On the equivalence of two methods of determining fabric tensor

In this paper it is studied how three methods of quantifying structural anisotropy are related. Mean intercept length (MIL) method has been designed for the analysis of binary images. Autocorrelation function and the covariance matrix of the gray-level intensity gradient (GST method) are approaches designed for the analysis of gray-level data. It is shown here that the autocorrelation function and the MIL methods are not related in a general case. In contrast, an analytical proof is provided to show that MIL and GST methods are strictly equivalent. The standard definition of MIL is expressed in terms of a gradient field. Next it is shown that eigenvectors of the MIL fabric tensor are also eigenvectors of the GST fabric tensor and eigenvalues of the MIL fabric tensor can be determined if the eigenvectors of the GST fabric tensor are known. It follows from the study that the application of MIL in the assessing quality of trabecular bone can be replaced in all cases by the application of the GST method, which is more general (as defined for gray-level data), easier to implement and less computationally expensive.