The aim of this paper is to investigate superresolution in deconvolution driven by sparsity priors. The observed signal is a convolution of an original signal with a continuous kernel.With the prior knowledge that the original signal can be considered as a sparse combination of Dirac delta peaks, we seek to estimate the positions and amplitudes of these peaks by solving a finite dimensional convex problem on a computational grid. Because, the support of the original signal may or may not be on this grid, by studying the discrete deconvolution of sparse peaks using L1-norm sparsity prior, we confirm recent observations that canonically the discrete reconstructions will result in multiple peaks at grid points adjacent to the location of the true peak. Owning to the complexity of this problem, we analyse carefully the de-convolution of single peaks on a grid and gain a strong insight about the dependence of the reconstructed magnitudes on the exact peak location. This in turn allows us to infer further information on recovering the location of the exact peaks i.e. to perform super-resolution. We analyze in detail the possible cases that can appear and based on our theoretical findings, we propose an self-driven adaptive grid approach that allows to perform superresolution in one-dimensional and multi-dimensional spaces. With the view that the current study can provide a further step in the development of more robust algorithms for the detection of single molecules in fluorescence microscopy or identification of characteristic frequencies in spectral analysis, we demonstrate how the proposed approach can recover sparse signals using simulated clusters of point sources (peaks) of low-resolution in one and two-dimensional spaces.

中文翻译：

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稀疏峰解卷积中的自适应超分辨率

本文的目的是研究由稀疏先验驱动的反卷积中的超分辨率。观察到的信号是原始信号与连续核的卷积。根据原始信号可以被认为是 Dirac delta 峰值的稀疏组合的先验知识，我们试图通过求解一个有限的问题来估计这些峰值的位置和幅度计算网格上的维凸问题。因为，原始信号的支持可能在也可能不在这个网格上，通过使用 L1 范数稀疏先验研究稀疏峰的离散反卷积，我们确认了最近的观察结果，即离散重建将在相邻的网格点处产生多个峰到真峰所在的位置。由于这个问题的复杂性，我们仔细分析了网格上单峰的反卷积，并深入了解了重建幅度对精确峰位置的依赖性。这反过来又使我们能够推断出有关恢复精确峰位置的更多信息，即执行超分辨率。我们详细分析了可能出现的情况，并根据我们的理论发现，提出了一种自驱动自适应网格方法，允许在一维和多维空间中执行超分辨率。鉴于目前的研究可以为开发更强大的算法提供进一步的步骤，用于在荧光显微镜中检测单个分子或在光谱分析中识别特征频率，