superpose3d 1.5.0
Diamond's 1988 rotational superposition algorithm (+scale tranforms)
License: MIT
Keywords: registration, 3d, structure-comparison, molecular-structure, clem

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
                # ---- 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 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:
```
    R = a rotation matrix    (a 3x3 numpy array representing the rotation. |R|=1)
    T = a translation vector (a 1-D numpy array containing x,y,z displacements)
    c = a scalar             (a number, 1 by default)
```
This function returns a 4-tuple containing the optimal values of:
```
   (RMSD, R, T, c)
```
*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*.)

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 the
[quaternion corresponding to rotation *R*](https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation),
from which the rotation angle and rotation axis can be easily 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