code  /  papers  /  links

One of the central tenets of signal processing is the Shannon/Nyquist sampling theory: the number of samples needed to capture a signal is dictated by its bandwidth. Very recently, an alternative theory of "compressive sampling"has emerged. By using nonlinear recovery algorithms (based on convex optimization), super-resolved signals and images can be reconstructed from what appears to be highly incomplete data. Compressive sampling shows us how data compression can be implicitly incorporated into the data acquisition process, a gives us a new vantage point for a diverse set of applications including accelerated tomographic imaging, analog-to-digital conversion, and digital photography.

See examples of compressive sampling in action.


L1-MAGIC is a collection of MATLAB routines for solving the convex optimization programs central to compressive sampling. The algorithms are based on standard interior-point methods, and are suitable for large-scale problems.

Download the code (including User's Guide)
Download the User's Guide (pdf)



A nonlinear sampling theorem

"Robust uncertainty principles: Exact recovery from highly incomplete Fourier information"
by: Emmanuel Candes, Justin Romberg, and Terence Tao
To appear in IEEE Transactions on Information Theory, February 2006.

The central result from this paper is that a sparse vector can be recovered exactly from a small number of
Fourier domain observations. More precisely, let f be a length-N discrete signal which has B nonzero
components (we stress that the number and locations of the components are unknown a priori). We collect
samples at K different frequencies which are randomly selected. Then for K on the order of B log N,
we can recover f perfectly (with very high probability) through l1 minimization.

Near-optimal signal recovery and the Uniform Uncertainty Principle

"Near-optimal signal recovery from random projections and universal encoding strategies"
by: Emmanuel Candes and Terence Tao
Submitted to IEEE Transactions on Information Theory, November 2004.

This paper derives precise conditions for when an arbitrary sparse signal f can be recovered from a
fixed set of linear measurements y = Mf. If M obeys what is terms the Uniform Uncertainty Principle
for sets of size S (which essentially means that all submatrices formed by taking S columns from M are
approximate isometries), then every signal f with no more than S nonzero components can be recovered
from its measurements y=Mf via an l1 minimization program. It is shown that if M is generated randomly,
it will obey the UUP with high probability for sets of size S ~ K log (N/K), where K is the number of rows
of M. Using this result, it is shown that if f is compressible rather than sparse (meaning that the sorted
components of f decay quickly), then the l1 recovery is near-optimal: the recovery error goes to zero as
we add more measurements almost as fast as the nonlinear approximation error of the original signal.


"Stable signal recovery from incomplete and inaccurate measurements"
by: Emmanuel Candes, Justin Romberg, and Terence Tao
To appear in Communications on Pure and Applied Mathematics, 2006.

This paper demonstrates that the recovery procedure is stable. Given that the measurement matrix
satisfies the UUP, we show that we can recover a sparse or compressible signal f from corrupted measurements
y = Mf+e, where the size of e is less than epsilon, to error within epsilon. The proof is short and clean, and
encompasses the previous recovery results for the noiseless case.

Statistical Estimation

"The Dantzig selector: Statistical estimation when p is much smaller than n"
by: Emmanuel Candes and Terence Tao
Submitted to IEEE Transactions on Information Theory, June 2005.

When the errors made in the measurement process are Gaussian, much more can be said about the precision
of the recovery. This paper shows that a sparse signal can be estimated from an incomplete set of measurements
corrupted by additive white Gaussian noise just as well as from observing the entire noisy signal by itself
(and thresholding). The estimation process, which is again a certain type of l1 minimization program, is termed
the Dantzig Selector.

Linear Decoding

"Decoding by Linear Programming"
by: Emmanuel Candes and Terence Tao
IEEE Transactions on Information Theory, December 2005.

This paper demonstrates that in addition to recovering sparse signals, l1 minimization can be used to detect
and correct sparse errors. A codeword c is generated by applying an MxN coding matrix A to a message m:
c = Am. It is shown that if A obeys a type of uncertainty principle, then c can be recovered even if m is
deviously altered in qM unknown locations (where q is a constant).

Finding Sparse Decompositions

"Quantitative robust uncertainty principles and optimally sparse decompositions"
by: Emmanuel Candes and Justin Romberg
To appear in Foundations of Computational Mathematics, 2006.

This paper revisits the now classical application of l1 minimization to finding sparse representations in unions
of bases. The spike-sinusoid system is studied in detail: it is shown that if a signal is comprised of a superposition
of ~ N/sqrt(log N) spikes and sinusoids, then the sparsest (in the sense of minimum support) decomposition can be
found via l1 minimization. This paper makes explicit use of novel uncertainty principles between the time and
frequency domains. The extension to finding sparse decompositions in general pairs of bases is also discussed.



David Donoho's publications, including work on Compressed Sensing and Sparse Recovery (with Jared Tanner)

The Rice University DSP group's Compressed Sensing resources page; see in particular the very recent work on building a CS camera

Robert Nowak and Jarvis Haupt's paper on Signal Reconstruction from Noisy Random Projections.

Terence Tao's summary of the current state of compressive sampling theory

Joel Tropp's web page at the California Institute of Technology, see in particular his work on reconstruction using greedy algorithms (with Anna Gilbert).

Martin Strauss and Anna Gilbert, at the University of Michigan, and their papers on fast algorithms for estimating sparse Fourier transforms

Martin Vetterli and Irena Maravic's work on sampling signals with "finite rate of innovation"

David Brady's Duke Integrated Sensing and Processing page