed. See docs/gate0_novelty.md for the full audit, including a named competitive risk.

Workflow

Gate 0  Novelty audit ........................... PASS (narrowed)
Gate 1  Does mismatch cause plausible-but-wrong output? .. PASS
Gate 2  Is the trivial residual already enough? ... FAILED above 100 photons
Gate 3  Does the PSF-family profile beat it? ...... PASS (23-26% tighter)
Gate 4  Does shift-aware calibration repair coverage? .... COMPLETE
        [generalised beyond Richardson-Lucy; second instrument added]

The ordering is the point. Gates 1-3 cost CPU-hours; Gate 4 and the reconstructor sweep cost ~95 T4-hours. Establishing the phenomenon first is what prevents spending those hours on a dead hypothesis -- and Gate 2 did kill the original one.

Layout

src/optics/         zernike (OSA/ANSI, RMS-normalised), psf (sampling guards),
                    forward_model (Poisson-Gaussian, exact adjoint),
                    aberrations (Marechal-anchored severity ladder)
src/reconstruction/ rl (Richardson-Lucy with the true H^T 1 normaliser)
src/reliability/    features (physics-consistency profile, spectra)
src/data/           mrc (dependency-free reader), biosr (cell-blocked splits)
src/metrics/        image (SSIM, PSNR, radial spectrum, scale match)
scripts/            inspect_objects, gate1_phenomenon, gate3_deployment,
                    extract_features, gate4_calibration, summarize_gate1
tests/              134 tests pinning the physics and the analysis

Reproducing

pip install -e ".[dev]"
python -m pytest -q
python scripts/inspect_objects.py
python scripts/extract_features.py --photons 100 --n-iter 75 --out experiments/features/photons_100.npz
python scripts/gate4_calibration.py --features experiments/features/photons_100.npz

Data is not committed; see data/README.md for fetch commands, licensing and the verified acquisition parameters. Measurements are regenerated from deterministic per-image seeds rather than stored, because the full shift matrix would run to hundreds of GB against a 20 GB Kaggle working disk.

Still open

• Anticipating coverage loss before it happens. Conformity-score shift diagnoses degradation after the fact; the label-free observables tested here do not reliably predict it in advance on real measurements.
• A limitation to report: the profile degrades more than the plain residual under content shift (ood rho 0.49 vs 0.76), so richer features are more tied to training-structure content.

Citation

Please cite the archived release; metadata is in CITATION.cff.

T. T., S. (2026). Reliability of Conformal Reconstruction-Error Bounds Under Forward-Operator Mismatch in Computational Microscopy (v1.0.0). Zenodo. https://doi.org/10.5281/zenodo.22750126