CIF: Small: Multidimensional Remaindering Theory and Applications

CIF：小：多维余数理论与应用

• 批准号: 2246917
• 负责人: Xiang-Gen Xia
• 金额: $ 39.71万
• 依托单位: University of Delaware
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246917&HistoricalAwards=false
• 关键词: CIF; Small; Multidimensional; Remaindering; Theory

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.

许多实际应用，例如高光谱成像、生物医学传感和多输入多输出合成孔径雷达（SAR），通常涉及多维信号/数据的分析和处理。由于时间、空间、成本、存储、带宽和计算能力均受到限制，多维海量数据集的获取、处理、传输和存储具有挑战性。应用中国剩余定理（CRT），其中一些问题可以通过将大型任务划分为许多可以并行执行的小但独立的子任务来解决。此外，利用CRT，雷达等传感器的目标探测能力可能会得到显着提高。传统的 CRT 允许根据一组小整数（称为模）的余数重建大整数。 CRT对于余数误差不具有鲁棒性，因为余数中的小误差可能会导致大的重构误差，这会降低实际应用中的性能。近年来，研究小组开发了一种改进的 CRT，它对剩余错误具有鲁棒性，因此比传统 CRT 提高了性能。研究发现，解决方案依赖于素数和模因子，它们的选择有限，因此可能限制性能的提高。对于多维信号，他们最近获得了多维中国余数定理（MD-CRT），该定理提供了一种算法，可以根据一组作为模的整数矩阵从余数向量中唯一地重建整数值向量。此外，还使用特殊的整数矩阵模获得了其鲁棒版本的特殊情况。 MD-CRT和鲁棒MD-CRT都依赖于互素整数矩阵和模因子矩阵。与传统标量CRT和鲁棒CRT中素数和因子的选择相比，互素整数的选择更多MD-CRT 和稳健 MD-CRT 中的矩阵。 该项目旨在系统地研究 MD-CRT 和鲁棒 MD-CRT 对于一组通用的整数矩阵模（比整数矩阵交换对更通用）。它研究了重建单个实数向量和多个整数/实数向量的概括。它还研究了雷达中的新应用、从多通道模数转换器 (ADC) 稳健地恢复矢量值信号，以及使用平面天线阵列在 SAR 成像中进行移动目标检测和估计。该项目中 MD-CRT 的系统性和更通用的结果以及稳健的 MD-CRT 可能会导致上述应用中的性能改进。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值进行评估，被认为值得支持以及更广泛的影响审查标准。