任意常维超分辨率鲁棒稀疏连续傅里叶变换：近似线性时间和样本复杂度
Super-resolution and Robust Sparse Continuous Fourier Transform in Any Constant Dimension: Nearly Linear Time and Sample Complexity
摘要：
分辨被成像物体的细节的能力，以分辨率命名，是成像系统的核心参数。超分辨率是一种能够提高成像系统分辨率，甚至超越系统衍射极限的技术。尽管在应用中取得了巨大的成功，但在理论方面，特别是对于任何维度的$d \geq 2$，超分辨率还没有得到很好的理解。具体地，为了恢复$k$稀疏信号，所有先前的结果受到聚$(k)$样本或运行时间的任一者/两者的影响。我们在发展稀疏傅里叶变换（Sparse FT）中的一些新技术的基础上，设计了强噪声模型下任意（常）维的鲁棒算法，如鲁棒线性系统的反演、“蛋壳”采样方案以及高维的划分和投票方法。这些算法是第一个实现运行时间和样本复杂度（几乎）线性的源点的数量和对数的带宽为任何恒定的维度，我们相信在工作中开发的技术可以找到他们的进一步应用的超分辨率和稀疏FT问题。
英文摘要：
The ability to resolve detail in the object that is being imaged, named by resolution, is the core parameter of an imaging system. Super-resolution is a class of techniques that can enhance the resolution of an imaging system and even transcend the diffraction limit of systems. Despite huge success in the application, super-resolution is not well understood on the theoretical side, especially for any dimension $d \geq 2$. In particular, in order to recover a $k$-sparse signal, all previous results suffer from either/both poly$(k)$ samples or running time. We design robust algorithms for any (constant) dimension under a strong noise model based on developing some new techniques in Sparse Fourier transform (Sparse FT), such as inverting a robust linear system, "eggshell" sampling schemes, and partition and voting methods in high dimension. These algorithms are the first to achieve running time and sample complexity (nearly) linear in the number of source points and logarithmic in bandwidth for any constant dimension, and we believe the techniques developed in the work can find their further applications on the Super-resolution and Sparse FT problem.
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