Functional framework for directional and anisotropic texture operators

Develop the functional framework for directional or anisotropic texture operators constructed by replacing the isotropic Gaussian kernel with anisotropic kernels or directional diffusion processes, so that texture orientation and scale can be distinguished.

Background

The pointwise multiscale texture operator studied in the paper is isotropic because it is generated by the isotropic Gaussian kernel. The authors note that many natural textures exhibit preferred orientations or other anisotropic structures, which the current operator cannot explicitly distinguish.

The proposed extension would use anisotropic kernels or directional diffusion processes to encode orientation alongside scale. The associated functional-analytic theory, including the appropriate texture spaces and their structural properties, is explicitly identified as unresolved.