🚀 Increase performance of axis-angle conversions

[Theo-Cheynel]

Motivations and context

When converting from axis-angle to matrix and back, current rotation_conversions.py does an intermediate step by converting to quaternions. There exists a faster way using Rodrigues' rotation formula .

Changes

Used the formulas for both axis-angle to matrix and matrix to axis-angle conversions.
The changes work with any number of batch dimensions, just like the original functions.

Effect

There is a 2x speedup when converting from axis-angle to matrix, and this speedup is consistent regardless of device and batch size.
There is a 5-8x speedup when converting from matrix to axis-angle.

The only difference in the results happens in matrix_to_axis_angle, when you give as input a matrix that is not a rotation matrix. This behaviour is undocumented anyway and the results do not make sense either way.
If this is really an issue, we can always merge only the first part of the changes, or we could check that the matrix is orthogonal and run the previous code if it isn't (for backwards compatibility)

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[bottler]

I'm worried about very small angles here. The current implementations delegate to other functions which are being really careful about nasty angles.

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[alex-bene]

@bottler sorry for reviving this old PR, but is there no interest in merging this? Testing locally (m1 max) the pytorch3d implementation for axis-angles to rotation matrix conversion with the intermediate step vs a conversion directly using the [Rodrigues' rotation formula, there is more than 10x difference in performance.

[bottler]

I guess it could be added as a "fast" option? The default can't be this unless we have some way to check corner cases.

[alex-bene]

Hey @bottler , thanks for the prompt response, I made a new PR (#1948 ) to make improvements to the proposed changes from this PR. (I couldn't find a way to make a commit here)

I'm worried about very small angles here. The current implementations delegate to other functions which are being really careful about nasty angles.

• I believe this is also addressed within these changes, but let me know if there is something we need to improve further.

[bottler]

@alex-bene Can we be really conservative: add fast: bool=False, eps: float = 1e-6 as parameters and then the code would be

if not fast: 
    return quaternion_to_matrix(axis_angle_to_quaternion(axis_angle))
<<<your code>>>

?

[alex-bene]

@bottler we can also close this PR based on #1948 and 7a3c0cb