On the Equivariance of the Orientation and the Tensor Field Representation

The tensor representation has proven a successful tool as a mean to describe local multi-dimensional orientation. In this respect, the tensor representation is a map from the local orientation to a second order tensor. This paper investigates how variations of the orientation are mapped to variation of the tensor, thereby giving an explicit equivariance relation. The results may be used in order to design tensor based algorithms for extraction of image features defined in terms of local variations of the orientation, e.g. multidimensional curvature or circular symmetries. It is assumed that the variation of the local orientation can be described in terms of an orthogonal transformation group. Under this assumption a corresponding orthogonal transformation group, acting on the tensor, is constructed. Several correspondences between the two groups are demonstrated.