Complex Exponential Signal Reconstruction with Deep Matrix Factorization
Respondent： Jinkui Zhao
Supervisor： Xiaobo Qu Professor/Doctoral supervisor
Chairman of the Reply Committee：
Jiyang Dong, Professor Doctora/supervisor, Department of Electronic Science, Xiamen University
Member of the Defense Committee：
Lijun Bao, Associate Professor/Master supervisor, Department of Electronic Science, Xiamen University
Jianzhong Lin, Chief Physician/Master supervisor, Department of Imaging, Zhongshan Hospital, Xiamen University
Secretary of Defense:
Xin Wang, Senior Engineer, Department of Electronic Science
Time：May 25, 2021 at 9:00 AM
Place：Xiamen University Haiyun Park Research-II Building 313
Objective: Complex exponential function is a fundamental signal form in signal processing. Representative application fields of exponential signals include communication, imaging, electronic system and Nuclear Magnetic Resonance (NMR) spectroscopy. However, due to the objective limitations such as hardware conditions, the acquisition process of complex exponential signal in application is time consuming. A typical way to speed up the acquisition is to obtain partial data with non-uniform sampling, and then reconstruct the under sampled signal. Therefore, how to reconstruct under sampled complex exponentials is one of the fundamental problems and frontiers in signal processing that the academia and industry pay close attention to.
Methods: The traditional iterative reconstruction methods introduce the prior of the signal to complete the target task. The process is highly interpretable and can perform well when the signal characteristics meet the expectation. However, its reconstruction process is often time-consuming. Although the current deep learning complex exponential reconstruction methods can greatly accelerate the reconstruction process, they are all end-to-end "black box" learning networks with weak interpretability. And the reconstruction error of signals, especially on low-intensity spectral peaks, still needs to be reduced. By referring to the traditional iterative reconstruction method, low rank Hankel matrix factorization, we build a deep Hankel matrix factorization network (DHMF) by using the low rank property of Hankel matrix generated from complex exponential signals, which combines the advantages of traditional methods and deep learning together.
Results: Analysis on singular values of the intermediate results of the network shows that the proposed method can well constrain the rank of the Hankel matrix, making the network more interpretable. Experiments show that the network yields much lower reconstruction errors, the mean square error of the proposed method was reduced from 0.137 to 0.069 on a typical dataset when the acceleration factor is 4. In addition, the network can preserve the low-intensity signals better. Last but not least, the method is also applied to the reconstruction of fast NMR spectrum. Experimental results show that the reconstruction results of proposed method is closer to the fully sampled signal than the compared methods.
Conclusion: The proposed DHMF combines the advantages of traditional iterative algorithms and deep learning methods, making use of the low rank characteristics of complex exponential signal and the learning ability of deep learning based on large dataset, and has lower reconstruction error. At the same time, as the first network to discuss the low rank property of complex exponential signal, DHMF provides a new idea to approach the rank of target signal through matrix decomposition, which also has important guiding significance for other tasks involving low rank signal. Experiments on two-dimensional magnetic resonance spectroscopy also verify its practicability. Moreover, the method can be extended to a series of complex exponential signal application scenarios, such as radar array in communication, fluorescence microscope image, analog-to-digital signal conversion in electronic system, etc.
Key words: complex exponential function; deep learning; under sampled; Hankel matrix; low-rank reconstruction