scipy.spatial.transform.RigidTransform.

mean#

RigidTransform.mean(weights=None)[source]#

Get the mean of the transforms.

The mean of a set of transforms is the same as the mean of its rotation and translation components.

The mean used for the rotation component is the chordal L2 mean (also called the projected or induced arithmetic mean) [1]. If A is a set of rotation matrices, then the mean M is the rotation matrix that minimizes the following loss function:

\[L(M) = \sum_{i = 1}^{n} w_i \lVert \mathbf{A}_i - \mathbf{M} \rVert^2 ,\]

where \(w_i\)’s are the weights corresponding to each matrix.

Parameters:
weightsarray_like shape (…, N), optional

Weights describing the relative importance of the transforms. If None (default), then all values in weights are assumed to be equal. If given, the shape of weights must be broadcastable to the transform shape. Weights must be non-negative.

Returns:
meanRigidTransform instance

Single transform containing the mean of the transforms in the current instance.

Notes

Array API Standard Support

mean has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

Library

CPU

GPU

NumPy

✅

n/a

CuPy

n/a

✅

PyTorch

✅

✅

JAX

✅

✅

Dask

⛔

n/a

See Support for the array API standard for more information.

References

[1]

Hartley, Richard, et al., “Rotation Averaging”, International Journal of Computer Vision 103, 2013, pp. 267-305.

Examples

>>> import numpy as np
>>> from scipy.spatial.transform import RigidTransform as Tf
>>> from scipy.spatial.transform import Rotation as R
>>> rng = np.random.default_rng(seed=123)

The mean of a set of transforms is the same as the mean of the translation and rotation components:

>>> t = rng.random((4, 3))
>>> r = R.random(4, rng=rng)
>>> tf = Tf.from_components(t, r)
>>> tf.mean().as_matrix()
array([[ 0.61593485, -0.74508342,  0.25588075,  0.66999034],
       [-0.59353615, -0.65246765, -0.47116962,  0.25481794],
       [ 0.51801458,  0.13833531, -0.84411151,  0.52429339],
       [0., 0., 0., 1.]])
>>> Tf.from_components(t.mean(axis=0), r.mean()).as_matrix()
array([[ 0.61593485, -0.74508342,  0.25588075,  0.66999034],
       [-0.59353615, -0.65246765, -0.47116962,  0.25481794],
       [ 0.51801458,  0.13833531, -0.84411151,  0.52429339],
       [0., 0., 0., 1.]])
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