Estimation of orientation tensors for simple signals by means of second-order filters

Tensors have become a popular tool for representation of local orientation and can be used also for estimation of velocity. A number of computational approaches have been presented for tensor estimation which, however, are difficult to analyze or compare since there has been no common framework in which analysis or comparisons can be made. In this article we propose such a framework based on second order filters and show how it applies to three different methods for tensor estimation. The framework contains a few conditions on the filters which are sufficient to obtain correctly oriented rank one tensors for the case of simple signals. It also allows the derivation of explicit expressions for the variation of the tensor across oriented structures which, e.g., can be used to formulate conditions for phase invariance.