One of the central tenets of signal processing and data acquisition is the Shannon/Nyquist sampling theory: the number of samples needed to capture a signal is dictated by its bandwidth. Very recently, an alternative sampling or sensing theory has emerged which goes against this conventional wisdom. This theory now known as "Compressive Sampling" or "Compressed Sensing" allows the faithful recovery of signals and images from what appear to be highly incomplete sets of data, i.e., from far fewer data bits than traditional methods use. Underlying this metholdology is a concrete protocol for sensing and compressing data simultaneously. Following this protocol would bypass the current wasteful acquisition process in which massive amounts of data are collected only to be—in large part—discarded at the compression stage, which is necessary for storage and transmission purposes. In the compressed sensing paradigm, one could translate analog data into already compressed digital form, obtaining super-resolved signals from just a few sensors.
The last two years have seen an explosion of research activity in the area of compressive sampling and our lectures will present the key mathematical ideas underlying this new sampling or sensing theory, which come from various subdisciplines within the mathematical sciences; namely, probability theory and especially random matrix theory, mathematical optimization, and analysis in high-dimensional Banach spaces. In addition, a beautiful thing about compressed sensing is that is has deep connections with many disciplines; with signal processing of course, but also with information theory, coding theory, theoretical computer science and statistics to name just a few. A good portion of these lectures will make these connections clear. Compressed sensing also offers a new vantage point for a diverse set of applications including accelerated tomographic imaging, analog-to-digital conversion, and digital photography. Interestingly, there are already many ongoing efforts to build a new generation of sensing devices based on compressed sensing and the lecturers will address remarkable recent progress in this area as well. Finally, while we will survey foundational results in compressive sampling, it is good to keep in mind that this is after all a very young field, which has a flurry of open problems.