Diamond's 1988 rotational superposition algorithm (+scale tranforms)
Project description
Register 3-D point clouds using rotation, translation, and scale transformations.
Usage
def Superpose3D(X, # <-- Nx3 array of coords for the "frozen" point cloud
x, # <-- Nx3 array of coords for the "mobile" point cloud
w = None, # <-- an optional array of N weights
# (If w=None, equal weights will be used)
allow_rescale=False, #<--attempt to rescale mobile point cloud?
q = None) # <-- optional: store the quaternion for rotation here
Superpose3D() takes two ordered lists (or numpy arrays) of xyz coordinates (of the same length, N) representing points in a point cloud (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 point cloud, where RMSD is defined as:
RMSD = sqrt( (Σ_n w[n] * Σ_i |X[n][i] - (Σ_j c*R[i][j]*x[n][j] + T[i])|^2) / (Σ_n w[n]) )
If w=None, equal weights are used. In that case:
RMSD = sqrt( (Σ_n Σ_i |X[n][i] - (Σ_j c*R[i][j]*x[n][j] + T[i])|^2) / N )
...where:
T = a translation vector (a 1-D numpy array containing x,y,z displacements),
R = a rotation matrix (a 3x3 numpy array whose determinant = 1),
c = a scalar (a number, 1 by default)
This function returns a 4-tuple containing the optimal values of:
(RMSD, T, R, c)
If the rotation angle and axis are also needed, the caller can specify an optional q argument (an array of size 4). After invoking Superpose(), q will store the quaternion corresponding to rotation R, from which the rotation angle and axis can be determined.
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 stereoisomer.
Note: A C++ version of this repository is available at https://github.com/jewettaij/superpose3d_cpp
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