CIF: Small: Inverse Methods for Parametric Mixture Models

在新兴的科学和工程应用中，许多成像模式涉及对来自不同点扩展函数编码的混合成员的点源的超分辨率低分辨率观测。一个值得注意的例子是超分辨率荧光显微镜，它的重要性得到了2014年诺贝尔化学奖的认可，因为它能够在纳米尺度上对复杂的生物过程进行非侵入性成像。下一个前沿是三维超分辨率单分子显微镜，它允许从二维图像重建三维结构，使用工程化的点扩散函数来编码不同分子的轴向信息，例如通过引入柱面透镜。因此，算法上的挑战是同时从它们的叠加中分离和分解尽可能多的分子，以提高成像的时间分辨率。该研究计划将开发一个统一的框架，以了解在这种混合模型中何时可能同时进行分离和超分辨率，以及开发计算高效、可证明正确和对噪声具有鲁棒性的算法。算法将与俄亥俄州立大学多萝西·M·戴维斯心肺研究所的合作者在三维超分辨率单分子成像的真实数据上实施。该项目为学生提供跨学科的培训机会，学生将在数学信号处理、优化和生物医学数据分析方面发展专业知识。这个项目的结果将被整合到俄亥俄州立大学关于反问题和高维数据分析的研究生课程中。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Many imaging modalities in emerging science and engineering applications involve super-resolving low-resolution observations of point sources coming from mixed memberships encoded by different point spread functions. A notable example is super-resolution fluorescence microscopy, whose importance is recognized by the 2014 Nobel Prize in Chemistry, due to its ability of noninvasive imaging of complex biological processes at the nanometer scale. The next frontier, which is three-dimensional super-resolution single-molecule microscopy, allows reconstruction of three-dimensional structures from two-dimensional images, using engineered point spread functions to encode the axial information of different molecules, e.g. by introducing a cylindrical lens. The algorithmic challenge is therefore to simultaneously separate and resolve as many molecules as possible from their superposition in order to enhance the time resolution of imaging.This research program will develop a unified framework to understand when separation and super-resolution in such mixture models is simultaneously possible, as well as develop algorithms that are computationally efficient, provably correct, and robust to noise. Algorithms will be implemented on real data of three-dimensional super-resolution single-molecule imaging with collaborators at the Dorothy M. Davis Heart and Lung Research Institute at OSU. The project provides interdisciplinary opportunities for students training, where students will develop expertise in mathematical signal processing, optimization, and biomedical data analysis. The results of this project will be integrated into graduate-level courses on inverse problems and high-dimensional data analysis at OSU.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.