In this paper, we study the solving of the gridless sparse optimization problem and its application to 3D image deconvolution. Based on the recent works of (Denoyelle et al, 2019) introducing the Sliding Frank-Wolfe algorithm to solve the Beurling LASSO problem, we introduce an accelerated algorithm, denoted BSFW, that preserves its convergence properties, while removing most of the costly local descents. Besides, as the solving of BLASSO still relies on a regularization parameter, we introduce an homotopy algorithm to solve the constrained BLASSO that allows to use a more practical parameter based on the image residual, e.g. its standard deviation. Both algorithms benefit from a finite termination property, i.e. they are guaranteed to find the solution in a finite number of step under mild conditions. These methods are then applied on the problem of 3D tomographic diffractive microscopy images, with the purpose of explaining the image by a small number of atoms in convolved images. Numerical results on synthetic and real images illustrates the improvement provided by the BSFW method, the homotopy method and their combination.

中文翻译：

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一种用于无网格稀疏恢复的快速同伦算法

在本文中，我们研究了无网格稀疏优化问题的求解及其在 3D 图像反卷积中的应用。基于 (Denoyelle et al, 2019) 引入 Sliding Frank-Wolfe 算法来解决 Beurling LASSO 问题的近期工作，我们引入了一种加速算法，称为 BSFW，它保留了其收敛特性，同时去除了大部分代价高昂的局部下降. 此外，由于 BLASSO 的求解仍然依赖于正则化参数，我们引入了一种同伦算法来求解受约束的 BLASSO，该算法允许使用基于图像残差的更实用的参数，例如其标准偏差。两种算法都受益于有限终止特性，即保证它们在温和条件下以有限步数找到解。然后将这些方法应用于 3D 断层扫描衍射显微镜图像的问题，目的是通过卷积图像中的少量原子来解释图像。合成图像和真实图像的数值结果说明了 BSFW 方法、同伦方法及其组合所提供的改进。