Quaternion Weighted Averaging
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.