The characterization and estimation of the Holder regularity of random fields has long been an important topic of Probability theory and Statistics. This notion of regularity has also been widely used in Image Analysis to measure the roughness of textures. However, such a measure is often not sufficient to characterize textures as it does account for their directional properties (e.g. isotropy and anisotropy). In this paper, we present an approach to further characterize directional properties associated to the Holder regularity of random fields. Using the spectral density, we define a notion of asymptotic topothesy which quantifies directional contributions of field high-frequencies to the Holder regularity. This notion is related to the topothesy function of the so-called anisotropic fractional Brownian fields, but is defined in a more generic framework of intrinsic random fields. We then propose a method based on multi-oriented quadratic variations to estimate this asymptotic topothesy. Eventually, we evaluate this method on synthetic data and apply it for the characterization of historical photographic papers.
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