A novel algorithm for the identification of dirac impulses from filtered noisy measurements

Abstract In this paper we address the recovery of a finite stream of Dirac pulses from noisy lowpass-filtered samples in the discrete-time setting. While this problem has been successfully addressed for the noise-free case using the concept of signals with finite rate of innovation, such techniques are not efficient in the presence of noise. In the FRI framework, the determination of the location of Dirac pulses is based on the singular value decomposition of a matrix whose rank in the noise-free case equals the number of Dirac pulses and the signal can be related to the non zero singular values. However, in noisy situations this matrix becomes full rank and the singular value decomposition is subject to subspace swap, meaning some singular values associated with noise become larger than some values related to the signal. This phenomenon has been recognized as the reason for performance breakdown in the method. The goal of this paper is to propose a novel algorithm that limits the alteration of these singular values in the presence of noise, thus significantly improving the estimation of Dirac pulses.

[1]  D. Donoho.  Superresolution via sparsity constraints , 1992 .

[2]  Arthur Jay Barabell,et al.  Improving the resolution performance of eigenstructure-based direction-finding algorithms , 1983, ICASSP.

[3]  Etienne Barnard,et al.  Two-dimensional superresolution radar imaging using the MUSIC algorithm , 1994 .

[4]  Pier Luigi Dragotti,et al.  Guaranteed Performance in the FRI Setting , 2015, IEEE Signal Processing Letters.

[5]  Mohamed-Jalal Fadili,et al.  Sharp Support Recovery from Noisy Random Measurements by L1 minimization , 2011, ArXiv.

[6]  M. Genton,et al.  A skewed Kalman filter , 2005 .

[7]  James S. Duncan,et al.  Using skew Gabor filter in source signal separation and local spectral orientation analysis , 2005, Image Vis. Comput..

[8]  Emmanuel J. Candès,et al.  Super-Resolution of Positive Sources: The Discrete Setup , 2015, SIAM J. Imaging Sci..

[9]  James A. Cadzow,et al.  Signal enhancement-a composite property mapping algorithm , 1988, IEEE Trans. Acoust. Speech Signal Process..

[10]  Emmanuel J. Candès,et al.  Towards a Mathematical Theory of Super‐resolution , 2012, ArXiv.

[11]  Thierry Blu,et al.  Extrapolation and Interpolation) , 2022 .

[12]  M. Vetterli,et al.  Sparse Sampling of Signal Innovations , 2008, IEEE Signal Processing Magazine.

[13]  Tapan K. Sarkar,et al.  Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise , 1990, IEEE Trans. Acoust. Speech Signal Process..

[14]  Hayit Greenspan,et al.  Super-Resolution in Medical Imaging , 2009, Comput. J..

[15]  Yohann de Castro,et al.  Exact Reconstruction using Beurling Minimal Extrapolation , 2011, 1103.4951.

[16]  A. Azzalini.  A class of distributions which includes the normal ones , 1985 .

[17]  K. Puschmann,et al.  On super-resolution in astronomical imaging , 2005 .

[18]  Thierry Blu,et al.  FRI Sampling With Arbitrary Kernels , 2013, IEEE Transactions on Signal Processing.

[19]  Joel A. Tropp,et al.  Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit , 2007, IEEE Transactions on Information Theory.

[20]  Emmanuel J. Candès,et al.  Super-Resolution from Noisy Data , 2012, Journal of Fourier Analysis and Applications.

[21]  Gabriel Peyré,et al.  Support Recovery for Sparse Super-Resolution of Positive Measures , 2017 .

[22]  Mike E. Davies,et al.  Fast Non-Negative Orthogonal Matching Pursuit , 2015, IEEE Signal Processing Letters.

[23]  Thierry Blu,et al.  Sampling signals with finite rate of innovation , 2002, IEEE Trans. Signal Process..

[24]  Laurent Condat,et al.  Cadzow Denoising Upgraded: A New Projection Method for the Recovery of Dirac Pulses from Noisy Linear Measurements , 2015 .

[25]  Marco F. Duarte,et al.  Spectral compressive sensing , 2013 .

[26]  Gabriel Peyré,et al.  Exact Support Recovery for Sparse Spikes Deconvolution , 2013, Foundations of Computational Mathematics.

[27]  Laurent Demanet,et al.  Super-resolution via superset selection and pruning , 2013, ArXiv.

[28]  J. Schmee.  Applied Statistics—A Handbook of Techniques , 1984 .

[29]  Martin Vetterli,et al.  Sampling and reconstruction of signals with finite rate of innovation in the presence of noise , 2005, IEEE Transactions on Signal Processing.

[30]  Jari Lindberg,et al.  Mathematical concepts of optical superresolution , 2012 .

[31]  K. Bredies,et al.  Inverse problems in spaces of measures , 2013 .

[32]  R. E. Carlson,et al.  Monotone Piecewise Cubic Interpolation , 1980 .

[33]  Laurent Jacques,et al.  Sparse Support Recovery with Non-smooth Loss Functions , 2016, NIPS.

[34]  Stéphane Mallat,et al.  Matching pursuits with time-frequency dictionaries , 1993, IEEE Trans. Signal Process..

[35]  C W McCutchen,et al.  Superresolution in microscopy and the Abbe resolution limit. , 1967, Journal of the Optical Society of America.

[36]  Vivek K. Goyal,et al.  Estimating Signals With Finite Rate of Innovation From Noisy Samples: A Stochastic Algorithm , 2007, IEEE Transactions on Signal Processing.

[37]  C. Dossal.  A necessary and sufficient condition for exact recovery by l1 minimization. , 2012 .

[38]  G. Peyré,et al.  Sparse regularization on thin grids I: the Lasso , 2017 .

[39]  Andrew R. Teel,et al.  ESAIM: Control, Optimisation and Calculus of Variations , 2022 .

[40]  Jean-Jacques Fuchs,et al.  Sparsity and uniqueness for some specific under-determined linear systems , 2005, Proceedings. (ICASSP '05). IEEE International Conference on Acoustics, Speech, and Signal Processing, 2005..