A method to reconstruct a localized gradient-index field using the background-oriented schlieren (BOS) technique is proposed here. The BOS technique can measure deflection angles of light rays passing through the gradient-index field. When a gradient-index field is localized in a limited area, a scalar potential that is the integral of the gradient-index field over the light ray path can be transformed in the form of the integral of the deflection angle over the localized area. The gradient-index field is then derived from the scalar potential assuming that the field is axisymmetric. When the gradient-index is anisotropic, a light ray path depends on its polarization. The localized reconstruction method is applicable not only to the isotropic gradient-index field but also to such anisotropic field. An anisotropic gradient-index field can be experimentally generated by applying stress to a glass block with a resin sphere that contacts the glass block at a point. The anisotropic gradient-index field can be reconstructed using the localized reconstruction method, and it agrees with that of theoretical calculation using structural analysis software.