The rapid imaging renaissance: sparser samples, denser dimensions, and glimmerings of a grand unified tomography

The task of imaging is to gather spatiotemporal information which can be organized into a coherent map. Tomographic imaging in particular involves the use of multiple projections, or other interactions of a probe (light, sound, etc.) with a body, in order to determine cross-sectional information. Though the probes and the corresponding imaging modalities may vary, and though the methodology of particular imaging approaches is in constant ferment, the conceptual underpinnings of tomographic imaging have in many ways remained fixed for many decades. Recent advances in applied mathematics, however, have begun to roil this intellectual landscape. The advent of compressed sensing, anticipated in various algorithms dating back many years but unleashed in full theoretical force in the last decade, has changed the way imagers have begun to think about data acquisition and image reconstruction. The power of incoherent sampling and sparsity-enforcing reconstruction has been demonstrated in various contexts and, when combined with other modern fast imaging techniques, has enabled unprecedented increases in imaging efficiency. Perhaps more importantly, however, such approaches have spurred a shift in perspective, prompting us to focus less on nominal data sufficiency than on information content. Beginning with examples from MRI, then proceeding through selected other modalities such as CT and PET, as well as multimodality combinations, this paper explores the potential of newly evolving acquisition and reconstruction paradigms to change the way we do imaging in the lab and in the clinic.