A Compressive Introduction to Compressive Sensing

In recent years, we have come under mounting pressure to accommodate massive amounts of increasingly high-dimensional data. For example, witness the explosion in the quantity of high-resolution audio, imagery, video, and other sensed data produced by relatively inexpensive mobile devices. Similar scenarios arise in such diverse application areas as medical and scientific imaging, genomic data analysis, and digital communications. Despite extraordinary advances in computational power, such high-dimensional data continues to pose a number of challenges. Fortunately, in many cases these high-dimensional signals contain relatively little information compared to their ambient dimensionality. For example, signals can often be well-approximated as sparse in a known basis or using a low-dimensional manifold or parametric model. Similarly, ensembles of signals can often be approximated via low-rank matrices. Exploiting this structure is critical to designing efficient algorithms for processing such high-dimensional data.  

This talk will provide an overview of compressive sensing -- an emerging field that provides a flexible framework for designing hardware and algorithms that can tackle previously intractable problems involving the acquisition and processing of high-dimensional signals. As an important example, we will see that a sparse signal can be recovered from just a few linear, nonadaptive measurements as long as the number of measurements is proportional to the number of degrees of freedom in the signal, which can be much lower than its dimensionality. This result has applications in the design of novel cameras and imaging systems, tomography, biological testing, radio receivers, communication systems, and networks. This talk will provide an overview of the theory of compressive sensing and discuss a few example applications, setting the stage for the following talks which will highlight both practical applications in science and engineering as well as connections with deep mathematical theory that underlies the field.