{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T13:37:43Z","timestamp":1762522663560,"version":"build-2065373602"},"reference-count":44,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2021,12,6]],"date-time":"2021-12-06T00:00:00Z","timestamp":1638748800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"EUR DS4H","award":["ANR-17-EURE-0004"],"award-info":[{"award-number":["ANR-17-EURE-0004"]}]},{"name":"3IA","award":["ANR-19-P3IA-0002"],"award-info":[{"award-number":["ANR-19-P3IA-0002"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Imaging"],"abstract":"<jats:p>Gridless sparse spike reconstruction is a rather new research field with significant results for the super-resolution problem, where we want to retrieve fine-scale details from a noisy and filtered acquisition. To tackle this problem, we are interested in optimisation under some prior, typically the sparsity i.e., the source is composed of spikes. Following the seminal work on the generalised LASSO for measures called the Beurling-Lasso (BLASSO), we will give a review on the chief theoretical and numerical breakthrough of the off-the-grid inverse problem, as we illustrate its usefulness to the super-resolution problem in Single Molecule Localisation Microscopy (SMLM) through new reconstruction metrics and tests on synthetic and real SMLM data we performed for this review.<\/jats:p>","DOI":"10.3390\/jimaging7120266","type":"journal-article","created":{"date-parts":[[2021,12,6]],"date-time":"2021-12-06T22:18:42Z","timestamp":1638829122000},"page":"266","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Off-The-Grid Variational Sparse Spike Recovery: Methods and Algorithms"],"prefix":"10.3390","volume":"7","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2933-8171","authenticated-orcid":false,"given":"Bastien","family":"Laville","sequence":"first","affiliation":[{"name":"Universit\u00e9 C\u00f4te d\u2019Azur, CNRS, Inria, I3S, Morpheme Project, 06900 Sophia Antipolis, France"}]},{"given":"Laure","family":"Blanc-F\u00e9raud","sequence":"additional","affiliation":[{"name":"Universit\u00e9 C\u00f4te d\u2019Azur, CNRS, Inria, I3S, Morpheme Project, 06900 Sophia Antipolis, France"}]},{"given":"Gilles","family":"Aubert","sequence":"additional","affiliation":[{"name":"Universit\u00e9 C\u00f4te d\u2019Azur, CNRS, Inria, I3S, Morpheme Project, 06900 Sophia Antipolis, France"},{"name":"Universit\u00e9 C\u00f4te d\u2019Azur, CNRS, LJAD, 06000 Nice, France"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1109\/TIT.2005.862083","article-title":"Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information","volume":"52","author":"Candes","year":"2006","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_2","first-page":"1779","article-title":"SuperMix: Sparse Regularization for Mixtures","volume":"49","author":"Gadat","year":"2021","journal-title":"Ann. Stat."},{"key":"ref_3","unstructured":"Chizat, L., and Bach, F. (2018). On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport. arXiv."},{"key":"ref_4","unstructured":"Denoyelle, Q. (2018). Theoretical and Numerical Analysis of Super-Resolution without Grid. [Ph.D. Thesis, Universit\u00e9 Paris Sciences et Lettres]."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1111\/j.2517-6161.1996.tb02080.x","article-title":"Regression Shrinkage and Selection Via the Lasso","volume":"58","author":"Tibshirani","year":"1996","journal-title":"J. R. Stat. Soc. Ser. B Methodol."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"095008","DOI":"10.1088\/1361-6420\/aa7fce","article-title":"Sparse spikes super-resolution on thin grids II: The continuous basis pursuit","volume":"33","author":"Duval","year":"2017","journal-title":"Inverse Probl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"055008","DOI":"10.1088\/1361-6420\/aa5e12","article-title":"Sparse regularization on thin grids I: The Lasso","volume":"33","author":"Duval","year":"2017","journal-title":"Inverse Probl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1315","DOI":"10.1007\/s10208-014-9228-6","article-title":"Exact Support Recovery for Sparse Spikes Deconvolution","volume":"15","author":"Duval","year":"2014","journal-title":"Found. Comput. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1607","DOI":"10.1137\/151003714","article-title":"A Continuous Exact l0 penalty (CEL0) for least squares regularized problem","volume":"8","author":"Soubies","year":"2015","journal-title":"SIAM J. Imaging Sci."},{"key":"ref_10","first-page":"906","article-title":"Towards a Mathematical Theory of Super-resolution","volume":"67","year":"2013","journal-title":"Commun. Pure Appl. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1051\/cocv\/2011205","article-title":"Inverse problems in spaces of measures","volume":"19","author":"Bredies","year":"2012","journal-title":"ESAIM Control. Optim. Calc. Var."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"621","DOI":"10.1109\/TIT.2016.2619368","article-title":"Exact solutions to Super Resolution on semi-algebraic domains in higher dimensions","volume":"63","author":"Castro","year":"2017","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"014001","DOI":"10.1088\/1361-6420\/ab2a29","article-title":"The sliding Frank\u2013Wolfe algorithm and its application to super-resolution microscopy","volume":"36","author":"Denoyelle","year":"2019","journal-title":"Inverse Probl."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Federer, H. (1996). Geometric Measure Theory, Springer.","DOI":"10.1007\/978-3-642-62010-2"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Cohn, D.L. (2013). Measure Theory, Springer.","DOI":"10.1007\/978-1-4614-6956-8"},{"key":"ref_16","unstructured":"Temam, R. Fonction Convexe d\u2019une Mesure et Applications. S\u00e9minaire \u00c9quations aux D\u00e9riv\u00e9es Partielles (Polytechnique) dit Aussi \u201cS\u00e9minaire Goulaouic-Schwartz\u201d, Available online: http:\/\/www.numdam.org\/item\/SEDP_1982-1983____A10_0.pdf."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"336","DOI":"10.1016\/j.jmaa.2012.05.011","article-title":"Exact reconstruction using Beurling minimal extrapolation","volume":"395","author":"Gamboa","year":"2012","journal-title":"J. Math. Anal. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/j.acha.2014.03.004","article-title":"Spike detection from inaccurate samplings","volume":"38","author":"Azais","year":"2015","journal-title":"Appl. Comput. Harmon. Anal."},{"key":"ref_19","unstructured":"Beurling, A. Sur les int\u00e9grales de Fourier absolument convergentes et leur application \u00e0 une transformation fonctionnelle. Proceedings of the Ninth Scandinavian Mathematical Congress, Helsinki, Finland."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Ekeland, I., and T\u00e9mam, R. (1999). Convex Analysis and Variational Problems, Society for Industrial and Applied Mathematics.","DOI":"10.1137\/1.9781611971088"},{"key":"ref_21","first-page":"251","article-title":"Super-Resolution of Point Sources via Convex Programming","volume":"5","year":"2016","journal-title":"Inf. Inference J. IMA"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1153","DOI":"10.1007\/s00041-016-9502-x","article-title":"Support Recovery for Sparse Super-Resolution of Positive Measures","volume":"23","author":"Denoyelle","year":"2017","journal-title":"J. Fourier Anal. Appl."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Chizat, L. (2021). Sparse optimization on measures with over-parameterized gradient descent. Math. Program.","DOI":"10.1007\/s10107-021-01636-z"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"045003","DOI":"10.1088\/1361-6420\/ab5aa3","article-title":"The basins of attraction of the global minimizers of the non-convex sparse spike estimation problem","volume":"36","author":"Traonmilin","year":"2020","journal-title":"Inverse Probl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1110","DOI":"10.1109\/LSP.2020.3003241","article-title":"Projected gradient descent for non-convex sparse spike estimation","volume":"27","author":"Traonmilin","year":"2020","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"342","DOI":"10.1007\/BF01443605","article-title":"Ueber die Darstellung definiter Formen als Summe von Formenquadraten","volume":"32","author":"Hilbert","year":"1888","journal-title":"Math. Ann."},{"key":"ref_27","unstructured":"Dumitrescu, B.A. (2007). Positive Trigonometric Polynomials and Signal Processing Applications, Springer."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"796","DOI":"10.1137\/S1052623400366802","article-title":"Global Optimization with Polynomials and the Problem of Moments","volume":"11","author":"Lasserre","year":"2001","journal-title":"SIAM J. Optim."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0041-5553(66)90114-5","article-title":"Constrained minimization methods","volume":"6","author":"Levitin","year":"1966","journal-title":"USSR Comput. Math. Math. Phys."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1002\/nav.3800030109","article-title":"An algorithm for quadratic programming","volume":"3","author":"Frank","year":"1956","journal-title":"Nav. Res. Logist. Q."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1007\/s10107-014-0778-9","article-title":"Conditional gradient algorithms for norm-regularized smooth convex optimization","volume":"152","author":"Harchaoui","year":"2014","journal-title":"Math. Program."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Boyd, N., Schiebinger, G., and Recht, B. (2015, January 13\u201316). The alternating descent conditional gradient method for sparse inverse problems. Proceedings of the 2015 IEEE 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), Cancun, Mexico.","DOI":"10.1109\/CAMSAP.2015.7383735"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Beck, A., and Teboulle, M. (2009, January 19\u201324). A fast Iterative Shrinkage-Thresholding Algorithm with application to wavelet-based image deblurring. Proceedings of the 2009 IEEE International Conference on Acoustics, Speech and Signal Processing, Taipei, Taiwan.","DOI":"10.1109\/ICASSP.2009.4959678"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1190","DOI":"10.1137\/0916069","article-title":"A Limited Memory Algorithm for Bound Constrained Optimization","volume":"16","author":"Byrd","year":"1995","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_35","unstructured":"Ambrosio, L., Gigli, N., and Savare, G. (2008). Gradient Flows in Metric Spaces and in the Space of Probability Measures, Springer."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Santambrogio, F. (2016). Euclidean, Metric, and Wasserstein Gradient Flows: An overview. arXiv.","DOI":"10.1007\/s13373-017-0101-1"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Peyr\u00e9, G., and Cuturi, M. (2019). Computational Optimal Transport, Now Publishers Inc.","DOI":"10.1561\/9781680835519"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"2833","DOI":"10.1137\/140975528","article-title":"Imaging with Kantorovich-Rubinstein discrepancy","volume":"7","author":"Lellmann","year":"2014","journal-title":"SIAM J. Imaging Sci."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Denoyelle, Q., an Pham, T., del Aguila Pla, P., Sage, D., and Unser, M. (2021, January 13\u201316). Optimal-transport-based metric for SMLM. Proceedings of the 2021 IEEE 18th International Symposium on Biomedical Imaging (ISBI), Nice, France.","DOI":"10.1109\/ISBI48211.2021.9433874"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"2540","DOI":"10.1137\/16M1108807","article-title":"Sampling the Fourier transform along radial lines","volume":"55","author":"Dossal","year":"2017","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"717","DOI":"10.1038\/nmeth.3442","article-title":"Quantitative evaluation of software packages for single-molecule localization microscopy","volume":"12","author":"Sage","year":"2015","journal-title":"Nat. Methods"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1038\/nmeth0411-279","article-title":"DAOSTORM: An algorithm for high- density super-resolution microscopy","volume":"8","author":"Holden","year":"2011","journal-title":"Nat. Methods"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"458","DOI":"10.1364\/OPTICA.5.000458","article-title":"Deep-STORM: Super-resolution single-molecule microscopy by deep learning","volume":"5","author":"Nehme","year":"2018","journal-title":"Optica"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"887","DOI":"10.1016\/j.topol.2013.02.007","article-title":"The strict topology for the space of Radon measures on a locally compact Hausdorff space","volume":"160","author":"Javanshiri","year":"2013","journal-title":"Topol. Appl."}],"container-title":["Journal of Imaging"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2313-433X\/7\/12\/266\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:40:21Z","timestamp":1760168421000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2313-433X\/7\/12\/266"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,6]]},"references-count":44,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2021,12]]}},"alternative-id":["jimaging7120266"],"URL":"https:\/\/doi.org\/10.3390\/jimaging7120266","relation":{},"ISSN":["2313-433X"],"issn-type":[{"type":"electronic","value":"2313-433X"}],"subject":[],"published":{"date-parts":[[2021,12,6]]}}}