CIF: Small: Computationally Efficient Analytic Reconstructions via Embeddings and Sparsity for Non-Linear Dynamic Imaging Problems

随着低成本、高耐久性无人驾驶车辆的发展，现在可以使用多个分布式传感器在长时间和空间内持续获取成像数据。这种持久的多通道感测系统相对于单通道系统具有若干优点，包括监视动态介质的能力、更宽的空间覆盖范围、增加的分辨率以及获取三维或更高维度信息的能力。虽然这种持久的系统有可能在许多应用领域彻底改变传感和成像，但部署这些系统的根本瓶颈之一是图像形成方面的挑战：首先，在未知复杂环境中重建3D变形，速度场和成像是高度非线性的大规模逆问题。其次，这样的系统每天可以获取数百TB的数据，并形成一个标准的图像，即使是一个单程的数据，目前还不能在real-time.This project developed a unified mathematical framework and new classes of novel，computationally efficient analytic image reconstruction methods to address these large scale，computationally required，non-linear inverse problems.我们发展的核心是逆散射理论，稀疏信号恢复方法和微局部分析的协同组合。我们的研究涉及以下主题：（i）使用多个传感器的三维变形率成像，速度成像和未知非均匀环境中成像的精确正演模型的开发。(ii)通过将所产生的非线性问题嵌入到具有稀疏约束的更大维空间中，将其重新表述为线性问题。(iii)通过协同结合微局部分析和稀疏信号恢复方法的思想，设计分析的、计算高效的反演方法。

With the development of low cost, high endurance uninhabited vehicles, it is now possible to acquire imaging data persistently over long periods of time and space using multiple distributed sensors. Such persistent multi-pass sensing systems have several advantages over single-pass systems including the ability to monitor a dynamic medium, wider spatial coverage, increased resolution and the ability to acquire three or higher dimensional information. While such persistent systems have the potential to revolutionize sensing and imaging across many domains of applications, one of the fundamental bottlenecks for the deployment of these systems is the challenges in image formation: First, the reconstructions of 3D deformations, velocity fields and imaging in unknown complex environments are highly non-linear large scale inverse problems. Secondly, such systems can acquire hundreds of terabytes of data daily and forming a standard image using even a single-pass data cannot currently be done in real-time.This project develops a unified mathematical framework and new classes of novel, computationally efficient analytic image reconstruction methods to address these large scale, computationally demanding, non-linear inverse problems. Central to our development is the synergistic combination of inverse scattering theory, sparse signal recovery methods and microlocal analysis. Our research involves the following topics: (i) Development of accurate forward models for 3D deformation-rate imaging, velocity imaging and imaging in unknown heterogenous environments using multiple sensors. (ii) Reformulation of the resulting nonlinear problems as linear problems by embedding them into larger dimensional spaces coupled with sparsity constraints. (iii) Design of analytic, computationally efficient methods for inversion by synergistically combining ideas from microlocal analysis and sparse signal recovery methods.