Generalized quaternion interpolation

Generalized quaternion interpolation is an interpolation method used by the slerp algorithm. It is closed-form and fixed-time, but it cannot be applied to the more general problem of interpolating more than two unit-quaternions.

Definition of unconstrained interpolation

General interpolation of unconstrained values $left\{\; p\; ight\}$ with weights $left\{\; w\; ight\}$ is defined as the value $m$ that solves the sum:$sum\_i\; w\_i\; left(\; p\_i\; -\; m\; ight)\; =\; 0$.Because $m$ and $p$ values are unconstrained, this can be rewritten in the more familiar form of:$m\; =\; sum\_i\; w\_i\; p\_i.$

Quaternions, on the other hand, are constrained and the closed-form interpolation solution can not be applied to them.

Conversion to constrained interpolation

Because the unit-quaternion space is a closed Riemannian manifold, the difference between any two values on the manifold (in the tangent-space of the first value) can be defined as:$d\_\{0,1\}\; =\; log\; left(\; q\_0^\{-1\}\; q\_1\; ight)$where the logarithm is the hypercomplex logarithm. This difference can be applied to the value in which it is a tangent-space member as:$q\_0\; exp\; left(\; d\_\{0,1\}\; ight)\; =\; q\_1$where the hypercomplex exponential is used.

With these definitions in mind, the quaternion interpolation of values $left\{\; q\; ight\}$ with weights $left\{\; w\; ight\}$ can be defined (nearly identically to the unconstrained mean) as:$sum\_i\; w\_i\; log\; left(\; m^\{-1\}\; q\_i\; ight)\; =\; 0$which says that the weighted sum of all differences to $m$ (in $m$'s tangent-space) is zero.

Recursive formulation

The quaternion mean value defined above can be found in a recursive algorithm with some initial estimate (one of the points, for example) that will halt when the net-error is below some threshold or the algorithm has iterated beyond some time limit.

Each iteration of the algorithm is as follows, with an initial mean estimate of $m\_0$:$e\_\{k-1\}\; =\; sum\_i\; w\_i\; log\; left(\; m\_\{k-1\}^\{-1\}\; q\_i\; ight)$:$m\_\{k\}\; =\; m\_\{k-1\}\; exp\; left(\; e\_\{k-1\}\; ight)$

as iteration index $k$ increases, the value $m\_k$ will approach the true weighted-mean of the points.

References

* Xavier Pennec, "Computing the mean of geometric features - Application to the mean rotation," Tech Report 3371, Institut National de Recherche en Informatique et en Automatique, March 1998.