Analysing superimposed oriented patterns

Estimation of local orientation in images is often posed as the task of finding the minimum variance axis in a local neighborhood. The solution is given as the eigenvector belonging to the smaller eigenvalue of a 2×2 tensor. Ideally, the tensor is rank-deficient, i.e., the smaller eigenvalue is zero. A large minimal eigenvalue signals the presence of more than one local orientation. We describe a framework for estimating such superimposed orientations. Our analysis of superimposed orientations is based on the eigensystem analysis of a suitably extended tensor. We show how to carry out the eigensystem analysis efficiently using tensor invariants. Unlike in the single orientation case, the eigensystem analysis does not directly yield the orientations, rather, it provides so-called mixed orientation parameters. We therefore show how to decompose the mixed orientation parameters into the individual orientations. These, in turn, allow the superimposed patterns to be separated.