Generalized second-order invariance in texture modeling

In image processing, micro-textures are generally represented as homogeneous random fields, the term "homogeneous" indicating a second-order stationary random process. However, such a formulation is restrictive, and does not allow for the processing of anisotropic textures. The aim of this paper is to study a generalization of second-order stationarity to second-order invariance under a group of transforms, in order to apply this generalization to texture modeling and analysis. The general formulation of second-order homogeneity or G-invariance is given in relation to the framework of group theory. Two approaches are derived, taking into consideration transitive groups and generalized translations. For the latter approach, an important particular case is outlined, in which a second-order G-invariant random field X can be one-to-one associated to a second-order stationary random field. Some examples of interesting groups of transforms are given. Finally, Cholesky factorization is applied for the synthesis of random fields showing the generalized invariance property.