Characterization of Second-Order Isotropic Fractional Brownian Fields

A first-order isotropic fractional Brownian field (IFBF) is generated by integrating a 2-dimensional (2-D) Gaussian field that has been obtained by passing a 2-D white Gaussian noise through a 2-D isotropic fractal filter. This first-order IFBF is characterized by a single parameter, the Hurst exponent H ∈ ]0,1[, similar to their 1-dimensional (1-D) counterpart. This paper presents a theoretical framework for the extension of a first-order IFBF to a second-order IFBF with H ∈ ]1,2[. Statistical properties such as covariance functions of these fields are investigated. We observe that the Hurst exponent of a number of real life images belongs to the range ]1,2[ suggesting that these images can be approximated as members of this class.