Marta Lazzaretti, Claudio Estatico, Alejandro Melero, Luca Calatroni
{"title":"Off-the-grid regularisation for Poisson inverse problems.","authors":"Marta Lazzaretti, Claudio Estatico, Alejandro Melero, Luca Calatroni","doi":"10.1007/s10589-025-00688-7","DOIUrl":null,"url":null,"abstract":"<p><p>Off-the-grid regularisation has been extensively employed over the last decade in the context of ill-posed inverse problems formulated in the continuous setting of the space of Radon measures <math><mrow><mi>M</mi> <mo>(</mo> <mi>Ω</mi> <mo>)</mo></mrow> </math> . These approaches enjoy convexity and counteract the discretisation biases as well the numerical instabilities typical of their discrete counterparts. In the framework of sparse reconstruction of discrete point measures (sum of weighted Diracs), a Total Variation regularisation norm in <math><mrow><mi>M</mi> <mo>(</mo> <mi>Ω</mi> <mo>)</mo></mrow> </math> is typically combined with an <math><msup><mi>L</mi> <mn>2</mn></msup> </math> data term modelling additive Gaussian noise. To assess the framework of off-the-grid regularisation in the presence of signal-dependent Poisson noise, we consider in this work a variational model where Total Variation regularisation is coupled with a Kullback-Leibler data term under a non-negativity constraint. Analytically, we study the optimality conditions of the composite functional and analyse its dual problem. Then, we consider an homotopy strategy to select an optimal regularisation parameter and use it within a Sliding Frank-Wolfe algorithm. Several numerical experiments on both 1D/2D/3D simulated and real 3D fluorescent microscopy data are reported.</p>","PeriodicalId":55227,"journal":{"name":"Computational Optimization and Applications","volume":"91 2","pages":"827-860"},"PeriodicalIF":2.0000,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12085367/pdf/","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computational Optimization and Applications","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10589-025-00688-7","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2025/5/4 0:00:00","PubModel":"Epub","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
Off-the-grid regularisation has been extensively employed over the last decade in the context of ill-posed inverse problems formulated in the continuous setting of the space of Radon measures . These approaches enjoy convexity and counteract the discretisation biases as well the numerical instabilities typical of their discrete counterparts. In the framework of sparse reconstruction of discrete point measures (sum of weighted Diracs), a Total Variation regularisation norm in is typically combined with an data term modelling additive Gaussian noise. To assess the framework of off-the-grid regularisation in the presence of signal-dependent Poisson noise, we consider in this work a variational model where Total Variation regularisation is coupled with a Kullback-Leibler data term under a non-negativity constraint. Analytically, we study the optimality conditions of the composite functional and analyse its dual problem. Then, we consider an homotopy strategy to select an optimal regularisation parameter and use it within a Sliding Frank-Wolfe algorithm. Several numerical experiments on both 1D/2D/3D simulated and real 3D fluorescent microscopy data are reported.
期刊介绍:
Computational Optimization and Applications is a peer reviewed journal that is committed to timely publication of research and tutorial papers on the analysis and development of computational algorithms and modeling technology for optimization. Algorithms either for general classes of optimization problems or for more specific applied problems are of interest. Stochastic algorithms as well as deterministic algorithms will be considered. Papers that can provide both theoretical analysis, along with carefully designed computational experiments, are particularly welcome.
Topics of interest include, but are not limited to the following:
Large Scale Optimization,
Unconstrained Optimization,
Linear Programming,
Quadratic Programming Complementarity Problems, and Variational Inequalities,
Constrained Optimization,
Nondifferentiable Optimization,
Integer Programming,
Combinatorial Optimization,
Stochastic Optimization,
Multiobjective Optimization,
Network Optimization,
Complexity Theory,
Approximations and Error Analysis,
Parametric Programming and Sensitivity Analysis,
Parallel Computing, Distributed Computing, and Vector Processing,
Software, Benchmarks, Numerical Experimentation and Comparisons,
Modelling Languages and Systems for Optimization,
Automatic Differentiation,
Applications in Engineering, Finance, Optimal Control, Optimal Design, Operations Research,
Transportation, Economics, Communications, Manufacturing, and Management Science.