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On the Stable Resolution Limit of Total Variation Regularization for Spike Deconvolution

机译：关于穗碎屑总变化正规化的稳定分辨率限制

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摘要

The stability of spike deconvolution, which aims at recovering point sources from their convolution with a point spread function (PSF), is known to be related to the separation between those sources. When the observations are noisy, it is critical to ensure support stability, where the deconvolution does not lead to spurious, or oppositely, missing estimates of the point sources. In this paper, we study the resolution limit of stably recovering the support of two closely located point sources using the Beurling-LASSO estimator, which is a convex optimization approach based on total variation regularization. We establish a sufficient separation criterion between the sources, depending only on the PSF, above which the Beurling-LASSO estimator is guaranteed to return a stable estimate of the point sources, with the same number of estimated elements as that of the ground truth. Our result highlights the impact of PSF on the resolution limit in the noisy setting, which was not evident in previous studies of the noiseless setting. Towards the end, we show that the same resolution limit applies to resolving two close-located sources in conjunction of other well-separated sources.
机译：峰值去卷积的稳定性，其目的是从它们的点扩散函数（PSF）的卷积恢复点源（PSF），与这些来源之间的分离有关。当观察结果是嘈杂的时，确保支持稳定性至关重要，其中折卷积不会导致虚假或相反，缺少点源的估计。在本文中，我们使用Beurling-Lasso估计器研究了稳定地恢复了两个紧密位置源的支持的分辨率限制，这是基于总变化正则化的凸优化方法。我们在源之间建立了足够的分离标准，仅根据PSF取决于PSF，保证Beurling-Lasso估计器保证返回点源的稳定估计，与地面真理的估计元素相同。我们的结果突出了PSF对嘈杂环境中的分辨率限制的影响，这在对无噪声环境的研究中并不明显。在最后，我们表明，同样的分辨率限制适用于解决其他分隔良好的来源的两个近距离来源。

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