Joint Low-rank and Sparse Inversion for Multidimensional Simultaneous Random/Erratic Noise Attenuation and Interpolation

Several methods have been proposed to improve the signal-to-noise ratio by attenuating incoherent noise, including prediction error filtering (Canales 1984), projection filtering (Soubaras 1995), and more recently rank reduction filtering. In this last category, we can differentiate eigenimage filtering (Trickett 2003), Cadzow / Singular Spectrum Analysis (SSA) filtering (Trickett 2009, Sacchi 2009) and tensor methods (Kreimer and Sacchi 2012, Trickett 2013, Da Silva and Herrmann 2014). Also, these latter methods have been extended to robust noise attenuation to deal with erratic noise, or data interpolation for binned data within a defined grid (Trickett et al. 2010, 2012, Oropeza and Sacchi 2011, Chen and Sacchi 2013). Here, we propose a systematic formulation of the simultaneous random plus erratic noise attenuation and data interpolation problem as a convex optimization program, which can be solved efficiently. We model the coherent signal via its low-rank trajectory matrix in the spirit of Cadzow/SSA filtering, and the erratic noise as a sparse component of the input data. The signal component is recovered by solving a Joint Low-Rank and Sparse Inversion (JLRSI) thanks to a joint minimization of the nuclear and L1 norms of the low-rank and sparse components respectively.