The view-independent global illumination problem is rephrased as one determining a radiance function across each surface in the environment. A new methodology for diffuse environments, based on the sampling and reconstruction of these functions is introduced. Within this context, the following problems are investigated: (i) where the radiance functions should be samples; (ii) how to evaluate a radiance function at each sample; and (iii) how to reconstruct a radiance function for the set of samples. The new methodology relaxes some of the assumptions built into current radiosity algorithms. Results are presented which show that the new methodology yields significantly higher accuracy than existing radiosity methods.