MultiDimensional Sparse Super-Resolution

This paper studies sparse super-resolution in arbitrary dimensions. More precisely, it develops a theoretical analysis of support recovery for the so-called BLASSO method, which is an off-the-grid generalisation of l1 regularization (also known as the LASSO). While super-resolution is of paramount importance in overcoming the limitations of many imaging devices, its theoretical analysis is still lacking beyond the 1-dimensional (1-D) case. The reason is that in the 2-dimensional (2-D) case and beyond, the relative position of the spikes enters the picture, and different geometrical configurations lead to different stability properties. Our first main contribution is a connection, in the limit where the spikes cluster at a given point, between solutions of the dual of the BLASSO problem and Hermite polynomial interpolation ideals. Polynomial bases for these ideals, introduced by De Boor, can be computed by Gaussian elimination, and lead to an algorithmic description of limiting solutions to the dual problem. With this construction at hand, our second main contribution is a detailed analysis of the support stability and super-resolution effect in the case of a pair of spikes. This includes in particular a sharp analysis of how the signal-to-noise ratio should scale with respect to the separation distance between the spikes. Lastly, numerical simulations on different classes of kernels show the applicability of this theory and highlight the richness of super-resolution in 2-D.

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

[2]  C. D. Boor,et al.  The least solution for the polynomial interpolation problem , 1992 .

[3]  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..

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

[5]  Michael J Rust,et al.  Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM) , 2006, Nature Methods.

[6]  J. Lippincott-Schwartz,et al.  Imaging Intracellular Fluorescent Proteins at Nanometer Resolution , 2006, Science.

[7]  Laurent Jacques,et al.  A Geometrical Study of Matching Pursuit Parametrization , 2008, IEEE Transactions on Signal Processing.

[8]  Ulrich von der Ohe,et al.  A multivariate generalization of Prony's method , 2015, 1506.00450.

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

[10]  Richard M. Leahy,et al.  Electromagnetic brain mapping , 2001, IEEE Signal Process. Mag..

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

[12]  Peyman Milanfar,et al.  Statistical and Information-Theoretic Analysis of Resolution in Imaging , 2006, IEEE Transactions on Information Theory.

[13]  Rémi Gribonval,et al.  Compressive Statistical Learning with Random Feature Moments , 2017, Mathematical Statistics and Learning.

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

[15]  Didier Henrion,et al.  Exact Solutions to Super Resolution on Semi-Algebraic Domains in Higher Dimensions , 2015, IEEE Transactions on Information Theory.

[16]  M. Viberg,et al.  Two decades of array signal processing research: the parametric approach , 1996, IEEE Signal Process. Mag..

[17]  A. Ouamri,et al.  Performance of high resolution frequencies estimation methods compared to the Cramer-Rao bounds , 1989, IEEE Trans. Acoust. Speech Signal Process..

[18]  C. D. Boor,et al.  Computational aspects of polynomial interpolation in several variables , 1992 .

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

[20]  Gongguo Tang.  Atomic Decomposition of Mixtures of Translation-Invariant Signals , 2013 .

[21]  R. Tibshirani.  Regression Shrinkage and Selection via the Lasso , 1996 .

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

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

[24]  Louis L. Scharf,et al.  Two-dimensional modal analysis based on maximum likelihood , 1994, IEEE Trans. Signal Process..

[25]  Ankur Moitra,et al.  Super-resolution, Extremal Functions and the Condition Number of Vandermonde Matrices , 2014, STOC.

[26]  Petre Stoica,et al.  Spectral Analysis of Signals , 2009 .

[27]  Gabriel Peyré,et al.  Sparse Spikes Super-resolution on Thin Grids I: the LASSO , 2016 .

[28]  Benjamin Recht,et al.  The alternating descent conditional gradient method for sparse inverse problems , 2015, 2015 IEEE 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP).

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

[30]  Tomas Sauer.  Prony’s method in several variables , 2017, Numerische Mathematik.

[31]  Nikos D. Sidiropoulos,et al.  Almost-sure identifiability of multidimensional harmonic retrieval , 2001, IEEE Trans. Signal Process..

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

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

[34]  M. Vetterli,et al.  Sparse Sampling of Signal Innovations: Theory, Algorithms and Performance Bounds , 2007 .

[35]  R. O. Schmidt,et al.  Multiple emitter location and signal Parameter estimation , 1986 .

[36]  Randolph L. Moses,et al.  Two-dimensional Prony modeling and parameter estimation , 1993, IEEE Trans. Signal Process..

[37]  J. Claerbout,et al.  Robust Modeling With Erratic Data , 1973 .

[38]  J. Frédéric Bonnans,et al.  Perturbation Analysis of Optimization Problems , 2000, Springer Series in Operations Research.

[39]  Benjamin Recht,et al.  Atomic norm denoising with applications to line spectral estimation , 2011, Allerton.

[40]  I. Johnstone,et al.  Maximum Entropy and the Nearly Black Object , 1992 .

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

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

[43]  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..

[44]  Wenjing Liao,et al.  MUSIC for Single-Snapshot Spectral Estimation: Stability and Super-resolution , 2014, ArXiv.

[45]  Emmanuel J. Candès,et al.  Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information , 2004, IEEE Transactions on Information Theory.

[46]  Michael A. Saunders,et al.  Atomic Decomposition by Basis Pursuit , 1998, SIAM J. Sci. Comput..

[47]  Marcus Carlsson,et al.  ESPRIT for Multidimensional General Grids , 2018, SIAM J. Matrix Anal. Appl..

[48]  Martin Jaggi,et al.  Revisiting Frank-Wolfe: Projection-Free Sparse Convex Optimization , 2013, ICML.

[49]  F. Gamboa,et al.  Spike detection from inaccurate samplings , 2013, 1301.5873.

[50]  F. Santosa,et al.  Linear inversion of ban limit reflection seismograms , 1986 .

[51]  David L Donoho,et al.  Compressed sensing , 2006, IEEE Transactions on Information Theory.

[52]  Jens Haueisen,et al.  Time-frequency mixed-norm estimates: Sparse M/EEG imaging with non-stationary source activations , 2013, NeuroImage.

[53]  Thomas Kailath,et al.  ESPRIT-estimation of signal parameters via rotational invariance techniques , 1989, IEEE Trans. Acoust. Speech Signal Process..

[54]  R. A. Lorentz,et al.  Multivariate Hermite interpolation by algebraic polynomials: a survey , 2000 .

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

[56]  S. Levy,et al.  Reconstruction of a sparse spike train from a portion of its spectrum and application to high-resolution deconvolution , 1981 .