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

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. Program summaryTitle of program: particleTiltCatalogue identifier: ADYW_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADYW_v1_0Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandComputers: IBM 2 compatible desktop PC; SGI2 OctaneOperating system: Red Hat 2 WS 3 Linux (with 2.4.21-40.EL kernel); IRIX 6.5 IP30Program language used: Fortran 90No. of bits in a word: 32No. of processors used: oneHas the code been vectorized: noNo. of lines in distributed program, including test data, etc.: 2397No. of bytes in distributed program, including test data, etc.: 47 017Distribution format: tar.gzPeripherals used: oneTypical 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.