Fast Volume Alignment by Frequency-Marched Newton

We develop a fast and accurate method for 3D alignment, recovering the rotation and translation that best align a reference volume with a noisy observation. Classical matched filtering evaluates cross-correlation over a large discretized transformation space; we show that high-precision alignment can be achieved far more efficiently by treating pose estimation as a continuous optimization problem. Our starting point is a band-limited Wigner-$D$ expansion of the rotational correlation, which enables rapid evaluation and efficient closed-form gradients and Hessians. Combined with analytical control of the complexity of trigonometric-polynomial landscapes, this makes second-order optimization practical in a setting where it is often avoided due to nonconvexity and noise sensitivity. We show that Newton-type refinement is stable and effective when initialized at low angular bandwidth: a coarse low-resolution $\mathrm{SO}(3)$ search provides robust candidates, which are then refined by iterative frequency marching and Newton steps, with translations updated via FFT in an alternating scheme. We provide a deterministic convergence guarantee showing that, under verifiable spectral-decay and gap conditions, the frequency-marching scheme returns a near-optimal solution whose suboptimality is controlled by the Newton tolerance. On synthetic rotation-estimation benchmarks, the method attains sub-degree accuracy while substantially reducing runtime relative to exhaustive $\mathrm{SO}(3)$ search. Integrated into the subtomogram-averaging pipeline of RELION5, it matches the baseline reconstruction quality, reaching local resolution at the Nyquist limit, while reducing pose-refinement time by more than an order of magnitude.