Non-uniform refinement: Adaptive regularization improves single particle cryo-EM reconstruction

Ali Punjani; Haowei Zhang; David J. Fleet

where w ρ (x) is positive and integrates to one, with spatial extent ρ. This allows one to estimate θ at each voxel independently, while the overlapping neighborhoods ensure that θ(x) varies smoothly. This approach also provides a natural way to allow for variable neighbourhood sizes, where ρ(x) depends on location x, so both rigid regions and transition regions are well modeled. Importantly, we want ρ(x) to be large enough to reliably estimate θ(x), but small enough to enable local transitions. A reasonable balance can be specified in terms of the highest frequency with significant power as captured by the regularization parameter θ(x). In particular, as a heuristic it suffices to ensure that ρ(x) > γ θ(x) where γ, the adaptive window factor (AWF), is a constant. 1 This constraint yields the final computational problem solved in non-uniform refinement to regularize 3D electron density at each iteration; i.e.,