In this article we propose a new framework for capturing light transport data of a real scene, based on the recently developed theory of compressive sensing. Compressive sensing offers a solid mathematical framework to infer a sparse signal from a limited number of nonadaptive measurements. Besides introducing compressive sensing for fast acquisition of light transport to computer graphics, we develop several innovations that address specific challenges for imagebased relighting, and which may have broader implications. We develop a novel hierarchical decoding algorithm that improves reconstruction quality by exploiting interpixel coherency relations. Additionally, we design new nonadaptive illumination patterns that minimize measurement noise and further improve reconstruction quality. We illustrate our framework by capturing detailed high-resolution reflectance fields for image-based relighting.