Stress-induced birefringence is created by the manufacturing process of injection-molded lenses. Characterizing the degradation in optical performance is important information to guide manufacturing improvements. The 3D birefringence distribution is proportional to the inhomogenous stress field. Birefringence is an anistropic property therefore retardance measurements change if the optical component is rotated. Reconstruction of a 3D birefringence distribution from a series of retardance measurements poses a challenge due to the extension of tomographic algorithms to tensor-valued quantities. Our approach is a closed-form forward model to linearly relate the polarimetric measurements to a line projection through an discrete arrangement of index ellipsoids. The rank of this linear system is investigated for varying arrangements of index ellipsoids. For example, given an homogenous birefringence distribution, only three non-parallel line projections are required for a full-rank operator that reconstructs the index ellipsoid of a planar slice. These three line projections are full-rank for some arrangements of three unique index ellipsoids and rank deficient for others. This mathematical framework is developed to design a tomographic polarimeter for inspecting the stress-induced 3D birefringence distribution of injection-molded optics.