Implications of invariance and uncertainty for local structure analysis filter sets

The paper discusses which properties of filter sets used in local structure estimation that are the most important. Answers are provided via the introduction of a number of fundamental invariances. Mathematical formulations corresponding to the required invariances leads up to the introduction of a new class of filter sets termed loglets. Loglets and are polar separable and have excellent uncertainty properties. The directional part uses a spherical harmonics basis. Using loglets it is shown how the concepts of quadrature and phase can be defined in n-dimensions. It is also shown how a reliable measure of the certainty of the estimate can be obtained by finding the deviation from the signal model manifold. Local structure analysis algorithms are quite complex and involve a lot more than the filters used. This makes comparisons difficult to interpret from a filter point of view. To reduce the number ‘free’ parameters and target the filter design aspects a number of simple 2D experiments have been carried out. The evaluation supports the claim that loglets are preferable to other designs. In particular it is demonstrated that the loglet approach outperforms a Gaussian derivative approach in resolution and robustness. keywords: generalized phase; invariance; uncertainty; quadrature filters; local structure tensor; spherical harmonics; orientation estimation; velocity estimation