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，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(SARS)，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、带宽和计算能力都是有限的，多维海量数据的获取、处理、传输和存储都是具有挑战性的。应用中国剩余定理(CRT)，可以通过将一个大任务划分成多个可以并行执行的小而独立的子任务来解决其中的一些问题。此外，有了CRT，雷达等传感器的目标探测能力可能会得到显著提高。传统的CRT允许从其余数相对于一组小整数(称为模数)重建大整数。CRT对于余数误差的健壮性不强，因为余数中的小误差可能导致大的重建误差，这将在实际应用中恶化性能。近年来，课题组开发了一种改进的CRT，它对剩余误差具有鲁棒性，从而改善了传统CRT的性能。研究发现，这些解依赖于素数和模因子，而素数和模因子的选择有限，从而限制了性能的提高。对于多维信号，他们最近得到了多维中国剩余定理(MD-CRT)，该定理提供了一种算法，从整值向量关于作为模数的一组整数矩阵的余数向量唯一地重构。此外，还利用一种特殊的整数矩阵模，得到了它的稳健形式的一个特例。MD-CRT和稳健MD-CRT都依赖于模的余素整数矩阵和因子矩阵，与传统标量CRT和稳健CRT中素数和因子的选择相比，MD-CRT和稳健MD-CRT有更多的余素整数矩阵的选择。这个项目的目的是系统地研究MD-CRT和关于整数矩阵模集(比整数矩阵交换对更一般)的稳健MD-CRT。它研究了重建单个实数向量和多个整数/实数向量的一般方法。它还研究了在雷达、多通道模数转换器(ADC)中矢量值信号的稳健恢复以及平面天线阵列在合成孔径雷达成像中的运动目标检测和估计等方面的新应用。该项目中关于MD-CRT和稳健MD-CRT的系统和更一般的结果可能会导致上述应用程序的性能改进。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。