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

随着低成本、高耐久性无人车的发展，现在可以使用多个分布式传感器在长时间和长空间内持久地获取成像数据。这种永久性多通道传感系统与单通道系统相比具有几个优点，包括能够监控动态介质、更宽的空间覆盖范围、更高的分辨率以及获取三维或更高维信息的能力。虽然这种持久性系统有可能在许多应用领域带来传感和成像的革命性变化，但这些系统部署的根本瓶颈之一是成像方面的挑战：首先，未知复杂环境中的三维变形、速度场和成像的重建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，并且使用即使是单遍数据也不能实时地形成标准图像。该项目开发了一个统一的数学框架和新型的、计算高效的解析图像重建方法来解决这些大规模的、计算要求高的非线性逆问题。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结合。我们的研究主要涉及以下几个方面：(1)建立精确的正演模型，用于三维形变速率成像、速度成像和未知非均匀环境下的多传感器成像。(2)将所得到的非线性问题重新表述为线性问题，方法是将其嵌入更大的维度空间，并加上稀疏约束。(3)协同结合微局部分析和稀疏信号恢复方法的思想，设计解析的、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.