We present a framework for calculating low-dimensional bases to represent image irradiance from surfaces with isotropic reflectance under arbitrary illumination. By representing the illumination and the bidirectional reflectance distribution function (BRDF) in frequency space, a model for the image irradiance is derived. This model is then reduced in dimensionality by analytically constructing the principal component basis for all images given the variations in both the illumination and the surface material. The principal component basis are constructed in such a way that all the symmetries (Helmholtz reciprocity and isotropy) of the BRDF are preserved in the basis functions. Using the framework we calculate a basis using a database of natural illumination and the CURET database containing BRDFs of real world surface materials.