Identifying the full-stress tensor over the surface of a vibrating structure is essential for design validation and vibration-fatigue-life estimation, yet current experimental methods provide only pointwise or incomplete stress data. Infrared cameras can provide non-contact, full-field surface-stress measurements on vibrating structures via the thermoelastic effect; however, they measure only the first stress invariant (i.e., the sum of the normal stresses at the free surface: σ_xx+σ_yy). The individual stress components (σ_xx, σ_yy, τ_xy) required for a complete characterization of the surface-stress state remain inaccessible from a single thermoelastic measurement. This article introduces a method that recovers the full-field, in-plane-stress power spectral density (PSD) field from thermoelastic camera data on a base-excited structure, demonstrated for thin, isotropic, metallic, plate-like structures under adiabatic conditions. First, a characterization measurement identifies the hybrid stress-mode shapes containing all the plane-stress components by merging thermoelastic camera data with a simplified FE model through SEMM. Then, during operation under broadband random base excitation, the camera response is projected onto these mode shapes and expanded to the full-stress-tensor PSD. The method is validated experimentally on a clamped aluminium plate, identifying spatially resolved stress PSDs for all three in-plane components (σ_xx, σ_yy, τ_xy). The full-field first invariant σ_xx+σ_yy is measured experimentally; the individual components σ_xx, σ_yy and τ_xy are reconstructed through an experimentally corrected modal expansion informed by an FE stress basis, which supplies the inter-component ratios.