Symmetry and the salience of textures

In this paper we investigate the role of symmetry in visual stimuli designed to probe human sensitivity to image statistics. Our starting point is a recently published parameter space, a point in which defines a family of binary texture images displaying a prescribed content of \nth{1}- to \nth{4}-order correlations among pixels in 2x2 patches. We show that this parameter space can be represented by fewer variables, namely the \emph{orbit invariants} obtained by exploiting texture symmetry. Next we show how a class of locally countable texture statistics, the Minkowski functionals -- recently shown to be a proxy for human performance in texture discrimination tasks - can be written as a linear combination of the dihedral orbit invariants. Furthermore, by recasting these functionals as a combination of dihedral invariants, a generalization of these functionals can be obtained for textures of any number of grey-levels, patch sizes, or lattice types -- greatly reducing the number of dimensions/parameters needed to characterize the generated images. Orbit invariants may therefore provide a clue on the discrimination of these richer textures, as the ordinary Minkowski functionals do for binary textures.