Hankel Matrix Nuclear Norm Regularized Tensor Completion for $N$-dimensional Exponential Signals
This addresses a bottleneck in spectroscopy for chemistry, biology, and medical imaging, where fast data acquisition leads to incomplete samples, and it is incremental as it extends recovery to higher dimensions (N≥3) using a hybrid method.
The paper tackles the problem of recovering N-dimensional exponential signals (N≥3) from partial observations, which existing methods cannot efficiently handle, by formulating it as a low-rank tensor completion problem with exponential factor vectors. The proposed approach successfully recovers full signals from very limited samples and is robust to estimated tensor rank, as shown in experiments on simulated and real magnetic resonance spectroscopy data.
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 $N\geq 3$. In this paper, we study the problem of recovering N-dimensional (particularly $N\geq 3$) 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.