Methods for sparse signal recovery using Kalman filtering with embedded pseudo-measurement norms and quasi-norms

  • Authors:

  • Avishy Carmi;Pini Gurfil;Dimitri Kanevsky

  • Affiliations:

  • Department of Engineering, University of Cambridge, U.K.;Faculty of Aerospace Engineering, Technion-Israel Institute of Technology, Haifa, Israel;IBM T. J. Watson Research Center, Yorktown, NY

  • Venue:
  • IEEE Transactions on Signal Processing
  • Year:
  • 2010

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Abstract

We present two simple methods for recovering sparse signals from a series of noisy observations. The theory of compressed sensing (CS) requires solving a convex constrained minimization problem. We propose solving this optimization problem by two algorithms that rely on a Kalman filter (KF) endowed with a pseudo-measurement (PM) equation. Compared to a recently-introduced KF-CS method, which involves the implementation of an auxiliary CS optimization algorithm (e.g., the Dantzig selector), our method can be straightforwardly implemented in a stand-alone manner, as it is exclusively based on the well-known KF formulation. In our first algorithm, the PM equation constrains the l1 norm of the estimated state. In this case, the augmented measurement equation becomes linear, so a regular KF can be used. In our second algorithm, we replace the norm by a quasi-norm lp, 0 ≤ p