Construction of a Large Class of Deterministic Sensing Matrices That Satisfy a Statistical Isometry Property

Compressed Sensing aims to capture attributes of k-sparse signals using very few measurements. In the standard compressed sensing paradigm, the N × C measurement matrix ¿ is required to act as a near isometry on the set of all k-sparse signals (restricted isometry property or RIP). Although it is kn...

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Bibliographic Details
Published in:IEEE journal of selected topics in signal processing 2010-04, Vol.4 (2), p.358-374
Main Authors: Calderbank, R., Howard, S., Jafarpour, S.
Format: Article
Language:English
Subjects:
Algorithm design and analysis
Algorithms
Chirp
Compressed
Compressed sensing
Construction
Criteria
Delsarte-Goethals codes
Detection
deterministic compressed sensing
finite groups
martingale sequences
Matching pursuit algorithms
Mathematical analysis
Matrices
Matrix methods
McDiarmid inequality
Measurement standards
Particle measurements
Performance analysis
Pursuit algorithms
Signal analysis
Signal processing
Signal processing algorithms
statistical near isometry
Studies
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Summary:Compressed Sensing aims to capture attributes of k-sparse signals using very few measurements. In the standard compressed sensing paradigm, the N × C measurement matrix ¿ is required to act as a near isometry on the set of all k-sparse signals (restricted isometry property or RIP). Although it is known that certain probabilistic processes generate N × C matrices that satisfy RIP with high probability, there is no practical algorithm for verifying whether a given sensing matrix ¿ has this property, crucial for the feasibility of the standard recovery algorithms. In contrast, this paper provides simple criteria that guarantee that a deterministic sensing matrix satisfying these criteria acts as a near isometry on an overwhelming majority of k-sparse signals; in particular, most such signals have a unique representation in the measurement domain. Probability still plays a critical role, but it enters the signal model rather than the construction of the sensing matrix. An essential element in our construction is that we require the columns of the sensing matrix to form a group under pointwise multiplication. The construction allows recovery methods for which the expected performance is sub-linear in C, and only quadratic in N, as compared to the super-linear complexity in C of the Basis Pursuit or Matching Pursuit algorithms; the focus on expected performance is more typical of mainstream signal processing than the worst case analysis that prevails in standard compressed sensing. Our framework encompasses many families of deterministic sensing matrices, including those formed from discrete chirps, Delsarte-Goethals codes, and extended BCH codes.
ISSN:1932-4553
1941-0484
DOI:10.1109/JSTSP.2010.2043161