In this paper, we develop a new technique to obtain improved estimates for the computational resolution limits in two-dimensional super-resolution problems and present a new idea for developing two-dimensional super-resolution algorithms. To be more specific, our main contributions are fourfold: (1) Our work improves the resolution estimates for number detection and location recovery in two-dimensional super-resolution problems; (2) As a consequence, we derive a stability result for a sparsity-promoting algorithm in two-dimensional super-resolution problems [or direction of arrival Problems (DOA)]. The stability result exhibits the optimal performance of sparsity promoting in solving such problems; (3) Inspired by the new techniques, we propose a new coordinate-combination-based model order detection algorithm for two-dimensional DOA estimation and theoretically demonstrate its optimal performance, and (4) we also propose a new coordinate-combination-based MUSIC algorithm for super-resolving sources in two-dimensional DOA estimation. It has excellent performance and enjoys some advantages compared to the conventional DOA algorithms.

在本文中，我们开发了一种新技术来获得二维超分辨率问题中计算分辨率极限的改进估计，并提出了开发二维超分辨率算法的新思路。更具体地说，我们的主要贡献有四个：（1）我们的工作改进了二维超分辨率问题中数字检测和位置恢复的分辨率估计； (2) 因此，我们得出了二维超分辨率问题[或到达方向问题（DOA）]中稀疏性促进算法的稳定性结果。稳定性结果展示了稀疏性促进在解决此类问题中的最优性能； （3）受新技术的启发，我们提出了一种新的基于坐标组合的模型阶次检测算法，用于二维 DOA 估计，并从理论上证明了其最佳性能，（4）我们还提出了一种新的基于坐标组合的 MUSIC二维 DOA 估计中超分辨源的算法。它具有优异的性能，与传统的DOA算法相比具有一定的优势。