Seed-independent Nonuniform Sampling A general framework for sparse sampling

# Introduction

## Gap sampling

It is becoming more and more common for NMR spectroscopists to nonuniformly sparsely subsample their signals in the time domain, a technique referred to as Nonuniform Sampling, or NUS. While I was playing around with one of the more widely used sampling methods, called Poisson-gap, I ended up describing a highly general multidimensional gap sampling framework. Under gap sampling, each sampled data point is determined from the previously sampled data point in the current trace by a simple recurrence relation:

where $g(x_i)$ is referred to as a ‘gap equation’. Multidimensional gap sampling schedules are built up recursively from lower-dimensional sub-schedules according to the above relation. The popular Poisson-gap method obeys a stochastic gap equation by drawing from a Poisson distribution at each evalution of its gap equation. The related sine-gap and sine-burst methods, developed in parallel with the framework, obey deterministic gap equations and require no pseudorandom number generators.

The generality of the gap sampling framework was so tempting that I decided to write a sampling program that could use any user-supplied gap equation. The gaputil program was the result, and wraps the Julia mathematical programming language to interpret almost any conceivable gap equation during schedule generation.

## Density sampling

Following on the heels of gap sampling, I decided to investigate if probability density-based sampling methods could be stripped of their pseudorandom seed dependence. Indeed, using Sub-random number sequences, it was possible.

In fact, sub-random numbers could be used to augment the deterministic gap equations for added incoherence in the resulting NUS schedules. The rejutil and jitutil programs were the fruits of this labor, and the gaputil program was also expanded to admit sub-random numbers into its deterministic gap equations.

# Selected results

To show that seed-independent sampling can produce schedules that perform similarly to stochastic methods, a Monte Carlo simulation was run. An ensemble of Poisson-gap samples was constructed. Synthetic and real time-domain data were then subsampled using both the ensemble of Poisson-gap schedules and the gap-framework schedules, and residuals from Iterative Soft Thresholding (IST) reconstruction were computed and compared.

The results were interesting. Sine-gap and sine-burst sampling using the general framework yielded residuals that were comparable or better to Poisson-gap sampling. This is a first for seed-independent sampling methods, which have until now been regarded as inferior to pseudorandomly sampled methods.

# The code

The nusutils programs are all fairly portable C, but require Julia for equation interpretation. They’re hosted on GitHub:

geekysuavo.github.io