A polar Fourier geometric approach to volume-free single-particle 3D reconstruction.

We introduce a compact mathematical formulation for the inverse single-particle 3D reconstruction problem, a high-dimensional inverse problem in which millions of parameters are estimated from extremely noisy experimental measurements. Given a collection of noisy 2D projection images (particles) of an unknown 3D charge-density distribution, the objective is to infer the unknown particle orientations and thereby enable ab initio 3D reconstruction via tomographic methods. We develop a method for generating regularized reprojections directly from the noisy particles that does not rely on explicit 3D density reconstruction. Instead, we recast the ab initio orientation-recovery problem in polar Fourier coordinates through discretization of the rotation group SO(3). The directions of projection are mapped onto slices intersecting the origin of the 3D Fourier transform. The rotations in the plane normal to a projection direction are mapped onto radial lines in the 2D Fourier transforms of the particles. An optimization procedure jointly estimates particle projection directions, in-plane rotation angles and rotational origin offsets. Regularized reprojections are computed by averaging along lines in the polar Fourier representation, exploiting data redundancy to suppress noise and improve stability. We present the mathematical framework in detail and provide initial benchmarks demonstrating the performance and robustness of the approach.