Novel Image Reconstruction Methods in the Frequency Domain

PI将开发更快，更准确的数学算法，用于生物医学成像，遥感和新型光学器件设计。这些图像构建问题通常涉及大型且复杂的系统。PI将建立在最近的理论突破，以构建简化系统，允许更直接，更有效地将数据映射到未知的物理量。 简化模型的新用途将首次允许仅从外部数据生成内部物理场，为更广泛的模态和实验设置提供桥梁。一旦从一个小的数据集生成了简化模型，就可以为任意大的数据集或以其他格式给出的数据找到内部场。内场的知识极大地简化了成像问题，并且对于某些应用，例如医疗消融，感兴趣的是内场本身。 几个学生将在这项研究的过程中进行培训，包括两个全职博士生。本科生将在这个研究问题上全职工作六个月作为他们的合作培训。PI继续明确致力于数学的多样性，并将尽一切努力让妇女和/或代表性不足的少数民族参与合作社的经验。PI将产生新的重建方法，并进一步发展在医学成像，遥感和无损检测中至关重要的理论。主要目标是（i）使用降阶模型从边界数据生成内部解，（ii）使用这些边界数据生成的内部解来解决较大类别的数据集的逆问题，（iii）导出、分析和应用适应于非线性的新的逆Born级数，以及（iv）使用边界校正和透射特征值来成像周期性和近周期性微结构。介绍了一种新的降阶模型（ROM）用于反问题的方法，即通过将ROM嵌入到连续问题中，仅由边界数据生成内部场。高精度的内场将被用于将ROM应用于大型数据集，以产生一种全新的、完全非线性的、计算成本低的反演方法。内场在医学消融等应用中具有重要意义，它们为经典反问题和多物理场混合方法之间提供了桥梁。在这个过程中，一个关键的正交化步骤将得到严格的证明。新的逆散射系列将使我们能够重建非线性散射体，而无需使用优化或前向求解器，除了参考介质。我们将分析级数，展示收敛估计，并充分理解级数的行为。对于微结构介质，渐近的前向解决方案将使我们能够捕捉精细尺度的功能。边界校正将用于成像媒体沿着与传输eigenvalues.This奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的知识价值和更广泛的影响审查标准。

The PI will develop faster and more accurate mathematical algorithms for use in biomedical imaging, remote sensing, and novel optical device design. These image construction problems generally involve large and complex systems. The PI will build on recent theoretical breakthroughs to construct reduced systems which allow for more straightforward and efficient mapping of the data to the unknown physical quantities. The novel usage of the reduced models will for the first time allow for the generation of internal physical fields from exterior data only, providing a bridge to a broader range of modalities and experimental settings. Once the reduced model is generated from a small data set, the interior fields can be found for an arbitrarily large data set or for data given in other formats. The knowledge of interior fields greatly simplifies the imaging problems, and for certain applications such as medical ablation, it is the interior fields themselves that are of interest. Several students will be trained in the course of this research, including two full time Phd students. Undergraduate students will work for six months full time on this research problem as their co-op training. The PI continues to have a clear commitment to diversity in mathematics and will make every effort to involve women and/or underrepresented minorities in the co-op experience.The PI will generate new reconstruction methods and develop further theories which are crucial in medical imaging, remote sensing and nondestructive testing. The main goals are to (i) use reduced order models to generate interior solutions from boundary data, (ii) use these boundary data generated interior solutions to solve inverse problems for larger classes of data sets, (iii) derive, analyze and apply a new inverse Born series adapted to nonlinearity, and (iv) use boundary corrections and transmission eigenvalues to image periodic and nearly periodic microstructures. A new way of using reduced order models (ROMs) for inverse problems is introduced, that is, by embedding the ROM back into the continuous problem and generating interior fields from boundary data only. Highly accurate interior fields will be used to apply the ROM to large data sets to yield a completely new, fully nonlinear inversion method with low computational cost. Interior fields are of interest in their own right for applications such as medical ablation, and they provide a bridge between classical inverse problems and multi-physics hybrid methods. A crucial orthogonalization step in the procedure will be justified rigorously. New inverse scattering series will allow us to reconstruct nonlinear scatterers without the use of optimization or forward solvers except for that of the reference medium. We will analyze the series, show convergence estimates, and fully understand the series behavior. For microstructured media, asymptotics of the forward solution will enable us to capture fine scale features. Boundary corrections will be used to image the media along with transmission eigenvalues.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.