Stable Calculation of Krawtchouk Functions from Triplet Relations

Brinker, A.C. den; Albertus, C

doi:10.3390/math9161972

Cited by 6 publications

(2 citation statements)

References 10 publications

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“…N ♢ with respect to the binomial distribution were introduced by Mykhailo Krawtchouk, hence the denomination Krawtchouk functions. They are also called normalized Krawtchouk polynomials [3], [22], [26], [27] in analogy with Hermite functions referring the fact that Krawtchouk polynomials are the discrete analogues of Hermite polynomials [1], [15], or weighted Krawtchouk polynomials [28], [29], [30], weighted and normalized Krawtchouk polynomials [31], [32], [33] or Krawtchouk functions [45].…”

Section: The Model Of Shift Invariant Image Recognition Of the Visual...mentioning

confidence: 99%

A method for determining the time-frequency characteristics of localized impulse components in continuous signals

Filimonova,

Specovius-Neugebauer,

Friedmann

2024

Preprint

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Background: We propose a method for analyzing the dynamics of signals of different nature. The method aims to find the time-frequency characteristics of the local impulse components for continuous signals. Thereby, we separate the information about the impulse shape from the time of its appearance. The method is based on a mathematical model of invariant pattern recognition by a visual system combined with a wavelet analysis method. Here we use Krawtchouk functions to model the response of the receptive fields of the lateral geniculate nucleus and, in addition, as the mother wavelet. Results: The inclusion of saccades in the model of the visual system allows us to extract invariant features of the signals. In addition, a parameter is included in the model that controls the front-to-back ratio of the basis functions. This means that the signal decomposition is not solely based on one particular set of basis functions, but on a family of basis functions parametrized by the shift transformation and an asymmetry parameter, which allows a better approximation of asymmetric impulse components. Conclusion: Our method makes it possible to study both the time of emergence of the impulse components of continuous signals and their spectral characteristics, which in turn can be used for their identification. Applying the method to biological signals of various nature, we could identify and localize a blinking, a muscle response due to emotions and find times of changes in a driver’s stress state, etc. Potentially, other applications of nonstationary signals with impulse components can also be considered.

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Section: The Model Of Shift Invariant Image Recognition Of the Visual...mentioning

confidence: 99%

A method for determining the time-frequency characteristics of localized impulse components in continuous signals

Filimonova,

Specovius-Neugebauer,

Friedmann

2024

Preprint

View full text Add to dashboard Cite

show abstract

“…OPs include several types, such as the Krawtchouk polynomial (KP) [18] and the Tchebichef polynomial (TP) [8]. The OP kernels of the KP and TP are used to construct the discrete Krawtchouk transform (DKT) and the discrete Tchebichef transform (DTT).…”

Section: Theoretical Background a Orthogonal Polynomialsmentioning

confidence: 99%

EEG Signal Classification Based on Orthogonal Polynomials, Sparse Filter and SVM Classifier

S. Radeaf,

Mohammed Z. Al-Faiz

2024

Iraqi Journal of ICT

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This work implements an Electroencephalogram (EEG) signal classifier. The implemented method uses Orthogonal Polynomials (OP) to convert the EEG signal samples to moments. A Sparse Filter (SF) reduces the number of converted moments to increase the classification accuracy. A Support Vector Machine (SVM) is used to classify the reduced moments between two classes. The proposed method’s performance is tested and compared with two methods by using two datasets. The datasets are divided into 80% for training and 20% for testing, with 5 -fold used for cross-validation. The results show that this method overcomes the accuracy of other methods. The proposed method’s best accuracy is 95.6% and 99.5%, respectively. Finally, from the results, it is obvious that the number of moments selected by the SP should exceed 30% of the overall EEG samples for accuracy to be over 90%.

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Fast Computation of Hahn Polynomials for High Order Moments

et al. 2022

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Discrete Hahn polynomials (DHPs) and their moments are considered to be one of the efficient orthogonal moments and they are applied in various scientific areas such as image processing and feature extraction. Commonly, DHPs are used as object representation; however, they suffer from the problem of numerical instability when the moment order becomes large. In this paper, an operative method to compute the Hahn orthogonal basis is proposed and applied to high orders. This paper developed a new mathematical model for computing the initial value of the DHP and for different values of DHP parameters (α and β). In addition, the proposed method is composed of two recurrence algorithms with an adaptive threshold to stabilize the generation of the DHP coefficients. It is compared with state-of-the-art algorithms in terms of computational cost and the maximum size that can be correctly generated. The experimental results show that the proposed algorithm performs better in both parameters for wide ranges of parameter values α and β, and polynomial sizes.INDEX TERMS Hahn polynomials, Hahn moments, propagation error, numerical error.

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Stable Calculation of Krawtchouk Functions from Triplet Relations

Cited by 6 publications

References 10 publications

A method for determining the time-frequency characteristics of localized impulse components in continuous signals

A method for determining the time-frequency characteristics of localized impulse components in continuous signals

EEG Signal Classification Based on Orthogonal Polynomials, Sparse Filter and SVM Classifier

Fast Computation of Hahn Polynomials for High Order Moments

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