Regression on Lie groups and its application to affine motion tracking

In this chapter, we present how to learn regression models on Lie groups and apply our formulation to visual object tracking tasks. Many transformations used in computer vision, for example orthogonal group and rotations, have matrix Lie group structure. Unlike conventional methods that proceed by directly linearizing these transformations, thus, making an implicit Euclidean space assumption, we formulate a regression model on the corresponding Lie algebra that minimizes a first order approximation to the geodesic error. We demonstrate our method on affine motions , however, it generalizes to any matrix Lie group transformations.