# Geometric Anisotropy Estimation

Non-parametric Identification of Anisotropic (Elliptic) Correlations in Spatially Distributed Data Sets, by Arsenia Chorti and Dionissios T. Hristopulos

The relevant paper was published in *IEEE Transactions on Signal Processing* (October 2008).

Random fields are useful models of spatially variable quantities, such as those occurring in environmental processes and medical imaging. The fluctuations obtained in most natural data sets are typically anisotropic. The parameters of anisotropy are often determined from the data by means of empirical methods or the computationally expensive method of maximum likelihood. In this paper we propose a systematic method for the identification of geometric (elliptic) anisotropy parameters of scalar fields. The proposed Covariance Hessian Identity (CHI) method is computationally efficient, non-parametric, non-iterative, and it applies to differentiable random fields with normal or lognormal probability density functions. Our approach uses sample based estimates of the random field spatial derivatives that we relate through closed form expressions to the anisotropy parameters. This paper focuses on two spatial dimensions. We investigate the performance of the method on synthetic samples with Gaussian and Matern correlations, both on regular and irregular lattices. The systematic anisotropy detection provides an important pre-processing stage of the data. Knowledge of the anisotropy parameters, followed by suitable rotation and rescaling transformations restores isotropy thus allowing classical interpolation and signal processing methods to be applied.

To use the matlab code, decompress the zip archive and place the files in a folder that is in the Matlab path.

The .m files include comments that can be read using the Matlab help function, e.g., by typing help aniso_cc_grid.

In interpreting results keep in mind the symmetries: (R, THETA) --> (1/R, 90 +/- THETA).**rf_gen:** This program simulates Gaussian random fields with specified geometric anisotropy.

**aniso_cc_grid:** This program uses finite centered differences to estimate the geometric anisotropy of random fields sampled on a regular grid.

**aniso_sg_grid:** This program uses Savitsky - Golay derivatives to estimate the geometric anisotropy of random fields sampled on a regular grid.

**aniso_interp_scatter:** This program uses interpolation followed by finite centered differences to estimate the geometric anisotropy of random fields sampled on a scattered grid. Scattered samples are generated using **rf_gen** and then randomly removing a number of points from the grid.

The anisotropy estimation code has been extended to the case of clustered data by means of the clustered CHI method. The relevant paper (Spiliopoulos et al. 2011) is published in *Computers and Geosciences*. The main idea is to use image processing filters to define clusters based on the variations of the sampling density, to estimate the anisotropy in each cluster, and then --if so desired -- to derive a coarse-grained estimate of the anisotropy for the entire area.

The R code implementing the clustered CHI method can be downloaded as a component of the Intamap packaged from *INTAMAP* or it can be obtained by emailing me at: dionisi at mred.tuc.gr