Reaching the sparse-sampling limit for reconstructing a single peak in a 2D NMR spectrum using iterated maps

Non-Uniform Sampling in NMR Spectroscopy and the Preservation of Spectral Knowledge in the Time and Frequency Domains

“…Generating the frequency-domain spectrum from the time-domain FID is straightforward for US using the DFT. It is more complicated for NUS sampled data, and considerable effort is currently underway to develop robust methods to accomplish this. ,,− Here, we consider two approaches that are applicable to dense, non-uniformly sampled time-domain data: direct FT of the NUWS FID acquired on the Nyquist grid and BFT of NUS data acquired off the Nyquist grid. We demonstrate how with appropriately modified versions of each the gains in intrinsic spectral knowledge realized with NUS in the time domain can be rigorously retained in the construction of the frequency-domain spectrum.…”

“…One challenge when working with data acquired under NUS is that direct Fourier transformation of the time-domain data to generate the frequency-domain spectrum is not always possible. A vast array of post-Fourier processing techniques such as maximum entropy reconstruction, , forward Max Ent (FM), maximum entropy interpolation (MINT), , iterative soft thresholding (IST/RIST/hmsIST), , SMILE, NESTA, FFT-CLEAN, SCRUB, DiffMap, MDD, and NUS-trained deep neural networks have been developed but lack the fundamental correspondence between the time and frequency domains enshrined by the FT. Most disconcerting, perhaps, is that the spectral reconstruction process is typically nonlinear; so the signal and the noise may not both be faithfully reproduced in the frequency domain.…”