vampire.amath.get_rotation_matrix

vampire.amath.get_rotation_matrix(A, B)[source]

Returns optimal rotation matrix to align 2D coordinates A to B.

Parameters
Andarray

Matrix to be rotated, with shape (2, n).

Bndarray

Matrix to be aligned to, with shape (2, n).

Returns
Rndarray

Optimal rotation matrix to be applied to A, with shape (2, 2).

See also

scipy.spatial.transform.Rotation.align_vectors

Aligns 3D coordinates.

Notes

We want to align 2D coordinates of object A to that of object B, represented by the matrices

\[\begin{split}\mathbf{A} = \begin{bmatrix} — & \mathbf{x}_A & — \\ — & \mathbf{y}_A & — \\ \end{bmatrix}, \mathbf{B} = \begin{bmatrix} — & \mathbf{x}_B & — \\ — & \mathbf{y}_B & — \\ \end{bmatrix},\end{split}\]

respectively, where \(\mathbf{x}_A, \mathbf{x}_B\) are the x-coordinates, and \(\mathbf{y}_A, \mathbf{y}_B\) are the y-coordinates.

It is equivalent to finding the optimal rotation matrix \(\mathbf{R}\) such that \(\mathbf{A}\) after rotation has the minimum sum of squared distance loss \(L(\mathbf{R})\) with \(\mathbf{B}\):

\[L(\mathbf{R}) = \Vert \mathbf{RA} - \mathbf{B} \Vert_2^2.\]

Kabsch algorithm

The optimal rotation matrix can be found using the Kabsch algorithm [1], [2], [3]:

  1. Compute the covariance matrix

\[\mathbf{C = AB}^T\]

  1. Compute the SVD of the covariance matrix

\[\mathbf{C = U\Sigma V}^T\]

  1. The optimal rotation matrix is

\[\mathbf{R = VU}^T\]

References

1

Lydia E. Kavraki, Molecular Distance Measures. OpenStax CNX. (2007) http://cnx.org/contents/1d5f91b1-dc0b-44ff-8b4d-8809313588f2@23

  • The section “Optimal Alignment for lRMSD Using Rotation Matrices” describes and proves the Kabsch algorithm.

2

Dryden, I. L. & Mardia, K. V. Statistical Shape Analysis, with Applications in R 2nd edn. https://doi.org/10.1002/9781119072492 (2016).

  • Section 4.1.1 “Procrustes distances” describes and proves the Kabsch algorithm with Lemma 4.1.

3

Wu, PH., Phillip, J., Khatau, S. et al. Evolution of cellular morpho-phenotypes in cancer metastasis. Sci Rep 5, 18437 (2016). https://doi.org/10.1038/srep18437

  • Supplemental information section “Decomposition of 2-dimensional shape and identification of shape mode” describes the implementation of the Kabsch algorithm in the context of VAMPIRE methodology.

vampire.amath.mean_center vampire.amath.pca