Exponential Signal Reconstruction With Deep Hankel Matrix Factorization

Exponential function is a basic form of temporal signals, and how to fast acquire this signal is one of the fundamental problems and frontiers in signal processing. To achieve this goal, partial data may be acquired but result in severe artifacts in its spectrum, which is the Fourier transform of exponentials. Thus, reliable spectrum reconstruction is highly expected in the fast data acquisition in many applications, such as chemistry, biology, and medical imaging. In this work, we propose a deep learning method whose neural network structure is designed by imitating the iterative process in the model-based state-of-the-art exponentials’ reconstruction method with the low-rank Hankel matrix factorization. With the experiments on synthetic data and realistic biological magnetic resonance signals, we demonstrate that the new method yields much lower reconstruction errors and preserves the low-intensity signals much better than compared methods.

I. Introduction

The exponential function is a basic signal form in signal processing. In many practical applications, signals can be approximated by the superposition of a few exponential functions. Examples include antenna signals in telecommunication [1]–[3], images in fluorescence microscopy [4], analog-to-digital conversion in electronic systems [5], signal functions in the theory of the finite rate of innovation [6], planar polygon recovering [7], and time-domain signals in biological nuclear magnetic resonance (NMR) spectroscopy [8]–[14]. Thus, achieving high-quality recovery of the exponential signals has great significance. The signal of interest is modeled as the sum of exponentials as follows: $$\begin{equation*} x(n\Delta t)=\sum _{j=1}^{J} {\left ({A_{j} e^{i\phi _{j}}}\right)e^{-\frac {n\Delta t}{\tau _{j}}}e^{i2\pi f_{j} n\Delta t}}\tag{1}\end{equation*}$$ where ) is the normalized frequency, ) is the phase, is the amplitude, is the damping factor, is the number of exponentials, , is the data acquisition (also called sampling) interval, and is the index of fully sampled data points in the time domain with . Accordingly, the sampled data can be represented with a vector . By performing the Fourier transform on , one can obtain a full spectrum .