Given an image generated by the convolution of point sources with a band-limited function, the deconvolution problem involves reconstructing the source number, positions, and amplitudes. This problem is related to many important applications in imaging and signal processing. It is well known that it is impossible to resolve the sources when they are sufficiently close in practice. Rayleigh investigated this problem and formulated a resolution limit known as the Rayleigh limit for the case of two sources with identical amplitudes. However, many numerical experiments have demonstrated that stable recovery of the sources is possible even if the sources are separated below the Rayleigh limit. In this study, a mathematical theory for the deconvolution problem in one dimension is developed. The theory addresses the problem when the source number can be recovered exactly from noisy data. The key component is a new concept called the “computational resolution limit,” which is defined as the minimum separation distance between the sources such that exact recovery of the source number is possible. This new resolution limit is determined by the signal-to-noise ratio and the sparsity of sources, as well as the cutoff frequency of the image. Quantitative bounds for this limit are derived, and they demonstrate the importance of the sparsity and signal-to-noise ratio for the recovery problem. The stability of recovering the source positions is also analyzed under a condition on their separation distances. Moreover, a singular value thresholding algorithm is proposed to recover the source number for a cluster of closely spaced point sources and to verify our theoretical results regarding the computational resolution limit. The results are based on a multipole expansion method and a nonlinear approximation theory in Vandermonde space.

给定由带限函数的点源卷积生成的图像，解卷积问题涉及重建源数量、位置和幅度。这个问题与成像和信号处理中的许多重要应用有关。众所周知，当它们在实践中足够接近时，是不可能解析源的。Rayleigh 研究了这个问题，并为具有相同幅度的两个源的情况制定了一个分辨率极限，称为 Rayleigh 极限。然而，许多数值实验表明，即使源分离低于瑞利极限，源的稳定恢复也是可能的。在这项研究中，开发了一维反卷积问题的数学理论。当可以从嘈杂的数据中准确地恢复源编号时，该理论解决了这个问题。关键组成部分是一个称为“计算分辨率极限”的新概念，它被定义为源之间的最小间隔距离，以便可以精确恢复源编号。这个新的分辨率限制是由信噪比和源的稀疏性以及图像的截止频率决定的。推导出了此限制的定量界限，它们证明了稀疏性和信噪比对恢复问题的重要性。还分析了在它们间隔距离条件下恢复源位置的稳定性。而且，提出了一种奇异值阈值算法来恢复一组密集点源的源数，并验证我们关于计算分辨率限制的理论结果。结果基于 Vandermonde 空间中的多极展开方法和非线性逼近理论。