Generalized quaternion interpolation

Generalized quaternion interpolation is an interpolation method that extends the quaternion slerp algorithm. This generalized method can interpolate between more than two unit-quaternions, but is neither closed-form nor fixed-time.

Definition of unconstrained interpolation

General interpolation of unconstrained values $\left\{p\right\}$ with weights $\left\{w\right\}$ is defined as the value $m$ that solves the sum

$\sum _{i}w_{i}\left(p_{i}-m\right)=0$ and $\sum _{i}w_{i}=1.$

Because $m$ and $p$ values are unconstrained, this can be rewritten in the more familiar form of

$m=\sum _{i}w_{i}p_{i}.\,$

Unit 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

$d_{0,1}=\log \left(q_{0}^{-1}q_{1}\right)$

where the logarithm is the hypercomplex logarithm. This difference can be applied to the value in which it is a tangent-space member as

$q_{0}\exp \left(d_{0,1}\right)=q_{1}$

where the hypercomplex exponential is used.

With these definitions in mind, the quaternion interpolation of values $\left\{q\right\}$ with weights $\left\{w\right\}$ can be defined (nearly identically to the unconstrained mean) as

$\sum _{i}w_{i}\log \left(m^{-1}q_{i}\right)=0$

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 $m_{0}$

$e_{k-1}=\sum _{i}w_{i}\log \left(m_{k-1}^{-1}q_{i}\right)$

$m_{k}=m_{k-1}\exp \left(e_{k-1}\right)$

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.

This article "Generalized quaternion interpolation" is from Wikipedia. The list of its authors can be seen in its historical. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.

📰 Article(s) of the same category(ies)

• Classic energy problem in open-channel flow
• Mathematics and climate change
• Applied mathematics
• Cryptography
• Tonti diagram
• Applied mathematics in industry
• Statistical stability

---