Continuous In-Plane Refinement

Problem

The 2D and 3D search steps choose the in-plane rotation from a grid of nrots angles, nrots = 2 pftsz where pftsz ~ pi * mask radius is the number of angular samples of the polar Fourier transform. The grid spacing is therefore about one pixel of arc at the mask edge. Shifts are already continuous. This chapter describes how the committed pose is polished to a continuous angle jointly with the shift, and why the polish is kept strictly separate from the search that selected the pose.

Model

Fix the particle, the class or projection reference A, and the CTF. Let c(phi, k) be the polar Fourier coefficients of the reference and x(phi, k) those of the particle. The whitened Euclidean loss at continuous rotation theta and shift s is

L(s, theta) = 1 + [ sum_k w_k ( |A|^2 CTF^2 - 2 Re x_s^* R_theta A ) ] / (2 nrots sum_k w_k |x|^2),
w_k = k / sigma2(k),

where x_s is the particle with the shift phase ramp applied. Because the reference and particle are sampled on a ring in angle, the dependence on theta of the cross term is a circular correlation, which is a finite trigonometric series:

L(s, theta) = c_0(s) + sum_{m=1}^{pftsz-1} 2 Re[ c_m(s) e^{i m theta} ]   (+ Nyquist term),
dL/dtheta   = - sum_m 2 m Im[ c_m(s) e^{i m theta} ],
dL/ds_j     = b_0j(s) + sum_m 2 Re[ c^{(j)}_m(s) e^{i m theta} ].

The angular coefficients c_m are obtained by one FFT along the ring of the products x_s^* A; the shift derivative coefficients c^{(j)}_m come from the same FFT applied to x_s^* A multiplied by the shift-phase argument in x and y. The three series are computed in one batched FFT and evaluated at any real theta directly, so the loss and its full gradient at a fractional angle cost the same as at a grid angle and are exactly smooth in all three variables. The argument products must be formed before the angular transform; forming them after would require a convolution in angular frequency.

For the correlation objective, cc = N(theta)/sqrt(D(theta) C) with N the cross series, D the shift-independent reference power series, and C the particle power; its derivatives follow by the quotient rule, with the angle derivative (N' - N D'/(2D))/sqrt(DC).

Algorithm

With inpl_cont=yes, the search runs unchanged until a class (or state and direction) and an integer rotation index have been committed. Then one bounded joint solve polishes the pose:

1. seed at the committed integer rotation and its native shift;
2. bound the rotation variable to plus or minus two grid cells around the seed and the shifts to [-trs, trs]; other cells are never rescanned;
3. run L-BFGS-B on L(s, theta) with the analytic gradient (three variables, at most maxits_sh iterations);
4. accept the result only if it is finite, valid for the objective's range, improves on the seed's re-scored value by a relative tolerance of 1e-4 (with a 1 percent floor for the correlation objective), and does not sit within 1e-3 of the range of any bound. A solution pinned to a bound is a sign that the true optimum lies in a neighboring cell the search rejected, and is not trusted.

On acceptance the fractional angle is persisted (e3), the nearest integer index is kept for compatibility, and shift and score are taken from the same joint pose. On rejection the committed discrete pose stands with its re-scored value.

What the polish must not change

The joint solve runs exactly once per particle per iteration, after selection. Candidate scoring, probability-table construction, and shift seeding all use the discrete route. Two consequences are deliberate:

• Search trajectory identity. The sequence of classes and cells visited, and the assignment committed, is identical with and without the polish. Polishing candidates before selection would sharpen the score contrast the stochastic sampler sees and collapse exploration on difficult particles.
• Statistics parity. Every statistic that steers the schedule (in-plane angular change, class overlap, visited fraction) is computed from the discrete cells on both sides of the comparison. A fractional angle would otherwise register as sub-grid motion, read as premature convergence, and truncate the annealing that difficult particles need. The fractional angle is used only by reconstruction and persisted for the next iteration's seed.

The polish is available for the raw Euclidean and correlation objectives. It is not applied in time-series shift-only search.

Rationale

The alignment precision that reconstruction can use is limited by the noise, not by the grid, once the grid spacing is finer than the angular uncertainty. The polish therefore mainly removes the systematic quantization of the angle at the highest frequencies, where one pixel of arc at the mask edge is a full phase cycle. Separating polish from search keeps the search's annealing schedule, which was tuned on the discrete route, valid by construction.

Implementation

• Joint objective and gradients: src/main/pftc/simple_polarft_corr.f90.
• Joint optimizer and acceptance guards: src/main/pftc/simple_pftc_shsrch_grad.f90.
• 2D and 3D integration: src/main/strategies/search/simple_strategy2D_srch.f90, src/main/strategies/search/simple_strategy3D_srch.f90.
• Validation: the continuous in-plane and refine3D in-plane state sub-suites of simple_test_exec test=unit_pftc_align2D3D (simple_pftc_inplane_tester, simple_strategy3D_inplane_tester), which replaced the simple_test_continuous_inplane_rotation2D* programs in September 2026.
• Design note: doc/implementation_notes/continuous_inplane_rotation_polar.md.