The Sharp Plane Is Tilted: Anisotropy Priors Must Follow the Stored Affine, Not the Grid

Restoration models for anisotropic 3-D data need one piece of physics: the direction of worst resolution. Recent 3-D microscopy restoration methods hard-code it as the grid axis $z$. We show that this is a statement about the resampling that produced the stored volume: under a linear resample $A$ of the acquisition, the plane of best resolution $\mathcal{S}$ maps to $A\mathcal{S}$ and its normal ${\mathbf{a}}_0$ to $A^{-\top }{\mathbf{a}}_0$, so a grid-axis prior is exact only when the transported axis still lands on a grid axis. For stage-scanned light-sheet data $A$ is the deskew affine the file records; when it leaves the camera plane tilted, the worst axis is $\mathbf{n}=A^{-\top }{e}_{\mathrm{scan}}$ or $\mathbf{s}=A{e}_{\mathrm{scan}}$, both read from the file and ordered by the ratio of $A$'s column norms. On 107 blocks from 18 open-top light-sheet biopsies (7 patients) the spectrum's sharp lobe lies along the camera axis $A$ predicts in 103/107 blocks, and a sweep that uses no network finds $\mathbf{n}$ the cheapest direction to under-sample on the median over blocks. On held-out blocks decimated along $\mathbf{n}$, the restorer that assumes $z$ leaves them worse than unrestored in explained variance (EV) at decimation factors $k=4$ and $8$ (0/4 patient-disjoint folds improve them). The same network pointed at $\mathbf{n}$ improves them in 4/4 folds and beats the convention by $+.025$, $+.060$ and $+.040$ in EV at $k=4$, $8$ and $16$, and by $.047$ F1 on a nucleus-detection endpoint that does not use EV ($k=8$). The gap survives the published CARE recipe, and eight restorers trained at fixed axes from $-45^{\circ }$ to $90^{\circ }$ trace a curve that peaks at the file's angle ($k\le 8$). On a second instrument (oblique-plane microscopy) a degradation whose stored-frame direction the file sets (every second camera frame dropped before the deskew) is improved only by the restorer pointed at the file's scan direction once its kept-slice mask input is removed; on brain cone-beam CT with an injected oblique axis and a true reference the rotated prior beats the $z$ prior in 9/9 fold$\times$seed pairs. The fix is one signed angle per stored volume, read from the file.