A finite element-based method is developed to process the stress maps to obtain stress distributions that satisfy the mechanical constraints. The method is applied to stresses obtained from the electron backscatter diffraction maps of a stainless steel 316-L material. The initial findings reveal realistic distribution of stress while preserving the experimental measurements at best. For example, the application of traction-free constraint on the surfaces reduces the out-of-plane stress components dramatically while keeping it underneath the surface. The method is applicable to any other stress or lattice strain measurement technique and remain to be a promising tool to reduce the measurement errors by application of physical constraints.