Fig. 1.

Introducing the sample deformation in tilt-series alignment. (a) The three different coordinate systems involved in the projection model: microscope, sample and projection image. The tilt axis is the Y-axis of the microscope. In this scheme, the tilt axis is marked by a crosshair in the middle of the sample and runs perpendicular to the sheet. At the untilted position, the coordinate system of the microscope and the sample coincide. The axes of the image system are denoted by u and v. In this simplified scheme, post-projection rotations are ignored, and v is thus parallel to the tilt axis, running perpendicular to the sheet. (b) Fiducial-based alignment relies on determination of the 3D coordinates of the fiducials (black spots in the sample) along with the basic image parameters by minimizing the differences between the calculated projections of the fiducials (black spots in the images) and the measured positions. Standard alignment ignores the potential deformation that the sample may undergo during imaging. (c) Sample deformation at the acquisition of any image i can be modelled by means of polynomial surfaces (${\mathrm{\Delta }}_x^i\text{,}{\mathrm{\Delta }}_y^i\text{,}{\mathrm{\Delta }}_z^i$) that account for the 3D motion at each point of the sample. The calculated projections of the fiducials are expected to better approach the experimental measurements. (d) Sample deformation at an image i can alternatively be modelled by polynomial surfaces ($S_u^i\text{,}{S}_v^i$) that represent the 2D motion at the projection image level for each point of the sample. This alternative modelling stems from the fact that the projection operation prevents full 3D description of the sample deformation with the approach in (c). See main text for details.