Toward deterministic compressed sensing.

Over the past decade, compressed sensing has delivered significant advances in the theory and application of measuring and compressing data. Consider capturing a 10 mega pixel image with a digital camera. Emailing an image of this size requires an unnecessary amount of storage space and bandwidth. Instead, users employ a standard digital compression scheme, such as JPEG, to represent the image as a 64KB file. The compressed image is completely recognizable even though the dimension of the compressed version is a tiny fraction of the original 10 million dimensions. Compressed sensing takes this mathematical phenomenon one step further. Is it possible to capture the pertinent information, such as the 64KB image, without first measuring the full 10 million pixel values? If so, how should we perform the measurements? If we capture the important information, can we still reconstruct the image from this limited number of observations? Compressed sensing exploded in 2004 when Donoho (1,2) and Candes and Tao (3) definitively answered these questions by incorporating randomness in the measurement process. Since engineering a truly random process is impossible, a major open problem in compressed sensing is the search for deterministic methods for sparse signal measurement that capture the relevant information in the signal and permit accurate reconstruction. In this issue of PNAS, Monajemi et al. (4) provide a major step forward in understanding the potential for deterministic measurement matrices in compressed sensing. Capturing digital images on a camera is simple; however, there are many applications where the measurement process has a much greater underlying cost. Magnetic Resonance Imaging (MRI) is a prime example of a high‐impact compressed sensing application. For most MRI examinations, a patient is required to lie still in a confined space for roughly 45 minutes. In some situations, compressed sensing has generated diagnostic‐quality MR images using only 10% as many measurements (5). MRI is only a single example of compressed sensing applications which extend well beyond imaging and include computed tomography, electro‐cardiography, multispectral imaging, seismology, analog to digital conversion, radar, X‐ray holography, astronomy, DNA sequencing, genotyping, and more (6). Traditional signal processing procedures measure the full signal directly and apply standard compression routines for storage or transmission. When needed, the original signal can be reconstructed by inverting the linear compression procedure. Compressed sensing transfers the workload from the measurement process to the signal reconstruction. While the measurement process remains linear, the reduced number of measurements forces a …