Fast Bessel Matching

A key step in 3D single particle reconstruction, is the 2D image alignment kernel, which largely determines the efficiency and accuracy of the overall reconstruction procedure. In this work we present Fast Bessel Matching (FBM), a real-space, correlation-based approach to perform the 2D alignment step. By casting the matching problem into a formulation involving three angular parameters, the problem is reduced to the calculation of a single 3D fast Fourier transform. An extra speedup is accomplished by rapidly decaying factors in the formula for the correlation transform. We also consider variations of the RPC ("resampling to polar coordinates") and the FRM2D ("fast rotational matching in 2D") methods, including a radially dependent angular sampling. We perform the operation counts for these variations and for FBM, observing that FBM is more than 2 times faster than any of the others. We also note that the Fourier-space SCF (self-correlation function) method has, for usual values of the sampling parameters, an operation count comparable to that of FBM. Finally, we argue that FBM might be the optimal real-space correlation-based image-matching method in such applications as EM image reconstruction. This paper deals exclusively with the theory of our approach; implementation and applications will be presented elsewhere.