Spatio-temporal analysis using tensors

Abstract

Summary form only given. A fundamental issue in the problem of finding an efficient algorithm for estimation of 3D orientation is how 3D orientation should be represented. A representation is regarded as suitable if it meets the three basic requirements of uniqueness, uniformity, and polar separability. A tensor representation suitable in the above sense has been obtained. The uniqueness requirement implies a mapping that maps 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 map onto the same tensor. It has been demonstrated that the above mapping can be implemented by sampling the 3D space using a specific class of symmetrically distributed quadrature filters.<>