Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules

Impact of different classification schemes on discrimination of proteins with noise-contaminated spectra using laboratory-measured fluorescence data

Biological agents are important to detect and identify with respect to environmental contamination and public health. Noise contamination in fluorescent spectra is one of the contributors to the uncertainties of identification. In order to investigate the noise-tolerant capability provided by laboratory-measured excitation-emission matrix (EEM) fluorescence spectra that are used as a database, fluorescence properties of four proteinaceous biotoxin samples and ten harmless protein samples were characterized by EEM fluorescence spectra, and the predicting performance of models trained by laboratory-measured fluorescence data was tested and verified from validation data with noise-contaminated spectra. By means of peak signal of noise (PSNR) as an indicator of noise levels, the potential impact of noise contaminations on the characterization and discrimination of these samples was evaluated quantitatively. Different classification schemes utilizing multivariate analysis techniques of Principal Component Analysis (PCA), Random Forest (RF), and Multi-layer Perceptron (MPL) coupled with feature descriptors of differential transform (DT), Fourier transform (FT) and wavelet transform (WT) were conducted under different PSNR values. We systematically analyzed the performance of classification schemes by the case study at 20 PSNR and by statistical analysis from 1-100 PSNR. The results show that the spectral features with EEM-WT decreased the demanding number of input variables while retaining high performances in sample classification. The spectral features with EEM-FT presented the worst performance although having the largest number of features. The distributions of feature importance and contribution were found sensitive to noise contaminations. The classification scheme of PCA prior to MPL with EEM-WT as input presented an improvement in lower PSNR. These results indicate that robust features extracted by corresponding techniques are critical to enhancing the spectral differentiation capabilities among these samples and play an important role in eliminating the noise effect. The study of classification schemes for discriminating protein samples with noise-contaminated spectra presents tremendous potential for future developments in the rapid detection and identification of proteinaceous biotoxins based on three-dimensional fluorescence spectrometry.

Implications and Consequences of SL(2R) as Invariance Group in the Description of Complex Systems Dynamics from a Multifractal Perspective of Motion

Possible implications and consequences of using SL(2R) as invariance groups in the description at any scale resolution of the dynamics of any complex system are analyzed. From this perspective and based on Jaynes’ remark (any circumstance left unspecified in the description of any complex system dynamics has the concrete expression in the existence of an invariance group), in the present paper one specifies such unspecified circumstances that result directly from the consideration of the canonical formalism induced by the SL(2R) as invariance group. It follows that both the Hamiltonian function and the Guassian distribution acquire the status of invariant group functions, the parameters that define the Hamiltonian acquire statistical significances based on a principle of maximizing informational energy, the class of statistical hypotheses specific to Gaussians of the same average acts as transitivity manifolds of the group (transitivity manifolds which can be correlated with the multifractal-non-multifractal scale transitions), joint invariant functions induced through SL(2R) groups isomorphism (the SL(2R) variables group, and the SL(2R) parameters group, etc.). For an ensemble of oscillators of the same frequency, the unspecified circumstances return to the ignorance of the amplitude and phase of each of the oscillators, which forces the recourse to a statistical ensemble traversed by the transformations of the Barbilian-type group. Finally, the model is validated based on numerical simulations and experimental results that refer to transient phenomena in ablation plasmas. The novelty of our model resides in the fact that fractalization through stochasticization is imposed through group invariance, situation in which the group’s transitivity manifolds can be correlated with the scale resolution.

Discrete two dimensional Fourier transform in polar coordinates part II: numerical computation and approximation of the continuous transform

The theory of the continuous two-dimensional (2D) Fourier Transform in polar coordinates has been recently developed but no discrete counterpart exists to date. In the first part of this two-paper series, we proposed and evaluated the theory of the 2D Discrete Fourier Transform (DFT) in polar coordinates. The theory of the actual manipulated quantities was shown, including the standard set of shift, modulation, multiplication, and convolution rules. In this second part of the series, we address the computational aspects of the 2D DFT in polar coordinates. Specifically, we demonstrate how the decomposition of the 2D DFT as a DFT, Discrete Hankel Transform and inverse DFT sequence can be exploited for coding. We also demonstrate how the proposed 2D DFT can be used to approximate the continuous forward and inverse Fourier transform in polar coordinates in the same manner that the 1D DFT can be used to approximate its continuous counterpart.