Plant cell wall profiling by fast maximum likelihood reconstruction (FMLR) and region-of-interest (ROI) segmentation of solution-state 2D 1H–13C NMR spectra

Table 2 Basis functions and parameters used in FMLR

Basis functions
Name	Type	Expression	Derivative	Usage
Sinusoid	Complex	e ^iωt	ite ^iωt	Always
Damping function	Real	$e^{- a t^{η}}$	$- t^{η} e^{- a t^{η}}$	Used except along indirect dimensions of constant time experiments
Parameters
Parameter	Symbol	Variable	Basis function	Initial value	Constrained
Frequency	Ω	Yes	Sinusoid	From peak position	No
Decay Rate	Α	Yes	Damping function	From “prototype” signal	Yes
Decay Power	Η	No	Damping function	Assigned based on profiling of data sets. Fixed per analysis on single data set	No (fixed)

The time domain basis functions along each model dimension are the complex product of a sinusoid basis function with a damping function. The corresponding frequency domain functions are obtained from Fourier transformation using a digital operator derived from the acquisition and processing parameters. Multidimensional basis functions are derived from the product of the orthogonal component basis functions along each dimension. For gradient-based (non-linear) optimization of the parameters, the derivative basis functions are used. The exponent η appearing in the decay rate term is a value that modulates the signal between a Lorentzian (η = 1) and Gaussian (η = 2) decay profile. This value is adjusted to fit a similar class of peak shapes and is left constant throughout the optimization of any given data set. The corresponding derivative basis functions are used to calculate the derivative of the basis function with respect to the angular frequency (sinusoid) and decay rate (damping function).

ISSN: 2731-3654