This first chapter formulates the objectives of compressive sensing. It introduces the
standard compressive problem studied throughout the book and reveals its ubiquity in many …

This paper deals with the computational complexity of conditions which guarantee that the
NP-hard problem of finding the sparsest solution to an underdetermined linear system can …

A variety of practical methods have recently been introduced for finding maximally sparse
representations from overcomplete dictionaries, a central computational task in compressive …

This paper considers compressed sensing and affine rank minimization in both noiseless
and noisy cases and establishes sharp restricted isometry conditions for sparse signal and …

This paper establishes a sharp condition on the restricted isometry property (RIP) for both
the sparse signal recovery and low-rank matrix recovery. It is shown that if the measurement …

Compressed sensing (CS) has the potential to reduce magnetic resonance (MR) data
acquisition time. In order for CS-based imaging schemes to be effective, the signal of interest …

Many practical methods for finding maximally sparse coefficient expansions involve solving
a regression problem using a particular class of concave penalty functions. From a Bayesian …

The ℓ0 minimization of compressed sensing is often relaxed to ℓ1, which yields easy
computation using the shrinkage map** known as soft thresholding, and can be shown to …

Convex optimization problems arising in applications, possibly as approximations of
intractable problems, are often structured and large scale. When the data are noisy, it is of …

This paper discusses new bounds for restricted isometry constants in compressed sensing.
Let Φ be an n× p real matrix and A; be a positive integer with k≤ n. One of the main results …