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

Ali Punjani; Haowei Zhang; David J. Fleet

where e is the negative log likelihood of the validation half-set given the regularized half-map. The second line simplifies this expression by using the raw reconstruction from the opposite half-set as a proxy for the actual observed images. Note that assumptions for "gold standard" refinement [28] are not broken in this procedure (see Section 5) . With the L2 norm, Eqn. 1 reduces to a sum of per-voxel squared errors, corresponding to white Gaussian noise between the half-set reconstructions. When the choice of θ causes r θ to remove too little noise from the raw reconstruction, the residual error E will be unnecessarily large. If θ causes r θ to over-regularize, removing too much structure from the raw reconstruction, then E increases as the structure retained by r θ no longer cancels corresponding structure in the opposite half-map. As such, minimizing E(θ) provides the regularizer that optimally separates signal from noise. Similar objectives have been used for image de-noising [18] . Finally, we note that this formulation can be extended to compare each half-set reconstruction against images directly (dealing appropriately with the latent pose variables) or to use error functions corresponding to different noise models.