Uncertainty principles and signal recovery
SIAM Journal on Applied Mathematics
Fractal functions and wavelet expansions based on several scaling functions
Journal of Approximation Theory
Atomic Decomposition by Basis Pursuit
SIAM Journal on Scientific Computing
Quantitative Robust Uncertainty Principles and Optimally Sparse Decompositions
Foundations of Computational Mathematics
Blind multiband signal reconstruction: compressed sensing for analog signals
IEEE Transactions on Signal Processing
Compressed sensing of analog signals in shift-invariant spaces
IEEE Transactions on Signal Processing
Robust recovery of signals from a structured union of subspaces
IEEE Transactions on Information Theory
Average case analysis of multichannel sparse recovery using convex relaxation
IEEE Transactions on Information Theory
Theoretical Results on Sparse Representations of Multiple-Measurement Vectors
IEEE Transactions on Signal Processing
Nonuniform Sampling of Periodic Bandlimited Signals
IEEE Transactions on Signal Processing - Part I
Reduce and Boost: Recovering Arbitrary Sets of Jointly Sparse Vectors
IEEE Transactions on Signal Processing - Part I
Sparse solutions to linear inverse problems with multiple measurement vectors
IEEE Transactions on Signal Processing
Uncertainty principles and ideal atomic decomposition
IEEE Transactions on Information Theory
A generalized uncertainty principle and sparse representation in pairs of bases
IEEE Transactions on Information Theory
On sparse representation in pairs of bases
IEEE Transactions on Information Theory
Sparse representations in unions of bases
IEEE Transactions on Information Theory
Greed is good: algorithmic results for sparse approximation
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Blind multiband signal reconstruction: compressed sensing for analog signals
IEEE Transactions on Signal Processing
Block sparsity and sampling over a union of subspaces
DSP'09 Proceedings of the 16th international conference on Digital Signal Processing
Robust recovery of signals from a structured union of subspaces
IEEE Transactions on Information Theory
Recovering signals from lowpass data
IEEE Transactions on Signal Processing
Time-delay estimation from low-rate samples: a union of subspaces approach
IEEE Transactions on Signal Processing
Block-sparse signals: uncertainty relations and efficient recovery
IEEE Transactions on Signal Processing
Coherence-based performance guarantees for estimating a sparse vector under random noise
IEEE Transactions on Signal Processing
Hi-index | 754.91 |
The past several years have witnessed a surge of research investigating various aspects of sparse representations and compressed sensing. Most of this work has focused on the finite-dimensional setting in which the goal is to decompose a finite-length vector into a given finite dictionary. Underlying many of these results is the conceptual notion of an uncertainty principle: a signal cannot be sparsely represented in two different bases. Here, we extend these ideas and results to the analog, infinite-dimensional setting by considering signals that lie in a finitely generated shift-invariant (SI) space. This class of signals is rich enough to include many interesting special cases such as multiband signals and splines. By adapting the notion of coherence defined for finite dictionaries to infinite SI representations, we develop an uncertainty principle similar in spirit to its finite counterpart. We demonstrate tightness of our bound by considering a bandlimited lowpass train that achieves the uncertainty principle. Building upon these results and similar work in the finite setting, we show how to find a sparse decomposition in an overcomplete dictionary by solving a convex optimization problem. The distinguishing feature of our approach is the fact that even though the problem is defined over an infinite domain with infinitely many variables and constraints, under certain conditions on the dictionary spectrum our algorithm can find the sparsest representation by solving a finite-dimensional problem.