Staged Ab Initio 3D Reconstruction

Problem

Infer particle orientations, translations, optional state labels, and 3D maps without a trusted starting model. The refine3D alternation converges to whatever local optimum is nearest its starting point, and from random orientations that optimum is usually a featureless blob or a symmetric artifact. Ab initio 3D is a continuation scheme with the same three control knobs as ab initio 2D: admitted bandwidth, stochasticity of the search, and number of particles per iteration, plus two that are specific to 3D: the angular sampling density and the point-group symmetry imposed.

Initialization

Particle orientations are drawn uniformly at random; with independent multi-state initialization, state labels are also drawn uniformly. The starting volume is deliberately uninformative: Gaussian noise of standard deviation 5 * box/box_crop plus a soft Gaussian sphere of radius box/4, written as the map and both half maps. The first iteration therefore aligns particles to a nearly isotropic reference and the reconstruction from those poses is the first real signal. Alternatives are a reconstruction from the random orientations (rndstart), class averages (abinitio3D_cavgs), or an external volume after a correlation-based pose-initialization pass.

Continuation schedule

The particle workflow has up to eight stages. Each stage is one refine3D run with its own bandwidth, angular sampling, crop, iteration cap, search mode, and sampling target:

stage	1	2	3	4	5	6	7	8
max iterations	20	20	17	15	12	12	12	25
nspace	500	1000	1000	1000	2500	2500	5000	5000
search	prob_neigh (shc)	prob_neigh (shc)	prob	prob	prob	prob_neigh (state)	prob_neigh (state)	prob_neigh (state)
shifts	0	0	on	on	on	on	on	on
ML regularization	no	no	yes	yes	yes	yes	yes	yes
symmetry	start group	start group	start group, then search	target	target	target	target	target

Bandwidth. The ladder is defined on FRC values, not resolutions. From the class-average FRC curve frc(k) that produced the input, the two end criteria are

crit_1 = max(frc(k at lp_start), 0.65),    crit_n = max(frc(k at lp_final), 0.03),

stepped linearly across stages, and each stage's low-pass is the resolution at which the FRC first drops below that stage's criterion, never finer than lp_final. lp_start lies in [10, 20] A, lp_final is the median of the three best class-average resolutions clamped to [4.5, 6] A. If the FRC is unusable the ladder degrades to a linear one in resolution. Stage 1 is never finer than 20 A. The stage-1 reference is band-limited by a Gaussian at its low-pass (as in 2D) rather than by an FSC filter.

Sampling grid. Each stage crops to smpd_target = max(2, lp/3) at the ends and interpolates the box in between, always preserving box * smpd = box_crop * smpd_crop; the shift bound is min(8, max(2, 12 A / smpd_crop)) pixels. When a single downscaled particle cache is used, every eligible stage adopts the final crop while the low-pass still marches.

Search stochasticity. Stages 1 and 2 use the stochastic direct neighborhood (first-improvement over a random subset of directions); stages 3 to 5 use the full probabilistic table with balanced assignment; stages 6 to 8 restrict the table to pooled coarse-peak neighborhoods so the late search is local and exhaustive within it. The coarse subspace is rescaled with nspace so its granularity stays constant. Independent multi-state runs use plain first-improvement hill climbing for the first two stages.

Particle subset. A fixed target nsample (default 10 000 per state) gives update_frac = nsample * nstates / N_active, capped at 0.9 so fractional updates stay on; if the target exceeds 90 percent of the active set the run switches to full updates and disables trailing. From stage 5 (4 for independent multi-state) the subset is redrawn stochastically each iteration and only the best-scoring half of each direction bin is eligible (frac_best = 0.5, rising to 0.85 in the last two stages), which is a soft outlier rejection.

Regularization. ML regularization starts at stage 3, once the halves are independent enough for the FSC-derived signal prior to mean something. From stage 6 the reference bandwidth may become spatially varying (nonuniform filtering); stage 8 adds the density envelope for FSC masking and, when requested, the lag-by-one reference mask. NU high-resolution shell extension is not used in ab initio.

Symmetry

If the target point group pgrp differs from the starting group pgrp_start (usually C1), stages up to 3 refine in the starting group. At stage 3 the symmetry axis is found by maximizing the mutual consistency of the symmetry mates of the volume's band-limited polar Fourier samples:

T_g = samples rotated by M_g R_axis,     S = sum_g T_g,
score(R_axis) = (1/n_sym) sum_g Re<S, T_g> / sqrt(||S||^2 ||T_g||^2),

over 1000 spiral directions on the northern hemisphere at a low-pass of at least 12 A, followed by simplex refinement of the 10 best from three restarts. The map is symmetrized by averaging its mates in the found frame, the orientations are transformed into that frame and reduced to the asymmetric unit, and a symmetric reconstruction becomes the next reference. In multi-state runs each state finds and applies its own axis.

Multi-state modes

• single: one map.
• independent: nstates maps refined from random labels and independent random starts. Its default five-stage schedule stops at 6 A, before the local, NU, trailing, and automask stages, because its purpose is inspection.
• docked: one shared map through stage 5, then a balanced cohort of particles (capped at 100 000) is randomly split into states and refined with shared geometric neighborhoods, so the states begin from one consensus geometry and diverge only in density.

Final reconstruction

The FSC during the run is diagnostic only: the halves share initialization and the ab initio search is not gold-standard. At the end, a fresh all-particle reconstruction is made at the original sampling with no staged search or trailing controls, after a fill-in pass has given every active particle at least one post-schedule pose. Docked mode first verifies that every active particle received a post-split update.

Rationale

• Starting from noise with random poses means the first reconstruction is the projection-slice average of the data itself, and the coarse bandwidth keeps that average from locking onto high-frequency coincidences.
• Angular sampling follows bandwidth: at 20 A a 4-degree grid oversamples the data, so nspace = 500 is enough and the search is cheap when it is most stochastic.
• Searching in C1 first and imposing symmetry only after the map has a shape avoids the classic failure where a symmetric start point is an unbreakable fixed point of the alternation.
• Balanced probabilistic assignment in the middle stages does for projection directions what it does for 2D classes: it prevents a few directions from absorbing the data before the map is informative.

Implementation

• Stage schedule and policies: src/main/abinitio/simple_abinitio_controller.f90; bandwidth ladder in src/main/abinitio/simple_abinitio_utils.f90 and src/utils/filter/simple_estimate_ssnr.f90.
• Orchestration: src/main/commanders/simple/simple_commanders_abinitio.f90.
• Symmetry axis search: src/main/volume/simple_volpft_symsrch.f90, src/main/volume/simple_symanalyzer.f90.
• Base estimator: src/main/commanders/simple/simple_commanders_refine3D.f90.
• Policy: doc/policies/abinitio3D_policy.md.