Abstract: Sampling a scene by tracing rays and reconstructing an image from such pointwise samples is fundamental to computer graphics. To improve the efficacy of these computations, we propose an alternative theory of sampling. In contrast to traditional formulations for image synthesis, which appeal to nonconstructive Dirac deltas, our theory employs constructive reproducing kernels for the correspondence between continuous functions and pointwise samples. Conceptually, this allows us to obtain a common mathematical formulation of almost all existing numerical techniques for image synthesis. Practically, it enables novel sampling based numerical techniques designed for light transport that provide considerably improved performance per sample. We exemplify the practical benefits of our formulation with three applications: pointwise transport of color spectra, projection of the light energy density into spherical harmonics, and approximation of the shading equation from a photon map. Experimental results verify the utility of our sampling formulation, with lower numerical error rates and enhanced visual quality compared to existing techniques.