Compressive sampling. (English) Zbl 1130.94013

Candès, Emmanuel J.

Sanz-Solé, Marta (ed.) et al., Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22–30, 2006. Volume III: Invited lectures. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-022-7/hbk). 1433-1452 (2006).

Summary: Conventional wisdom and common practice in acquisition and reconstruction of images from frequency data follow the basic principle of the Nyquist density sampling theory. This principle states that to reconstruct an image, the number of Fourier samples we need to acquire must match the desired resolution of the image, i.e., the number of pixels in the image. This paper surveys an emerging theory which goes by the name of “compressive sampling” or “compressed sensing”, and which says that this conventional wisdom is inaccurate. Perhaps surprisingly, it is possible to reconstruct images or signals of scientific interest accurately and sometimes even exactly from a number of samples which is far smaller than the desired resolution of the image/signal, e.g. the number of pixels in the image.
For the entire collection see [Zbl 1095.00006].

MSC:

94A20	Sampling theory in information and communication theory
62D05	Sampling theory, sample surveys
62H35	Image analysis in multivariate analysis
68P30	Coding and information theory (compaction, compression, models of communication, encoding schemes, etc.) (aspects in computer science)
94A08	Image processing (compression, reconstruction, etc.) in information and communication theory
68U10	Computing methodologies for image processing

Keywords:

Compressive sampling, sparsity, uniform uncertainty principle, underdetermined systems of linear equations, \(\ell_1\)-minimization, linear programming, signal recovery, error correction