Off-the-grid regularisation for Poisson inverse problems.

IF 2 2区 数学 Q2 MATHEMATICS, APPLIED
Marta Lazzaretti, Claudio Estatico, Alejandro Melero, Luca Calatroni
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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 M ( Ω ) . 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 M ( Ω ) is typically combined with an L 2 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.

泊松反问题的离网正则化。
在过去十年中,离网正则化在Radon测度M空间的连续设置中制定的不适定逆问题的背景下被广泛应用(Ω)。这些方法具有凸性,并抵消了离散化偏差以及离散化方法典型的数值不稳定性。在离散点测度(加权狄拉克和)的稀疏重建框架中,M (Ω)中的总变分正则化范数通常与建模加性高斯噪声的l2数据项相结合。为了评估在信号依赖泊松噪声存在下的离网正则化框架,我们在这项工作中考虑了一个变分模型,其中总变分正则化与非负性约束下的Kullback-Leibler数据项相耦合。分析地研究了复合泛函的最优性条件,并分析了它的对偶问题。然后,我们考虑了一种同伦策略来选择最优正则化参数,并将其用于滑动Frank-Wolfe算法中。对一维/二维/三维模拟和真实三维荧光显微镜数据进行了数值实验。
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来源期刊
CiteScore
3.70
自引率
9.10%
发文量
91
审稿时长
10 months
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.
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