Regression of 3D rigid transformations on real-valued vectors in closed form

In this paper, we present a regression for predicting 3D rigid transformations from real-valued vectors. We use a unit dual quaternion to represent the transformation. The regression is formulated as blending unit dual quaternions. To formulate it in a closed form, we introduce an approximation based on error metrics according to geometric algebra. Finally, we take an articulated motion and an elastic deformation as examples to present the descriptive power of our method in modeling the motion and the deformation.