报告题目：A Mathematical Theory of Computational Resolution Limit and Super-resolution

报 告 人：刘平

报告时间：2024年11月15日11:00-12:00

报告地点：格物楼数学研究中心528报告厅

报告摘要：Due to the physical nature of wave propagation and diffraction, there is a fundamental diffraction barrier in optical imaging systems which is called the diffraction limit or resolution limit. Rayleigh investigated this problem and formulated the well-known Rayleigh limit. However, the Rayleigh limit is empirical and only considers the resolving ability of the human visual system. On the other hand, resolving sources separated below the Rayleigh limit to achieve so-called “super-resolution” has been demonstrated in many numerical experiments.

In this talk, we will propose a new concept “computational resolution limit” which reveals the fundamental limits in super-resolving the number and locations of point sources from a data-processing point of view. We will quantitatively characterize the computational resolution limits by the signal-to-noise ratio, the sparsity of sources, and the cutoff frequency of the imaging system. As a direct consequence, it is demonstrated that optimization achieves the optimal order resolution in solving super-resolution problems. For the case of resolving two point sources, the resolution estimate is improved to an exact formula. We will also propose an optimal algorithm to distinguish images generated by single or two point sources. Generalization of our results to the imaging of positive sources and imaging in multi-dimensional spaces will be briefly discussed as well.

报告人简介：刘平，浙江大学数学科学学院百人计划研究员，2021-2024年在瑞士苏黎世联邦学院数学系任博士后， 2021年博士毕业于香港科技大学。 主要研究超分辨率成像和阵列信号处理的理论和算法，拓扑材料领域的一些物理现象。 学术成果发表在国际知名期刊Found. Comput. Math., Forum Math. Sigma., Appl. Comput. Harmon. Anal.,Stud. Appl. Math., SIAM系列以及IEEE-Trans系列等。