基于矩阵分解的深度学习复指数信号重建

Complex Exponential Signal Reconstruction with Deep Matrix Factorization

答辩人：赵金奎

指导老师：屈小波 教授/博导

答辩委员会主席：董继扬 教授 博导 厦门大学电子科学系

答辩委员会成员：包立君 副教授 硕导 厦门大学电子科学系

                    林建忠 主任医师 硕导 厦门大学附属中山医院影像科

答辩秘书：王 新 高级工程师 厦门大学电子科学系

时间：2021年05月25日上午9:00

地点：厦门大学海韵园科研二-313

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摘要

背景：复指数信号是信号处理领域中最重要的基本信号类型之一，在通信、成像、电子系统及核磁共振等应用中都有用到。在实际应用中，受硬件条件等客观因素所限，复指数信号的采集过程往往需要花费较长时间。一种典型的加速采集的方式是先通过非均匀的欠采样来获取部分数据，再通过特定方法将欠采样的信号重建成全采样信号。因此，如何对欠采样复指数信号进行重建是学术界与工业界密切关注的前沿问题。

方法：传统的迭代重建方法通过引入信号的先验假设，利用信号的数学特性以完成目标任务，该过程可解释性强，且当信号特性符合预期时重建表现良好，但其重建过程往往耗时极长。而当前的深度学习复指数重建方法，虽然可以大大加速重建过程，但都是端到端的“黑箱子”学习网络，可解释性较弱，且信号尤其是低强度谱峰的重建误差仍需降低。我们以传统模型中的迭代重建方法——基于低秩汉克尔矩阵分解的重建方法为指导，利用复指数信号构成的汉克尔矩阵的低秩特性，搭建了性能优越的深度汉克尔矩阵分解网络（DHMF），将传统方法与深度学习的优势结合了起来。

结果：对网络中间结果的奇异值分析表明，所提方法能够很好地约束重建信号的汉克尔矩阵的秩，使得网络具有一定可解释性。实验表明，该网络相对于对比方法在仿真信号重建中有着更低的重建误差，在加速4倍采样的典型数据集下，该网络将重建的均方误差由0.137降低至0.069。此外，该网络能更好地保护重建的弱峰。最后，所提方法还被应用到实测的快速磁共振波谱的信号重建。实验表明，所提方法的重建结果相对于对比方法更加接近全采样信号。

结论：本文所提的DHMF将传统迭代优化算法与深度学习方法的优势结合，利用了复指数信号的低秩特性与深度学习基于大量数据的学习能力，具有更佳的重建效果。同时，作为首个对复指数信号的低秩性进行讨论的网络，DHMF提供了一种通过矩阵分解来逼近目标信号的秩的新思路，对于其他涉及低秩信号的任务也有着重要的指导意义。方法在二维磁共振波谱上进行了重建实验，验证了其的实用性。同时，两种方法都可以拓展到通讯中的雷达阵列，荧光显微镜图像，电子系统中的模拟数字信号转换等一系列复指数信号的应用场景。

关键词：复指数信号；深度学习；欠采样；汉克尔矩阵；低秩重建

ABSTRACT

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

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