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Table 2 Basis functions and parameters used in FMLR

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

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  
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)
  1. 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).