3DEM-conventions ################

Here you can find a description of standard 3DEM conventions (as proposed by Heymann et al. JSB 151(2), 2005, p. 196-207 <http://dx.doi.org/10.1016/j.jsb.2005.06.001>_ with corrections <http://dx.doi.org/10.1016/j.jsb.2005.11.012>_) and conventions used by popular software packages if different.

.. contents:: Content :depth: 3

3DEM standard conventions

Coordinate and rotation convention

Coordinate convention is right-handed Cartesian coordinate system. Display convention is for x-axis to increase from left to right, y-axis to increase from bottom to top, and z-axis to increase from back to front (pointing at viewer) as shown:

.. image:: https://cloud.githubusercontent.com/assets/6952870/7271722/ab97fcf6-e8e5-11e4-8ff6-c23e85810ea9.jpg :width: 200px

.. image:: https://cloud.githubusercontent.com/assets/6952870/7271801/46dda0a8-e8e6-11e4-8a2d-b2441cf5af78.png :width: 400px

A positive rotation is defined as clockwise for the object. A positive rotation is defined as anti-clockwise for the coordinate system when viewed with the axis of rotation pointing at the viewer. For example, a rotation from the x-axis to the y-axis (about the z-axis) is positive. Note that the object will rotate clockwise when the coordinate system rotates anti-clockwise.

Euler angles

Traditionally in EM the direction of propagation of the electron beam is thought to coincide with z axis, which in addition uniquely specifies xy as the plane in which the data is collected. Thus, it is convenient to express the rotations with respect to the z axis. By using a ZYZ convention (rotation around the z axis, followed by a rotation around y, and another around z), one benefits from the fact that the description bears a simple relation to the description of a point on a sphere; that is to say, we can think of the decomposition as the description of a point on the sphere (the first two Eulerian angles YZ) and a final in-plane rotation (the final Z-Eulerian angle). We denote three respective Euler angles as phi (φ), theta (θ), psi (ψ) and the corresponding rotation in matrix notation as a product of three matrices:

RZ(ψ) RY(θ) RZ(φ)

Euler angles are three successive axial rotations:

#) phi, a rotation about the z-axis; #) theta, a rotation about the y'-axis; and #) psi, a rotation about the z''-axis.

Rotations of coordinate system (anti-clockwise):

.. image:: https://cloud.githubusercontent.com/assets/6952870/7271849/a9c70b28-e8e6-11e4-852c-52cfd4a8cd6a.jpg :width: 300px

Recall that it is the matrix at the rightmost end that is applied first. So one might write a 3DEM rotation as (φ, θ, ψ), where φ is applied first, θ second and ψ lastly.

Projection direction

Based on definition of Euler angles above, it is easy to see that first two Eulerian angles (φ, θ) define projection direction, while ψ defines rotation of projection in-plane of projection. Thus, as far as projections are concerned ψ is a trivial angle, as its change does not change the 'information content' of the projection.

Degeneracy of Euler angles

The range of possible Eulerian angles for an asymmetric structure is 0≤φ≤360, 0≤θ≤180, 0≤ψ≤360). However, for each projection whose direction is (φ, θ, ψ) there exists a projection that is related to it by an in-plane mirror operation along x-axis and whose direction is (180+φ, 180-θ, -ψ). Note the projection direction of the mirrored projection is also in the same range of Eulerian angles as all angles are given modulo 360 degrees (i.e., if say φ > 360, then φ = φ - 360, also if φ < 0, then φ = φ + 360.

Conversion between packages

Euler angles conversion

Relion <-> FReAlign

.. image:: https://user-images.githubusercontent.com/6952870/47355821-4da94480-d6ba-11e8-92f2-796a5c8432f3.png :width: 250px

CTF parameters conversion

• CTFFIND3 to IMAGIC <https://grigoriefflab.umassmed.edu/forum/software/ctffind_ctftilt/ctf_correction>_

.. code-block:: bash

DEFOCUS1=DFMID1
DEFOCUS2=DFMID2
DEFANGLE=90-ANGAST

• CTFFIND3 to SPIDER (now CTFFIND3 is available <https://spider.wadsworth.org/spider_doc/spider/docs/man/ctffind.html>_ from inside SPIDER)

.. code-block:: bash

defocus = (DFMID1 + DFMID2)/2;
astig = (DFMID2 - DFMID1);
angle_astig = ANGAST - 45;
if (astig < 0) {
    astig = -astig;
    angle_astig = angle_astig + 90;
}

• Relion to EMAN2 (use e2reliontoeman.py)

.. code-block:: bash

defocus=(rlnDefocusU+rlnDefocusV)/20000.0
dfang=rlnDefocusAngle
dfdiff=(rlnDefocusU-rlnDefocusV)/10000.0

• EMAN2 to Relion

  See e2refinetorelion2d <https://blake.bcm.edu/emanwiki/EMAN2/Programs/e2refinetorelion2d>_ and e2refinetorelion3d <https://blake.bcm.edu/emanwiki/EMAN2/Programs/e2refinetorelion3d>_

• SPIDER to FReAlign

.. code-block:: bash

df1 = spider.defocus - spider.magastig/2
df2 = spider.defocus + spider.magastig/2
angast = spider.angast + 45

Coordinate conversion

• EMAN1/2 boxer to XMIPP 2.4 Mark <http://web.archive.org/web/20130613235909/http://xmipp.cnb.csic.es/twiki/bin/view/Xmipp/BoxerToXmippMark>_
• SPIDER Web to XMIPP 2.4 <http://web.archive.org/web/20130715191432/http://xmipp.cnb.csic.es/twiki/bin/view/Xmipp/WebToXmippMark>_

File format conversion

• e2proc2d <http://blake.bcm.edu/emanwiki/EMAN2/Programs/e2proc2d>_ / e2proc3d <http://blake.bcm.edu/emanwiki/EMAN2/Programs/e2proc3d>_
• em2em <https://www.imagescience.de/em2em.html>_
• dm3 to other formats <http://web.archive.org/web/20150711065905/http://sites.bio.indiana.edu/~cryo/conversionFromDm3.html>_
• Various IMOD commands <https://bio3d.colorado.edu/imod/doc/program_listing.html>_
• relion_convert_to_tiff <https://relion.readthedocs.io/en/latest/Reference/MovieCompression.html#relion-convert-to-tiff>_

References

• 3DEM conventions <https://www.emdataresource.org/conventions.html>_
• Few info about Euler angles <https://web.archive.org/web/20200214125323/http://sparx-em.org/sparxwiki/Euler_angles>_
• The Transform Class in SPARX and EMAN2 <http://www.sciencedirect.com/science/article/pii/S1047847706002024>_
• Transform Python class in EMAN2 <http://blake.bcm.edu/emanwiki/Eman2TransformInPython>_
• Convert Euler angles for projections <http://blake.bcm.edu/emanwiki/EMAN2/FAQ/SpiderEuler>_
• https://github.com/alisterburt/eulerangles