Decomposing a RELION projection matrix for James Hooker

sympy_matrix_decomp.ipynb

Thanks Alister,

A few questions/comments:

1. I’ll take your word for it that the matrices are in the form you describe. Is there a specification somewhere that confirms this?

2. In the case that $\cos(\beta)$ is zero, $\beta$ could presumably be either $\pi / 2$ or $3 \pi / 2$, and -M[1, 2] / M[1, 1] could be either $\tan(\alpha - \gamma)$ or $\tan(\alpha + \gamma)$. Whether $\beta$ is $\pi / 2$ or $3 \pi / 2$ could be told from M[2, 0].

3. I don’t immediately see why in the case that $\cos(\beta)$ is zero, the choice of $\gamma$ should be arbitrary. Why not fix $\alpha$?

Hey James

1. no spec afaik, I actually made a small mistake in the matrices above, s1 should be made from the tilt-image center not the tomogram center: https://github.com/alisterburt/tomo_preprocessing/blob/4a47549a1e0bf350fd924820169d563adf78d5ca/tomography_preprocessing/tilt_series_alignment/_job_utils.py#L28-L69
   You will also need to be careful, IMOD uses N+1 / 2 (zero indexed) as the image rotation center whilst RELION uses N / 2

2. yep - Euler angles are not a singular representation, there will always be multiple degenerate solutions

3. Another arbitrary point, same as above - you could fix alpha