Clustered sparsity and Poisson-gap sampling

Paweł Kasprzak,Mateusz Urbańczyk,Krzysztof Kazimierczuk

doi:10.1007/s10858-021-00385-7

Abstract

Non-uniform sampling (NUS) is a popular way of reducing the amount of time taken by multidimensional NMR experiments. Among the various non-uniform sampling schemes that exist, the Poisson-gap (PG) schedules are particularly popular, especially when combined with compressed-sensing (CS) reconstruction of missing data points. However, the use of PG is based mainly on practical experience and has not, as yet, been explained in terms of CS theory. Moreover, an apparent contradiction exists between the reported effectiveness of PG and CS theory, which states that a “flat” pseudo-random generator is the best way to generate sampling schedules in order to reconstruct sparse spectra. In this paper we explain how, and in what situations, PG reveals its superior features in NMR spectroscopy. We support our theoretical considerations with simulations and analyses of experimental data from the Biological Magnetic Resonance Bank (BMRB). Our analyses reveal a previously unnoticed feature of many NMR spectra that explains the success of ”blue-noise” schedules, such as PG. We call this feature “clustered sparsity”. This refers to the fact that the peaks in NMR spectra are not just sparse but often form clusters in the indirect dimension, and PG is particularly suited to deal with such situations. Additionally, we discuss why denser sampling in the initial and final parts of the clustered signal may be useful.

Highlights

The main limiting factor in multidimensional NMR spectroscopy is the need for extensive sampling of indirect time dimensions
The distance between sampling points is imposed by the Nyquist–Shannon sampling theorem (Nyquist 1928), and often thousands of sampling points are needed in order to achieve evolution times that provide the desired spectral resolution (Szántay 2008)
We show that PG, with sinusoidal gap modulation, is superior to both weighted and unweighted random non-uniform sampling (NUS) when the spectrum is not just sparse but reveals clustered sparsity, that is to say, significant spectral points form a closely-spaced group

Summary

Introduction

The main limiting factor in multidimensional NMR spectroscopy is the need for extensive sampling of indirect time dimensions. We show that PG, with sinusoidal gap modulation, is superior to both weighted and unweighted random NUS when the spectrum is not just sparse but reveals clustered sparsity, that is to say, significant spectral points form a closely-spaced group. The panels show the summed results of 10,000 sampling schemes (64 points from the grid of 256) obtained using different seeds from a pseudo-random number generator: a gap size distributions, b summed sampling schemes (note that all schedules always contain 0th increment), and c averaged absolute values of PSFs (only the bottom part of a PSF is shown).

Results

Conclusion