Quaternion Weighted Averaging Proofs

Quaternion Averaging paper

1. Objective function

This paper use the element-wise matrix difference as the objective function.

Then extend the Frobenius norms we could found :

The new optimization problem becomes :

Which are equations (4) and (5) in the paper:

2. Further Derivative

Using the expression of the rotation matrix from quaternion :

Take the upper equation into the optimizaton objective function :

Further see the last term of the upper equation:

Finally, we have :

Which are equations (8) and (9) in the paper:

3. Using the rotation matrix expression

Further take the expression of rotation matrix into the upper equations. we could have :

Finally we reach the equation (13) in the paper:

Which could be solved efficiently.