Sparse Spikes Deconvolution on Thin Grids
Sparse Spikes Deconvolution on Thin Grids
复制标题
薄网格上的稀疏尖峰反卷积
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
G. Peyré
中科院分区：
文献类型：
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作者：
V. Duval;G. Peyré
本文分析了稀疏的尖峰对radon测量问题的稀疏尖峰脱卷积问题的恢复性能。我们在统一的框架中检查了L1正则化（通常称为Lasso或Basis-Pursuit）和连续的基础套件（C-BP）方法。套索是成像中反问题稀疏正规化的事实上的标准。它执行了采样网格上尖峰位置的最近邻居插值。 Ekanadham，Tranchina和Simoncelli引入的C-BP方法使用位置的线性插值来更好地对无限二维优化问题进行更好的近似，以实现积极度量。我们表明，在小的噪声状态下，这两种方法都估计尖峰数量是原始尖峰的数量。确实，我们表明他们都检测到原始尖峰位置周围的两个相邻尖峰。这些针对反卷积问题的结果基于对L1型问题解决方案的所谓扩展支持的抽象分析（包括特殊情况，即lasso和c-bp for Deonvolution），这具有独立的利益。当噪声较小并相应选择正则化参数时，它们精确地表征了解决方案的支持。我们说明了这些发现，当测量数量低于临界限制（文献中有良好的文献记录）时，首次分析压缩感应恢复的支持不稳定性，而支持证明是稳定的。
This article analyzes the recovery performance of two popular finite dimensional approximations of the sparse spikes deconvolution problem over Radon measures. We examine in a unified framework both the L1 regularization (often referred to as Lasso or Basis-Pursuit) and the Continuous Basis-Pursuit (C-BP) methods. The Lasso is the de-facto standard for the sparse regularization of inverse problems in imaging. It performs a nearest neighbor interpolation of the spikes locations on the sampling grid. The C-BP method, introduced by Ekanadham, Tranchina and Simoncelli, uses a linear interpolation of the locations to perform a better approximation of the infinite-dimensional optimization problem, for positive measures. We show that, in the small noise regime, both methods estimate twice the number of spikes as the number of original spikes. Indeed, we show that they both detect two neighboring spikes around the locations of an original spikes. These results for deconvolution problems are based on an abstract analysis of the so-called extended support of the solutions of L1-type problems (including as special cases the Lasso and C-BP for deconvolution), which are of an independent interest. They precisely characterize the support of the solutions when the noise is small and the regularization parameter is selected accordingly. We illustrate these findings to analyze for the first time the support instability of compressed sensing recovery when the number of measurements is below the critical limit (well documented in the literature) where the support is provably stable.
影响因子：
1.6
作者：
Amelunxen, Dennis;Lotz, Martin;Tropp, Joel A.
通讯作者：
Tropp, Joel A.