CIF: Small: Multidimensional Remaindering Theory and Applications

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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.