Representing Local Structure Using Tensors

The fundamental problem of finding a suitable representation of the orientation of 3D surfaces is considered. A representation is regarded suitable if it meets three basic requirements: Uniqueness, Uniformity and Polar separability. A suitable tensor representation is given. At the heart of the problem lies the fact that orientation can only be defined mod 180◦ , i.e the fact that a 180◦ rotation of a line or a plane amounts to no change at all. For this reason representing a plane using its normal vector leads to ambiguity and such a representation is consequently not suitable. The ambiguity can be eliminated by establishing a mapping between R3 and a higherdimensional tensor space. The uniqueness requirement implies a mapping that map all pairs of 3D vectors x and -x onto the same tensor T. Uniformity implies that the mapping implicitly carries a definition of distance between 3D planes (and lines) that is rotation invariant and monotone with the angle between the planes. Polar separability means that the norm of the representing tensor T is rotation invariant. One way to describe the mapping is that it maps a 3D sphere into 6D in such a way that the surface is uniformly stretched and all pairs of antipodal points maps onto the same tensor. It is demonstrated that the above mapping can be realized by sampling the 3D space using a specified class of symmetrically distributed quadrature filters. It is shown that 6 quadrature filters are necessary to realize the desired mapping, the orientations of the filters given by lines trough the vertices of an icosahedron. The desired tensor representation can be obtained by simply performing a weighted summation of the quadrature filter outputs. This situation is indeed satisfying as it implies a simple implementation of the theory and that requirements on computational capacity can be kept within reasonable limits. Noisy neigborhoods and/or linear combinations of tensors produced by the mapping will in general result in a tensor that has no direct counterpart in R3. In an adaptive hierarchical signal processing system, where information is flowing both up (increasing the level of abstraction) and down (for adaptivity and guidance), it is necessary that a meaningful inverse exists for each levelaltering operation. It is shown that the point in R3 that corresponds to the best approximation of a given tensor is given by the largest eigenvalue times the corresponding eigenvector of the tensor.