We show that, under spatially varying illumination, the light transport of diffuse scenes can be decomposed into direct, near-range (subsurface scattering and local inter-reflections) and far-range transports (diffuse inter-reflections). We show that these three component transports are redundant either in the spatial or the frequency domain and can be separated using appropriate illumination patterns. We propose a novel, efficient method to sequentially separate and acquire the component transports. First, we acquire the direct transport by extending the direct-global separation technique from floodlit images to full transport matrices. Next, we separate and acquire the near-range transport by illuminating patterns sampled uniformly in the frequency domain. Finally, we acquire the far-range transport by illuminating low-frequency patterns. We show that theoretically, our acquisition method achieves the lower bound our model places on the required number of patterns. We quantify the savings in number of patterns over the brute force approach. We validate our observations and acquisition method with rendered and real examples throughout.