Note: There is a C++ version of this repository here.
def Superpose3D(X, # <-- Nx3 array of coords for the "frozen" point cloud
x, # <-- Nx3 array of coords for the "mobile" point cloud
# ---- optional arguments: ----
w = None, # optional weights for the calculation of RMSD
allow_rescale=False, # attempt to rescale mobile point cloud?
report_quaternion=False) # report rotation angle and axis?Superpose3D() takes two lists (or numpy arrays) of xyz coordinates (of the same length, N) representing two ordered sets of points ("clouds", X and x). Treating them as rigid objects, "Superpose3D()" attempts to superimpose them using rotations, translations, and (optionally) scale transformations in order to minimize the root-mean-squared-distance (RMSD) between corresponding points from either cloud, where RMSD is defined as:
...where:
R_ij = a rotation matrix (a 3x3 numpy array representing the rotation. |R|=1)
T_j = a translation vector (a 1-D numpy array containing x,y,z displacements)
c = a scalar (a number. optional. 1 by default)
This function returns a 4-tuple containing the optimal values of:
(RMSD, R, T, c)
Note: The point clouds must contain the same number of points (N).
Note: This function does not attempt to determine which pairs of points from either cloud correspond. Instead, it infers them from the order of the arrays. (It assumes that the i'th point from X corresponds to the i'th point from x.) More generally, the open3d python module supports the IPC algorithm, and works with unordered point clouds of arbitrary size. (Usage example here. Also see: link1, link2, link3, link4(protein-alignment).)
If the rotation angle and axis are needed, then set the report_quaternion argument to True. In that case, the function will return this 4-tuple instead:
(RMSD, q, T, c)
...where q is a numpy array of size 4. The first element of q will store cos(θ/2) (where θ is the rotation angle). The remaining 3 elements of q form a vector (of length sin(θ/2)), pointing along the axis of rotation. Equivalently, q is the quaternion corresponding to rotation R.
A weighted version of the RMSD minimization algorithm is also available if the caller supplies an extra argument specifying the weight of every point in the cloud (wn). In that case, RMSD is defined as:
This function implements a more general variant of the method from this paper: R. Diamond, (1988) "A Note on the Rotational Superposition Problem", Acta Cryst. A44, pp. 211-216.
This version has been augmented slightly to support scale transformations. (I.E. multiplication by scalars. This can be useful for the registration of two different annotated volumetric 3-D images of the same object taken at different magnifications.)
Note that if you enable scale transformations (i.e. if allow_rescale=True), you should be wary if the function returns a negative c value. Negative c values correspond to inversions (reflections). For this reason, if you are using this function to compare the conformations of molecules, you should probably set allow_rescale=False. This will prevent matching a molecule with its stereoenantiomer.
pip install .
Later, you can uninstall superpose3d using:
pip uninstall superpose3d
superpose3d depends on numpy
superpose3d is available under the terms of the MIT license.