Hankel Matrix Nuclear Norm Regularized Tensor Completion for $N$-dimensional Exponential Signals

Abstract

Signals are generally modeled as a superposition of exponential functions in spectroscopy of chemistry, biology and medical imaging. For fast data acquisition or other inevitable reasons, however, only a small amount of samples may be acquired and thus how to recover the full signal becomes an active research topic. But existing approaches can not efficiently recover N -dimensional exponential signals with N ≥ 3. In this paper, we study the problem of recovering N -dimensional (particularly N ≥ 3) exponential signals from partial observations, and formulate this problem as a low-rank tensor completion problem with exponential factor vectors. The full signal is reconstructed by simultaneously exploiting the CANDECOMP/PARAFAC structure and the exponential structure of the associated factor vectors. The latter is promoted by minimizing an objective function involving the nuclear norm of Hankel matrices. Experimental results on simulated and real magnetic resonance spectroscopy data show that the proposed approach can successfully recover full signals from very limited samples and is robust to the estimated tensor rank.

Image reconstruction with low-rankness and self-consistency of k-space data in parallel MRI

“…Although a singular value decompositionfree algorithm is derived to reduce the computation load, faster algorithms are still expected in the future [28], [30]. Besides, the low rank Hankel matrix is highly related to exponential functions, and latest reconstruction methods of multi-dimensional exponentials [46], [47] may improve the Hankel-based MRI reconstruction or be extended into higher dimensional MRI in fast imaging.…”

Compressed sensing: Reconstruction of non‐uniformly sampled multidimensional NMR data

“…60,61 The low-rank approach attempts to reconstruct a spectrum with the fewest peaks, compared to CS which minimizes the number of nonzero values, and is independent of the line widths of the peaks. 60,61 The low-rank approach attempts to reconstruct a spectrum with the fewest peaks, compared to CS which minimizes the number of nonzero values, and is independent of the line widths of the peaks.…”

“…More recently, low-rank reconstruction has been proposed as a high-fidelity algorithm suitable for reconstructing NMR spectra, in particular for low intensity, broad peaks. 60,61 The low-rank approach attempts to reconstruct a spectrum with the fewest peaks, compared to CS which minimizes the number of nonzero values, and is independent of the line widths of the peaks. Low-rank reconstruction solves the following equation:…”

“…23 However, to date, available implementations of the low rank method are slower than other CS algorithms and limited to 2D data, although more recent algorithms have demonstrated extension of this approach to higher dimensional spectra ≥ 3D. 61…”

Large scale 2D spectral compressed sensing in continuous domain

“…Despite the 1D case, there are more applications involving the recovery of d-dimensional signals (d ≥ 2). For example, super-resolution imaging [9] and nuclear magnetic resonance (NMR) spectroscopy [10]. Chi and Chen [11,12] extended Tang's approach to 2D frequency models.…”