Using geometric algebra to create differentiable models for optimizing camera-based optical metrology systems

In the design process of camera-based optical metrology systems numerous intricate and
seemingly distinct optimization tasks emerge.A frequently occurring but crucial task in design or calibration is to optimize the spatial degrees of freedom of system components. Of course, modelling the poses of rigid bodies is long solved using rotation matrices and translation vectors, but when it comes to optimizing, this choice of model gets quite tedious to handle. Useful concepts such as homogeneous coordinates or (dual) quaternions have been introduced to overcome this, which however – lacking a unified framework – can quickly become difficult to maintain.
As an alternative, in this contribution it is shown how the unifying methods of geometric
algebra can be used as an advantage for gradient-based optimization of camera-based optical metrology and imaging systems – and how this can be done in a generalized way for seemingly different objectives with respect to system design and calibration.