Measurement-driven aberration calibration and a Fisher-information theory of optical identifiability for photon-limited fluorescence microscopy

Optical aberrations corrupt the point spread function (PSF) of a fluorescence microscope and silently degrade every downstream restoration, yet their measurement typically requires fluorescent beads, a wavefront sensor, or a training dataset. Here, we describe a measurement-driven calibration framework that estimates Zernike aberration coefficients directly from photon measurements of the sample itself, with no wavefront sensor, bead calibration, or training data: a differentiable scalar-diffraction forward model (Zernike pupil phase → PSF → Poisson photon counts) is fitted by maximum likelihood, and its Fisher information matrix quantifies how accurately the aberration parameters can be identified. The intensity PSF is exactly invariant to reversing the sign of the whole even-parity coefficient subvector (defocus, astigmatism, spherical) so that the sign is unidentifiable from a single frame, whereas the odd-parity coefficients (coma) are exempt from this degeneracy: the single-frame Fisher information of even-parity modes vanishes as |θ|2 when the aberration tends to zero, whereas odd-parity modes (coma) retain nonzero first-order information, a finite floor for the structures tested. This parity-resolved law is verified numerically over a factor of 300 in aberration magnitude, and defocus-diversity frames hold a constant information floor over the tested mode subspace, restoring identifiability at the 10−3-waves scale. The same framework predicts where information-limited breakdown occurs: across two real fluorescence structures and four aberration modes, the empirical estimation error tracks CRB; in 11 of the 12 measurable cells, the empirical crossing agrees with the Cramér–Rao prediction within a factor of three in either direction, and the remaining cell, an extrapolation below the scanned budget range, agrees within a factor of four (single-frame even-parity modes are scored over their equivalence class, one globally ambiguous sign for the even subvector); at a fixed total photon budget, diversity frames reach a given precision with 2.2 to 5.0 times fewer photons than a single focal frame for the degeneracy-limited modes (defocus, astigmatism). Validated on real fluorescence content (BioSR endoplasmic reticulum and microtubules, BioTISR clathrin-coated pits) and on deformable-mirror aberration ground truth (CoCoA beads), the differential z-stack estimate of the three mirror-applied modes (0.07 to 0.10 waves, exact axial-propagation frames) agrees with the applied coefficients to a mean absolute error of 0.023 waves and shows a two-dimensional blind failure onset near 0.09 waves root-mean-square (RMS) aberration. The structure-known axial-propagation working range extends well past the 2D one: no failure is observed at 0.15 waves RMS or below (0 of 17 runs), the first stochastic failures appear at ≈0.20 waves RMS (1 of 10 runs at each of the three transition levels), and failure dominates by 0.25 waves RMS (three of five cold starts at that level). What ends the axial range is a transition of the likelihood landscape, not an information shortage: at 0.25 waves RMS, the estimator can lock onto a near-sign-reversed basin, whereas the Cramér–Rao bound there is still four times below the target, and the Fisher matrix of the wrong solution is itself well conditioned, so the local bound cannot certify global uniqueness. Calibration reliability is therefore assessed by two complementary gates computed from the measurement itself: a Fisher gate for local precision and a restart-consensus gate, testing whether independent restarts agree on a single basin. Within the accepted regime, the calibrated PSF restores images to the oracle-PSF Richardson–Lucy level (within 0.05 dB of peak signal-to-noise ratio, PSNR) and improves spot-counting F1 from 0.03–0.04 to 0.73–0.75 in two of three tested noise realizations (the third leaves the calibrated working range).