A Universal Technique for Analysing Discrete Super-Resolution Algorithms

This leter develops a universal technique for analyzing discrete super-resolution algorithms with $\ell _1$-norm based objective function. Though the super-resolution problem with sparsity constraints is of intense research interest in the past decade, only a modified Dantzig selector has been non-asymptotically analyzed without additional structural information whereas this theoretical guarantee does not match the numerical results in the Gaussian noise case. More importantly, the relation between the analyses of discrete super-resolution problem and other underdetermined inverse problems in compressed sensing is still not clear. Using the proposed universal technique, this letter aims to close this gap in understanding the characteristics of discrete super-resolution problem. The theoretical claims are demonstrated by extensive numerical experiments.

[1]  Gongguo Tang,et al.  Near minimax line spectral estimation , 2013, 2013 47th Annual Conference on Information Sciences and Systems (CISS).

[2]  Wenjing Liao,et al.  Super-Resolution Limit of the ESPRIT Algorithm , 2019, IEEE Transactions on Information Theory.

[3]  P. P. Vaidyanathan,et al.  Pushing the Limits of Sparse Support Recovery Using Correlation Information , 2015, IEEE Transactions on Signal Processing.

[4]  Piya Pal,et al.  Super-Resolution with Noisy Measurements: Reconciling Upper and Lower Bounds , 2020, ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[5]  Weilin Li,et al.  Stable super-resolution limit and smallest singular value of restricted Fourier matrices , 2017, Applied and Computational Harmonic Analysis.

[6]  Terence Tao,et al.  The Dantzig selector: Statistical estimation when P is much larger than n , 2005, math/0506081.

[7]  Laurent Demanet,et al.  The recoverability limit for superresolution via sparsity , 2015, ArXiv.

[8]  Mark Bates,et al.  Three-Dimensional Super-Resolution Imaging by Stochastic Optical Reconstruction Microscopy , 2008, Science.

[9]  S. Geer,et al.  On the conditions used to prove oracle results for the Lasso , 2009, 0910.0722.

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

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

[12]  M. Lustig,et al.  Compressed Sensing MRI , 2008, IEEE Signal Processing Magazine.

[13]  Reinhard Heckel,et al.  Super-Resolution Radar , 2014, ArXiv.

[14]  Tamir Bendory,et al.  Robust Recovery of Positive Stream of Pulses , 2015, IEEE Transactions on Signal Processing.

[15]  Lihua Xie,et al.  On Gridless Sparse Methods for Line Spectral Estimation From Complete and Incomplete Data , 2014, IEEE Transactions on Signal Processing.

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

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

[18]  Piya Pal,et al.  Guaranteed Localization of More Sources Than Sensors With Finite Snapshots in Multiple Measurement Vector Models Using Difference Co-Arrays , 2019, IEEE Transactions on Signal Processing.

[19]  Sara van de Geer,et al.  Ecole d'été de probabilités de Saint-Flour XLV , 2016 .

[20]  Cishen Zhang,et al.  A Discretization-Free Sparse and Parametric Approach for Linear Array Signal Processing , 2013, IEEE Transactions on Signal Processing.

[21]  Jian Li,et al.  SPICE: A Sparse Covariance-Based Estimation Method for Array Processing , 2011, IEEE Transactions on Signal Processing.

[22]  P. Bickel,et al.  SIMULTANEOUS ANALYSIS OF LASSO AND DANTZIG SELECTOR , 2008, 0801.1095.

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

[24]  C. Giraud.  Introduction to High-Dimensional Statistics , 2014 .

[25]  Benjamin Recht,et al.  Superresolution without separation , 2015, 2015 IEEE 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP).

[26]  S. Geer,et al.  Oracle Inequalities and Optimal Inference under Group Sparsity , 2010, 1007.1771.

[27]  Florentina Bunea,et al.  The Group Square-Root Lasso: Theoretical Properties and Fast Algorithms , 2013, IEEE Transactions on Information Theory.

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

[29]  Moon Gi Kang,et al.  Super-resolution image reconstruction: a technical overview , 2003, IEEE Signal Process. Mag..

[30]  Trevor Hastie,et al.  Statistical Learning with Sparsity: The Lasso and Generalizations , 2015 .

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

[32]  Piya Pal,et al.  Unified analysis of co-array interpolation for direction-of-arrival estimation , 2017, 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[33]  Dmitry M. Malioutov,et al.  A sparse signal reconstruction perspective for source localization with sensor arrays , 2005, IEEE Transactions on Signal Processing.

[34]  Richard G. Baraniuk,et al.  Asymptotic Analysis of Complex LASSO via Complex Approximate Message Passing (CAMP) , 2011, IEEE Transactions on Information Theory.