| Compressive Sensing |
Compressive Sensing Richard Baraniuk, Rice Justin Romberg, Georgia Tech Michael Wakin, University of Michigan From reconstruction to compressive signal processing Applications of compressive sensing From decades of research in signal processing, we have learned that having a good signal representation is key for tasks such as compression, denoising, and restoration. The new theory of Compressive Sensing (CS) shows us how a good representation can fundamentally aid us in the acquisition (or sampling) process as well. The CS paradigm can be summarized neatly: the number of measurements (e.g., samples) needed to acquire a signal or image depends more on its inherent information content than on the desired resolution (e.g., number of pixels). The CS theory typically requires a novel measurement scheme that generalizes the conventional signal acquisition process: instead of making direct observations of the signal, for example, an acquisition device encodes it as a series of random linear projections. The theory of CS, while still in its developing stages, is far-reaching and draws on subjects as varied as sampling theory, convex optimization, source and channel coding, statistical estimation, uncertainty principles, and harmonic analysis. The applications of CS range from the familiar (imaging in medicine and radar, high-speed analog-to-digital conversion, and super-resolution) to truly novel image acquisition and encoding techniques. This short course will outline the theoretical foundations of CS, point out the important role played by the geometry of high-dimensional vector spaces, and discuss how the ideas can be applied in next-generation acquisition devices. Particular topics include sparse signal representations, convex optimization, random projections, Uniform Uncertainty Principles, n-widths, the Johnson-Lindenstrauss lemma, and Whitney's embedding theorem for manifolds. |