Frequency domain regularization of d-dimensional structure tensor-based directional fields

The present work is intended to address two of the major difficulties that can be found when tackling the estimation of the local orientation of the data in a scene, a task which is usually accomplished by means of the computation of the structure tensor-based directional field. On one hand, the orientation information only exists in the non-homogeneous regions of the dataset, while it is zero in the areas where the gradient (i.e. the first-order intensity variation) remains constant. Due to this lack of information, there are many cases in which the overall shape of the represented objects cannot be precisely inferred from the directional field. On the other hand, the orientation estimation is highly dependent on the particular choice of the averaging window used for its computation (since a collection of neighboring gradient vectors is needed to obtain a dominant orientation), typically resulting in vector fields which vary from very irregular (thus yielding a noisy estimation) to very uniform (but at the expense of a loss of angular resolution). The proposed solution to both drawbacks is the regularization of the directional field; this process extends smoothly the previously computed vectors to the whole dataset while preserving the angular information of relevant structures. With this purpose, the paper introduces a suitable mathematical framework and deals with the d-dimensional variational formulation which is derived from it. The proposed formulation is finally translated into the frequency domain in order to obtain an increase of insight on the regularization problem, which can be understood as a low-pass filtering of the directional field. The frequency domain point of view also allows for an efficient implementation of the resulting iterative algorithm. Simulation experiments involving datasets of different dimensionality prove the validity of the theoretical approach.

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► Variational regularization which extends and smooths the directional field.
► General formulation for both orientation estimation and regularization problems.
► Increase of insight on the regularization, now understood as a low-pass filtering.
► Efficient frequency domain implementation of the resulting iterative algorithm.