Statistical characterization of residual noise in the low-rank approximation filter framework, general theory and application to hyperpolarized tracer spectroscopy

TL;DR

This paper provides a statistical analysis of residual noise in low-rank approximation filters used in NMR spectroscopy, deriving their distribution, and demonstrating how to optimize signal estimation with a maximum likelihood approach.

Contribution

It introduces a theoretical framework for understanding residual noise in low-rank filters and applies it to improve signal estimation in hyperpolarized tracer spectroscopy.

Findings

Derived mean and covariance of residual noise

Developed a maximum likelihood estimator for signal recovery

Validated approach with Monte Carlo simulations

Abstract

The use of low-rank approximation filters in the field of NMR is increasing due to their flexibility and effectiveness. Despite their ability to reduce the Mean Square Error between the processed signal and the true signal is well known, the statistical distribution of the residual noise is still undescribed. In this article, we show that low-rank approximation filters are equivalent to linear filters, and we calculate the mean and the covariance matrix of the processed data. We also show how to use this knowledge to build a maximum likelihood estimator, and we test the estimator's performance with a Montecarlo simulation of a 13C pyruvate metabolic tracer. While the article focuses on NMR spectroscopy experiment with hyperpolarized tracer, we also show that the results can be applied to tensorial data (e.g. using HOSVD) or 1D data (e.g. Cadzow filter).