Fast Volume Alignment by Frequency-Marched Newton Method

Abstract

We introduce Matcha, a fast method for rotational pose estimation in three-dimensional alignment, and combine it with FFT-based translation updates for full pose estimation. Classical matched filtering evaluates cross-correlation over a large discretized transformation space; we instead treat rotational alignment as a continuous optimization problem on SO(3). Matcha starts from a bandlimited Wigner-D expansion of the rotational correlation, which enables rapid objective evaluation together with analytic gradients and Hessians. A low-bandwidth SOFFT search provides robust candidate rotations, which are then refined by frequency marching: the angular bandwidth is progressively increased, and candidates are updated by Newton steps at each level. This confines exhaustive search to a single low-frequency stage while allowing the final accuracy to be determined by continuous refinement rather than by the grid spacing. We prove a deterministic conditional guarantee showing that, under reasonable assumptions, Matcha returns a near-optimal solution for the final bandlimited objective. On synthetic rotation-estimation benchmarks, Matcha attains sub-degree accuracy while substantially reducing runtime relative to exhaustive SO(3) search. Integrated into a RELION-5 subtomogram-averaging workflow, it matches the baseline reconstruction quality on the tested dataset, reaching the same Nyquist-limited local resolution while reducing rotational pose-refinement time by more than an order of magnitude.