Nonuniform Filtering

Problem

A single FSC-derived low-pass cutoff filters the whole map at the resolution of its average voxel. A well-ordered core is then over-smoothed and a flexible periphery is left with noise. Nonuniform filtering estimates a spatially varying cutoff c(v) from the data themselves, so that each voxel is filtered at the resolution at which the two independent half maps still agree there.

The input is an unfiltered even/odd half pair on one grid. The output is a label field c(v) over a candidate bank of cutoffs, the filtered halves and merged map, and a local-resolution map.

Model

Let E and O be the raw halves and E_c, O_c their low-pass versions at candidate cutoff c from the static bank

20, 15, 12, 10, 8, 6, 5, 4 A,

capped at FSC0.143/1.5 of the raw pair (only members coarser than that are retained, never fewer than two); when ML regularization is active the regularized half pair is one more member beside the finest retained one, at the same Potts coordinate, so the unary alone decides between them.

If the true local resolution at voxel v is c, then E_c(v) predicts O(v) up to noise, and vice versa; filtering finer than c lets through noise that the other half does not share, and filtering coarser loses signal that it does. The evidence for candidate c at v is therefore the cross-half prediction error

r1_c(v) = [E(v) - O_c(v)] / sigma(|v|),
r2_c(v) = [E_c(v) - O(v)] / sigma(|v|),
C_c(v)  = H(r1_c(v)) + H(r2_c(v)),

with the Huber loss H(r) = r^2/2 for |r| <= 1.345, else 1.345(|r| - 0.6725). The transition 1.345 gives 95 percent efficiency at the Gaussian; the linear tail stops isolated large residuals (a stray strong voxel) from dominating.

Whitening. The noise scale sigma(r) is a function of real-space radius, estimated once from the raw difference E - O: in each radial shell,

sigma_j = 1.4826 * median | (E-O)_j - median(E-O)_j |,

with gap filling, radial smoothing, and linear interpolation between shell centers. A radial profile rather than a global scalar is required because the gridding correction and any tapered solve support make reconstruction noise grow toward the periphery; a global scale would put central and peripheral residuals in different Huber regimes.

Support. All statistics are evaluated inside the sphere of diameter mskdiam. The sphere, rather than a density mask, is used because it keeps the candidate objective comparable everywhere and retains a solvent population that later serves as a noise reference.

Algorithm

1. Candidate costs. For each c, compute C_c(v) on the support and smooth it with a mask-normalized tent kernel of radius min(1.5 c, 30 A). Smoothing at a scale proportional to c matches the spatial extent over which a cutoff of c is meaningful; voxels outside the support never influence in-support costs.
2. Initial labels. c(v) = argmin_c C_c^smoothed(v).
3. Ordered-label Potts smoothing. Minimize over the 26-neighbor lattice

text E(c) = sum_v C_{c(v)}(v) + sum_{v~w} phi( |rank(c(v)) - rank(c(w))| ) / deg(v),

where rank is the index in the retained bank and phi is zero for adjacent ranks and a linear-quadratic hinge for larger jumps. The prior is ordered: a 20-to-15 A boundary costs less than a 20-to-4 A jump. Degree normalization keeps the regularization strength uniform at the sphere boundary. Iterated conditional modes with eight-color sweeps (no two concurrently updated voxels are neighbors) and tie preservation. 4. Synthesis. At each voxel take E_{c(v)}(v) and O_{c(v)}(v); the merged map is their average; the local-resolution map stores c(v) in Angstrom inside the support and zero outside or beyond Nyquist.

When ML regularization is active, the regularized half pair (the closed-form voxelwise Wiener shrinkage of the raw pair) joins the bank as one more member beside the finest retained rung, at that rung's prior coordinate, and competes for every voxel like a hard rung: its cost is the cross-half prediction error of the regularized halves. Where it wins, the map keeps the estimator's own high-resolution content; where a hard rung wins, the estimator over-reached there. Its label resolution is the raw pair's FSC=0.143, and it joins the bank the moment that is at or beyond the ladder's finest rung; within the ladder the rungs compete alone (the cut at fsc/1.5 always keeps a rung finer than the FSC).

High-resolution extension (retired 2026-09-16)

A sequential shell walk (nu_refine=yes) used to challenge the frontier of the finest populated label with the next unrepresented Fourier shell. Its acceptance criterion was equivalent to a local FSC above 0.5, conservative for the map and no better than the FSC=0.143 extent for the matching band, and it never let the regularized pair compete. A generated dense ladder replaced it for two days (2026-09-16 to 18) and regressed PfCRT; the static ladder with the regularized pair beside the finest retained rung is the one competition for every workflow.

Handoff to matching

The FSC and the NU filter answer different questions: the FSC reports the average resolution, the NU field reports where the map is better than average. The matching low-pass for the next iteration is the finest member of the bank: the finest rung of the ladder cut at fsc/1.5, or the regularized pair once its FSC=0.143 is at or beyond the ladder's finest rung, bounded by any explicit lp and by lpstop. The same rule serves every workflow, and which labels won voxels decides the filter, never the band. In plain nonuniform mode the even and odd NU halves stay separate references; in nonuniform_lpset the merged NU map is used with a single band. No further low-pass is applied on top of an NU reference.

The band leads the FSC by 1.25-1.5x within the ladder, which is what carries a climb. With the band at the previous FSC=0.143 crossing the alignment never sees the shells beyond the crossing where the reference still holds signal; the crossing then moves at most about one shell per iteration and the orientation assignment crawls or freezes (PfCRT: 0/4 and 1/9 restarts converged, 2026-09-16/18). That is underfitting: the sigma2-weighted objective, the stochastic assignment and the evidence-limited NU reference guard against fitting noise, and the runs with a 1.5x lead (PfCRT 10/10, +0.47 A of FSC per stage-6 iteration against +0.19 A) show no sign of it. The uncapped July 2026 ladder led by about 2x and froze 2 of 10 restarts, so the lead is bounded by the cut.

The reproducibility envelope derived from the same candidate costs is a separate estimator: NU-evidence envelope masking.

Rationale

• Cross-half prediction error is a direct, model-free test of local resolution: it needs no assumption about the signal, only that the two halves have independent noise.
• Comparing candidates with one shared whitening profile makes the minimum-cost label meaningful; comparing candidates each smoothed at its own scale would bias the boundary.
• The ordered Potts prior encodes that resolution varies continuously in space, which is what distinguishes local resolution from voxelwise noise.

Implementation

• Bank, costs, labels: src/main/nu_filt/simple_nu_filter*.f90; the per-state driver: src/main/volume/simple_nu_state_filter.f90.
• Noise scale and Huber objective: src/main/image/simple_image_calc.f90.
• Integration into volume assembly: src/main/commanders/simple/simple_commanders_rec_distr.f90.
• Matching bandwidth handoff: src/main/strategies/search/simple_matcher_refvol_utils.f90.
• Policy: doc/policies/nonuniform_filtering_policy.md.