Blind inverse problems with isolated spikes

Assume that an unknown integral operator living in some known subspace is observed indirectly, by evaluating its action on a discrete measure containing a few isolated Dirac masses at an unknown location. Is this information enough to recover the impulse response location and the operator with a sub-pixel accuracy? We study this question and bring to light key geometrical quantities for exact and stable recovery. We also propose an in depth study of the presence of additive white Gaussian noise. We illustrate the well-foundedness of this theory on the challenging optical imaging problem of blind deconvolution and blind deblurring with unstationary operators.

[1]  Gábor Lugosi,et al.  Concentration Inequalities - A Nonasymptotic Theory of Independence , 2013, Concentration Inequalities.

[2]  Marc Teboulle,et al.  Proximal alternating linearized minimization for nonconvex and nonsmooth problems , 2013, Mathematical Programming.

[3]  O. Papaspiliopoulos.  High-Dimensional Probability: An Introduction with Applications in Data Science , 2020 .

[4]  G. Stewart.  On the Continuity of the Generalized Inverse , 1969 .

[5]  Levent Tunçel,et al.  Optimization algorithms on matrix manifolds , 2009, Math. Comput..

[6]  Pierre Weiss,et al.  A scalable estimator of sets of integral operators , 2018, Inverse Problems.

[7]  Peter Jung,et al.  Blind Demixing and Deconvolution at Near-Optimal Rate , 2017, IEEE Transactions on Information Theory.

[8]  Yanjun Li,et al.  Multichannel Sparse Blind Deconvolution on the Sphere , 2018, ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[9]  Liming Wang,et al.  Blind Deconvolution From Multiple Sparse Inputs , 2016, IEEE Signal Processing Letters.

[10]  Justin K. Romberg,et al.  Blind Deconvolution Using Convex Programming , 2012, IEEE Transactions on Information Theory.

[11]  Zhihui Zhu,et al.  Global Optimality in Low-Rank Matrix Optimization , 2017, IEEE Transactions on Signal Processing.

[12]  Jan Flusser,et al.  Multichannel blind iterative image restoration , 2003, IEEE Trans. Image Process..

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

[14]  Pierre Weiss,et al.  Approximation of Integral Operators Using Product-Convolution Expansions , 2017, Journal of Mathematical Imaging and Vision.

[15]  Emmanuel Soubies,et al.  The sliding Frank–Wolfe algorithm and its application to super-resolution microscopy , 2018, Inverse Problems.

[16]  Antonin Chambolle,et al.  On Representer Theorems and Convex Regularization , 2018, SIAM J. Optim..

[17]  Patrick L. Combettes,et al.  Proximal Splitting Methods in Signal Processing , 2009, Fixed-Point Algorithms for Inverse Problems in Science and Engineering.

[18]  Xiaodong Li,et al.  Rapid, Robust, and Reliable Blind Deconvolution via Nonconvex Optimization , 2016, Applied and Computational Harmonic Analysis.

[19]  Suliana Manley,et al.  Quantitative evaluation of software packages for single-molecule localization microscopy , 2015, Nature Methods.

[20]  Kristian Bredies,et al.  Tensor-Free Proximal Methods for Lifted Bilinear/Quadratic Inverse Problems with Applications to Phase Retrieval , 2019, Foundations of Computational Mathematics.

[21]  Pierre Weiss,et al.  Estimation of linear operators from scattered impulse responses , 2016, Applied and Computational Harmonic Analysis.

[22]  Michael Unser,et al.  Super-resolution fight club: Assessment of 2D & 3D single-molecule localization microscopy software , 2019, Nature Methods.

[23]  W. Moerner,et al.  Optical detection and spectroscopy of single molecules in a solid. , 1989, Physical review letters.

[24]  Holger Rauhut,et al.  Suprema of Chaos Processes and the Restricted Isometry Property , 2012, ArXiv.

[25]  Yuxin Chen,et al.  Convex and Nonconvex Optimization Are Both Minimax-Optimal for Noisy Blind Deconvolution Under Random Designs , 2020, Journal of the American Statistical Association.

[27]  Laurent Demanet,et al.  Leveraging Diversity and Sparsity in Blind Deconvolution , 2016, IEEE Transactions on Information Theory.

[28]  Wei Dai,et al.  Blind Two-Dimensional Super-Resolution and Its Performance Guarantee , 2018, IEEE Transactions on Signal Processing.

[29]  S. Mallat.  A wavelet tour of signal processing , 1998 .

[30]  R. Latala.  Estimates of moments and tails of Gaussian chaoses , 2005, math/0505313.

[31]  E. Tyrtyshnikov.  A brief introduction to numerical analysis , 1997 .

[32]  Jean-François Aujol,et al.  The basins of attraction of the global minimizers of the non-convex sparse spikes estimation problem , 2018, ArXiv.

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

[34]  Yanjun Li,et al.  Identifiability and Stability in Blind Deconvolution Under Minimal Assumptions , 2015, IEEE Transactions on Information Theory.

[35]  S.C. Douglas,et al.  Multichannel blind deconvolution and equalization using the natural gradient , 1997, First IEEE Signal Processing Workshop on Signal Processing Advances in Wireless Communications.

[36]  Ralf C Flicker,et al.  Anisoplanatic deconvolution of adaptive optics images. , 2005, Journal of the Optical Society of America. A, Optics, image science, and vision.

[37]  Felix Krahmer,et al.  Riemannian thresholding methods for row-sparse and low-rank matrix recovery , 2021, ArXiv.

[38]  Otmar Scherzer,et al.  Variational Methods in Imaging , 2008, Applied mathematical sciences.

[39]  Thomas Strohmer,et al.  Self-calibration and biconvex compressive sensing , 2015, ArXiv.

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

[41]  Ke Wei,et al.  Vectorized Hankel Lift: A Convex Approach for Blind Super-Resolution of Point Sources , 2020, IEEE Transactions on Information Theory.

[42]  Felix Krahmer,et al.  Optimal Injectivity Conditions for Bilinear Inverse Problems with Applications to Identifiability of Deconvolution Problems , 2016, SIAM J. Appl. Algebra Geom..

[43]  Felix Krahmer,et al.  On the Convex Geometry of Blind Deconvolution and Matrix Completion , 2019, Communications on Pure and Applied Mathematics.

[44]  M. Talagrand.  The Generic chaining : upper and lower bounds of stochastic processes , 2005 .

[45]  Paul Escande,et al.  Learning Low-Dimensional Models of Microscopes , 2020, IEEE Transactions on Computational Imaging.

[46]  Pierre Weiss,et al.  On the linear convergence rates of exchange and continuous methods for total variation minimization , 2019, Mathematical Programming.

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

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

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

[50]  Yuejie Chi,et al.  Guaranteed Blind Sparse Spikes Deconvolution via Lifting and Convex Optimization , 2015, IEEE Journal of Selected Topics in Signal Processing.