The structure tensor in projective spaces

Abstract

The structure tensor has been used mainly for representation of local orientation in spaces of arbitrary dimensions, where the eigenvectors represent the orientation and the corresponding eigenvalues indicate the type of structure which is represented. Apart from being local, the structure tensor may be referred to as "object centered" since it describes the corresponding structure relative to a local reference system. This paper proposes that the basic properties of the structure tensor can be extended to a tensor defined in a projective space rather than in a local Euclidean space. The result, the "projective tensor", is symmetric in the same way as the structure tensor, and also uses the eigensystem to carry the relevant information. However, instead of orientation, the projective tensor represents geometrical primitives such as points, lines, and planes (depending on dimensionality of the underlying space). Furthermore, this representation has the useful property of mapping the operation of forming the affine hull of points and lines to tensor summation, e.g., the sum of two projective tensors which represent two points amounts to a projective tensor that represent the line which passes through the two points, etc. The projective tensor may be referred to as "view centered" since each tensor, which still may be defined on a local scale, represents a geometric primitive relative to a global image based reference system. This implies that two such tensors may be combined, e.g., using summation, in a meaningful way over large regions