Equivariance and invariance-an approach based on Lie groups

The paper discusses the concepts of invariance and equivariance in the context of continuous operator groups, or Lie groups, acting on both the image and a descriptor of some image feature. The theory is illustrated by an example of how a 3D image neighbourhood that contains a linear structure can be mapped to a tensor that represents the orientation of the structure. A second order tensor is defined which represents both local orientation and local phase in a continuous way, the phase is given by the relation between the symmetric and anti-symmetric part of this tensor. By computing a tensor inner product of the tensor, a phase invariant orientations representation is obtained