An automated method for generating a fiber alignment map in tissues, tissue-equivalents, and other fibrillar materials exhibiting linear and circular optical properties and scattering is presented. This method consists of interrogating the sample with elliptically polarized light from a rotated quarter-wave plate and an effective circular analyzer, and implementing nonlinear regression techniques to estimate parameters defining the optical properties of the optic train and the sample. Thus, an account is made for imperfect and misaligned optic elements. The optic train was modeled using the Mueller matrix representation and the combined sample properties by an exponential matrix. Because a sample's Mueller matrix does not uniquely determine the linear, circular, or scattering properties, the circular properties and effective scattering are estimated for a matched isotropic sample to determine and correct for the linear birefringence of an aligned sample. The method's utility is demonstrated by generating an alignment map of an arterial media-equivalent, a relevant test case because of its circumferential alignment and thus showing the method's sample orientation independence.