A “mesostructure renderer” uses pre-computed multi-dimensional “generalized displacement maps” (GDM) to provide real-time rendering of general non-height-field mesostructures on both open and closed surfaces of arbitrary geometry. In general, the GDM represents the distance to solid mesostructure along any ray cast from any point within a volumetric sample. Given the pre-computed GDM, the mesostructure renderer then computes mesostructure visibility jointly in object space and texture space, thereby enabling both control of texture distortion and efficient computation of texture coordinates and shadowing. Further, in one embodiment, the mesostructure renderer uses the GDM to render mesostructures with either local or global illumination as a per-pixel process using conventional computer graphics hardware to accelerate the real-time rendering of the mesostructures. Further acceleration of mesostructure rendering is achieved in another embodiment by automatically reducing the number of triangles in the rendering pipeline according to a user-specified threshold for acceptable texture distortion.