Here you can find a description of standard 3DEM conventions (as proposed by Heymann et al. JSB 151(2), 2005, p. 196-207 with corrections) and conventions used by popular software packages if different.
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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:
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.
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):
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.
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.
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.
Relion <-> FReAlign
DEFOCUS1=DFMID1
DEFOCUS2=DFMID2
DEFANGLE=90-ANGAST- CTFFIND3 to SPIDER (now CTFFIND3 is available from inside SPIDER)
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)
defocus=(rlnDefocusU+rlnDefocusV)/20000.0
dfang=rlnDefocusAngle
dfdiff=(rlnDefocusU-rlnDefocusV)/10000.0EMAN2 to Relion
See e2refinetorelion2d and e2refinetorelion3d
SPIDER to FReAlign
df1 = spider.defocus - spider.magastig/2
df2 = spider.defocus + spider.magastig/2
angast = spider.angast + 45