Warp Tomographic Reconstruction with Defocus Correction

Warp takes the local defocus and sample distortion, as well as magnification anisotropy into account when reconstructing full or partial tomographic volumes. For a partial reconstruction at any position in the volume, the original 2D images are sampled at the following positions: $$\mathbf{\text{s}}={\mathbf{\text{R}}}_{Euler}\left(0,{\alpha }_{Tilt},{\psi }_{Tilt}\right)\cdot \mathbf{\text{p}}+{\mathbf{\text{o}}}_{Tilt},$$ where R_Euler is the rotation matrix for 3 Euler angles following the Xmipp convention^{55 (link)}, α is the stage tilt angle, ψ is the in-plane angle of the tilt axis, p is the particle position within the tomographic volume, and o is the in-plane offset of the tilt axis. The coordinates are centered within the volume and images. The CTF for each 2D image is calculated using a defocus of: $$z=z_{Tilt}+\frac{\mathbf{\text{s}}\ast {\mathbf{\text{n}}}_{Tilt}}{{\mathbf{\text{n}}}_{Tilt,z}},$$ where z_Tilt is the average defocus estimated for the tilt image, n_Tilt is the sample plane normal, n_Tilt,z is the z component of the normal, and * denotes the scalar product between two vectors. The reconstruction is performed in Fourier space using a gridding algorithm^{23 (link)}, with the data weighted by the respective CTF, and the dose- and tilt-dependent heuristic from RELION⁴⁹ , but without the final deconvolution step (i. e. the weights are inserted as |CTF|, not as CTF^{2 (link)}). To obtain a full tomogram, Warp reconstructs a uniform grid of small, cubical volumes with an overlap of 50%, and inserts the central 50% into the overall volume to account for artifacts associated with Fourier space reconstruction at the borders of the local volumes. This ensures the corrections can be applied with local precision and remain reasonably continuous between adjacent sub-volumes.