Wavelet-based orientation of localizable Gaussian fields

In this work we give a sense to the notion of orientation for self-similar Gaussian fields with stationary increments, based on monogenic wavelet analysis of these fields, with isotropic Riesz wavelets. We compute the covariance of the random wavelet coefficients, which leads to a new formulation for the structure tensor and provides an intrinsic definition of the orientation vector as eigenvector of this tensor. We show that the orientation vector does not depend on the choice of the mother wavelet, nor the scale, but only on the anisotropy encoded in the spectral density of the field. Then we generalize this definition to a larger class of random fields called localizable Gaussian fields, whose orientation is derived from the orientation of their tangent fields. Two classes of Gaussian models with prescribed orientation are studied in the light of these new analysis tools.

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