CIF: Small: Multidimensional Remaindering Theory and Applications

许多实际应用，如高光谱成像，生物医学传感和多输入多输出合成孔径雷达（SAR），往往涉及到多维信号/数据的分析和处理。由于时间、空间、成本、存储、带宽和计算能力等方面的限制，多维海量数据集的获取、处理、传输和存储都具有挑战性。应用中国剩余定理（CRT），这些问题中的一些可以通过将大任务划分为多个可以并行执行的小但独立的子任务来解决。此外，利用CRT，可以显著提高诸如雷达的传感器的目标可探测性。传统的CRT允许一个大整数的重构，从它的模相对于一组小整数（称为模）。CRT对余数误差的鲁棒性不强，因为余数中的小误差可能导致大的重建误差，这将使实际应用中的性能恶化。近年来，研究小组已经开发出一种改进的CRT，它对剩余误差具有鲁棒性，因此比传统的CRT提高了性能。结果发现，解决方案依赖于素数和模的因子，只有有限的选择，因此可能会限制性能的改善。对于多维信号，他们最近获得了一个多维中国剩余定理（MD-CRT），它提供了一个算法来唯一地重建一个整数值的向量从其剩余向量相对于一组整数矩阵，功能作为模。此外，它的鲁棒版本的一个特殊情况下已经得到，以及使用一个特殊的整数矩阵模。MD-CRT和鲁棒MD-CRT都依赖于模的互质整数矩阵和因子矩阵，与传统标量CRT和鲁棒CRT中素数和因子的选择相比，MD-CRT和鲁棒MD-CRT中互质整数矩阵的选择更多。 本项目旨在系统地研究MD-CRT和鲁棒MD-CRT相对于一组一般的整数矩阵模（比整数矩阵的交换对更一般）。它研究了重建单个真实的向量和多个整数/真实的向量的推广。它还研究了雷达中的新应用，从多通道模模数转换器（ADC）的矢量值信号的鲁棒恢复，以及使用平面天线阵列的SAR成像中的运动目标检测和估计。该项目中MD-CRT和鲁棒MD-CRT的系统性和更普遍的结果可能会导致上述应用的性能改进。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估来支持。

Many practical applications, such as hyperspectral imaging, biomedical sensing, and multiple-input multiple-output synthetic aperture radars (SARs), often involve the analysis and processing of multidimensional signals/data. As time, space, cost, storage, bandwidth, and computing power are all limited, the acquisition, processing, transmission, and storage of multidimensional massive data set are challenging. Applying the Chinese remainder theorem (CRT), some of these problems may be solved by partitioning a large task into a number of small but independent subtasks that can be performed in parallel. Furthermore, with the CRT, the target detectability of sensors, such as radar, may be significantly improved. The conventional CRT permits the reconstruction of a large integer from its remainders with respect to a set of small integers (called moduli). The CRT is not robust against remainder errors in the sense that a small error in a remainder may cause a large reconstruction error, which will deteriorate the performance in practical applications. In recent years, the investigative group has developed a modified CRT that is robust to remainder errors and therefore improves the performance over the traditional CRT. It was found that the solutions depend on prime numbers and factors of moduli, which have only limited choices and therefore may limit the performance improvement. For multidimensional signals, they have recently obtained a multidimensional Chinese remainder theorem (MD-CRT) which provides an algorithm to uniquely reconstruct an integer-valued vector from its remainder vectors with respect to a set of integer matrices that function as moduli. Moreover, a special case of its robust version has been obtained as well, using a special integer matrix moduli. Both MD-CRT and robust MD-CRT depend on co-prime integer matrices and factor matrices of moduli.Compared to the choices of prime numbers and factors in the conventional scalar CRT and robust CRT, there are many more choices of co-prime integer matrices in MD-CRT and robust MD-CRT. This project aims to systematically investigate MD-CRT and robust MD-CRT with respect to a general set of integer matrix moduli (more general than commutative pairs of integer matrices). It investigates the generalizations for reconstructing a single real vector and multiple integer/real vectors. It also investigates new applications in radar, robust recovery of vector-valued signals from multi-channel modulo analog to digital converters (ADCs), and moving target detection and estimation in SAR imaging using planar antenna arrays. The systematic and more general results on MD-CRT and robust MD-CRT in this project may lead to performance improvements in the above-mentioned applications.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.