Alignment of fiducial marks in a tomographic tilt series with an unknown rotation axis

Abstract

Alignment for tomography using a transmission electron microscopy frequently uses colloidal gold particles as fiducial reference marks. Typically, there is an implicit assumption that the tilt axis of the tomographic series is orthogonal to the beam direction. However, this may not be true, either intentionally, if a tilt-rotate stage is used, or unintentionally, because of mechanical errors in the rotation stage or the sample fixture. Here, we provide a computer code which takes as input a set of two-dimensional (2D) observations of fiducial reference marks at various tilt angles and the values of those tilt angles. It produces as output a three-dimensional model of the observations, 2D shifts for each view, and the tilt axis direction. 2 Commercial organizations identified in this paper are for the purpose of identification only and are not endorsed by NIST. The associated equipment or software identified is not necessarily the best available for the purpose. Program summary Title of program: particleTilt Catalogue identifier: ADYW_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADYW_v1_0 Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Computers: IBM 2 compatible desktop PC; SGI 2 Octane Operating system: Red Hat 2 WS 3 Linux (with 2.4.21-40.EL kernel); IRIX 6.5 IP30 Program language used: Fortran 90 No. of bits in a word: 32 No. of processors used: one Has the code been vectorized: no No. of lines in distributed program, including test data, etc.: 2397 No. of bytes in distributed program, including test data, etc.: 47 017 Distribution format: tar.gz Peripherals used: one Typical running time: 350 ms (larger included example, on 2.8 GHz 32-bit PC) Nature of problem: The program is used to assist the alignment step in tomography. The samples should be prepared with spherical particles (typically gold beads) which are observed in several views. (Not every particle need be observed in every view.) The program reports coordinates of a 3D model of the particles as well as the direction of the tilt axis as a point on the unit sphere. Method of solution: Our package minimizes an objective function whose free variables are a set of 3D model points and 2D shifts of the views as well as two parameters characterizing of tilt axis as a point on the unit sphere. The objective function is decomposed into a pure quadratic form which encompasses the model points and shifts, and a more complicated form which has only two degrees of freedom. The Broyden–Fletcher–Goldfarb–Shanno algorithm is applied alternately and iteratively to minimize the objective function with respect to the two sets.