Tomographic Reconstruction
Theory and Implementation
Prior to a 3D reconstruction projections must be aligned to a common coordinate system because despite the best efforts to keep the feature of interest in the center during acquisition movement is inevitable. Moreover, the electron optical lens system causes image rotations and even subtle focus-dependent magnification changes, which need to be compensated for.Currently, PyTom only supports alignment by fiducials (gold beads). The positions of markers in the projection images need to be provided in the form of a text file (result of interactive marker localisation in the PyTomGUI) or in an EM-file (This file can be obtained by interactive marker localization in the AV3/TOM or EM packages). There is also a function in AV3 to convert IMOD marker files to an EM file (av3_wimp2em.m) and a direct import function into PyTom.
The PyTom alignment function simply minimizes the residual error of the coordinates projected from a 3D model (i.e., each marker has 3 coordinates assigned) and the observed coordinates in the projections. In more detail, 3D coordinates of the marker model, the projection shifts, and the image rotation are first computed using an analytical approach, in which a constant rotation (=tilt axis) and magnification is assumed (TiltAlignment.computeCoarseAlignment). This function determines a tilt axis (to x-axis) between 0 and 180 degrees; by definition for a tilt axis x there is a second solution, x+180 deg, that would satisfy the marker equations equally well. You need prior knowledge to decide, which tilt axis is correct - assuming the wrong tilt axis will yield reconstructions with the inverted handedness. In the alignment it can be specified if the solution 180<x<360 deg is supposed to be correct rather than the default solution 0<x<180 (handflip=True).
To determine the 3D coordinate system of the markers the coordinates of one marker point need to be defined. Thus, a reference marker needs to be chosen (irefmark). Reference coordinates (r) can be assigned to this reference marker or default values can be chosen. The default value is r=(x_iref-center,y_iref-center,0). x_iref and y_iref are the coordinates of the marker in a reference projection iref. The choice of the coordinates will determine the determined shifts of projections. In other words: different choices of the coordinates of the reference marker will result in shifted 3D reconstructions!
After approximate determination of the image rotations, translations, and 3D coordinates of the markers an optimization algorithm (TiltAlignment.alignFromFiducials) determines the tilt-angle specific image rotation, tilt-angle specific magnification, and again translations and 3D coordinates. For the reference projection iref the magnification is defined as 1.
The projections are aligned by inverse application of the determined translations, rotations, and magnifications to each projection. The transformation is carried out as a single interpolation in real space (default interpolation method: 3rd order spline).
If a tomogram were reconstructed from the unprocessed projections the low-frequency information would be artificially enhanced as illustrated in Fig.1. To obtain a truthful 3D image the information below the so-called Crowther frequency need to be attenuated. This is accomplished by weighting the projections in Fourier space proportional to its frequency perpendicular to the tilt axis. This can be approximated as a 'ramp' function (analytical weighting). A more accurate weighting scheme would be to explicitly compute the overlapping information in Fourier space (exact weighting, not yet implemented in PyTom). In many cases it makes sense to furthermore apply a low-pass filter to the data to eliminate noise, which is dominant in the higher frequencies.