The determination of birefringence (magnitude and axis orientation) of optical materials is of significant interest in various fields. In the case of composite samples, this task becomes complicated and time-consuming; therefore, a partially automated procedure for reconstructing birefringence spatial distribution becomes valuable. Herein, we propose a procedure to reconstruct the spatial distributions of the retardance and optical axis orientation in a geological thin section from sparse quantitative birefringence measurements, using automatic boundary detection on cross-polarized light microscopy images. We examine two particular areas on the selected geological thin section: one that presents a uniaxial crystal with a circular cross-section of its refractive index ellipsoid and the other with grains of varying orientations. The measurement gives the orientation of the grain's optical axis both in and out of the plane of the thin section, which explains the qualitative observations with the cross-polarized light microscope. Future work will connect the measured orientation of the rock thin section with its 3D geological orientation in the field.