Robust phase unwrapping algorithm based on Zernike polynomial fitting and Swin-Transformer network

Phase unwrapping plays an important role in optical phase measurements. In particular, phase unwrapping under heavy noise conditions remains an open issue. In this paper, a deep learning-based method is proposed to conduct the phase unwrapping task by combining Zernike polynomial fitting and a Swin-Transformer network. In this proposed method, phase unwrapping is regarded as a regression problem, and the Swin-Transformer network is used to map the relationship between the wrapped phase data and the Zernike polynomial coefficients. Because of the self-attention mechanism of the transformer network, the fitting coefficients can be estimated accurately even under extremely harsh noise conditions. Simulation and experimental results are presented to demonstrate the outperformance of the proposed method over the other two polynomial fitting-based methods. This is a promising phase unwrapping method in optical metrology, especially in electronic speckle pattern interferometry.

Diagnosing Extrusion Process Based on Displacement Signal and Simple Decision Tree Classifier

The article presents an extensive analysis of the literature related to the diagnosis of the extrusion process and proposes a new, unique method. This method is based on the observation of the punch displacement signal in relation to the die, and then approximation of this signal using a polynomial. It is difficult to find in the literature even an attempt to solve the problem of diagnosing the extrusion process by means of a simple distance measurement. The dominant feature is the use of strain gauges, force sensors or even accelerometers. However, the authors managed to use the displacement signal, and it was considered a key element of the method presented in the article. The aim of the authors was to propose an effective method, simple to implement and not requiring high computing power, with the possibility of acting and making decisions in real time. At the input of the classifier, authors provided the determined polynomial coefficients and the SSE (Sum of Squared Errors) value. Based on the SSE values only, the decision tree algorithm performed anomaly detection with an accuracy of 98.36%. With regard to the duration of the experiment (single extrusion process), the decision was made after 0.44 s, which is on average 26.7% of the extrusion experiment duration. The article describes in detail the method and the results achieved.

Prediction of System Parameters of Carbon-Based Composite Structure for Different Carbon Fiber Orientations with Mode Information at Reference Angle Only

The prediction of system parameters is important for understanding the dynamic behavior of composite structures or selecting the configuration of laminated carbon in carbon-based composite (CBC) structures. The dynamic nature of CBC structures allows the representation of system parameters as modal parameters in the frequency domain, where all modal parameters depend on the carbon fiber orientations. In this study, the variation in the system parameters of a carbon fiber was derived from equivalent modal parameters, and the system parameters at a certain carbon fiber orientation were predicted using the modal information at the reference carbon fiber orientation only and a representative curve-fitted function. The target CBC structure was selected as a simple rectangular structure with five different carbon fiber orientations, and the modal parameters were formulated based on a previous study for all modes. Second-order curve-fitted polynomial functions were derived for all possible cases, and representative curve-fitting functions were derived by averaging the polynomial coefficients. The two system parameters were successfully predicted using the representative curve-fitting function and the modal information at only the reference carbon fiber orientation, and the feasibility of parameter prediction was discussed based on an analysis of the error between the measured and predicted parameters.

An observer-based output tracking controller for electrically driven cooperative multiple manipulators with adaptive Bernstein-type approximator

Thisarticle presents an observer-based output tracking control method for electrically actuated cooperative multiple manipulators using Bernstein-type operators as a universal approximator. This efficient mathematical tool represents lumped uncertainty, including external perturbations and unmodeled dynamics. Then, adaptive laws are derived through the stability analysis to tune the polynomial coefficients. It is confirmed that all the position and force tracking errors are uniformly ultimately bounded using the Lyapunov stability theorem. The theoretical achievements are validated by applying the proposed observer-based controller to a cooperative robotic system comprised of two manipulators transporting a rigid object. The outcomes of the introduced method are also compared to RBFNN, which is a powerful state-of-the-art approximator. The results demonstrate the efficacy of the introduced adaptive control approach in controlling the system even in the presence of disturbances and uncertainties.

Extended of TEA: A 256 bits block cipher algorithm for image encryption

<p>This paper introduces an effective image encryption approach that merges a chaotic map and polynomial with a block cipher. According to this scheme, there are three levels of encryption. In the first level, pixel positions of the image are scuffled into blocks randomly based on a chaotic map. In the second level, the polynomials are constructed by taking N unused pixels from the permuted blocks as polynomial coefficients. Finally, the third level a proposed secret-key block cipher called extended of tiny encryption algorithm (ETEA) is used. The proposed ETEA algorithm increased the block size from 64-bit to 256-bit by using F-function in type three Feistel network design. The key schedule generation is very straightforward through admixture the entire major subjects in the identical manner for every round. The proposed ETEA algorithm is word-oriented, where wholly internal operations are executed on words of 32 bits. So, it is possible to efficiently implement the proposed algorithm on smart cards. The results of the experimental demonstration that the proposed encryption algorithm for all methods are efficient and have high security features through statistical analysis using histograms, correlation, entropy, randomness tests, and the avalanche effect.</p>

Robust adaptive controller–observer scheme for robot manipulators: a Bernstein–Stancu approach

This article introduces a robust adaptive controller–observer structure for robotic manipulators such that the need for joints speed measurement is removed. Besides, it is presumed that the system model has uncertainties and is subject to disturbances, and the proposed method must eliminate the impact of these factors on the system response. According to this, for the first time in the robotics field, a model-free scheme is developed based on the Bernstein–Stancu polynomial. The universal approximation property of the Bernstein–Stancu polynomial enables it to accurately estimate the lumped uncertainty, including unmodeled dynamics and disturbances. Moreover, to increase the efficiency of the controller–observer structure, adaptive rules have been proposed to update polynomial coefficients. The boundedness of all system errors is proven using the Lyapunov theorem. Finally, the proposed robust Adaptive controller–observer is applied on a planer robot, and the results are precisely analyzed. The results of the proposed approach are also compared with two state-of-art powerful approximation methods.

On Sobolev bilinear forms and polynomial solutions of second-order differential equations

Given a linear second-order differential operator $${\mathcal {L}}\equiv \phi \,D^2+\psi \,D$$ L ≡ ϕ D 2 + ψ D with non zero polynomial coefficients of degree at most 2, a sequence of real numbers $$\lambda _n$$ λ n , $$n\geqslant 0$$ n ⩾ 0 , and a Sobolev bilinear form $$\begin{aligned} {\mathcal {B}}(p,q)\,=\,\sum _{k=0}^N\left\langle {{\mathbf {u}}_k,\,p^{(k)}\,q^{(k)}}\right\rangle , \quad N\geqslant 0, \end{aligned}$$ B ( p , q ) = ∑ k = 0 N u k , p ( k ) q ( k ) , N ⩾ 0 , where $${\mathbf {u}}_k$$ u k , $$0\leqslant k \leqslant N$$ 0 ⩽ k ⩽ N , are linear functionals defined on polynomials, we study the orthogonality of the polynomial solutions of the differential equation $${\mathcal {L}}[y]=\lambda _n\,y$$ L [ y ] = λ n y with respect to $${\mathcal {B}}$$ B . We show that such polynomials are orthogonal with respect to $${\mathcal {B}}$$ B if the Pearson equations $$D(\phi \,{\mathbf {u}}_k)=(\psi +k\,\phi ')\,{\mathbf {u}}_k$$ D ( ϕ u k ) = ( ψ + k ϕ ′ ) u k , $$0\leqslant k \leqslant N$$ 0 ⩽ k ⩽ N , are satisfied by the linear functionals in the bilinear form. Moreover, we use our results as a general method to deduce the Sobolev orthogonality for polynomial solutions of differential equations associated with classical orthogonal polynomials with negative integer parameters.

A combined methodology for the approximate estimation of the roots of the general sextic polynomial equation

In this article we present the methodology, according to which it is possible to derive approximate solutions for the roots of the general sextic polynomial equation as well as some other forms of sextic polynomial equations that normally cannot be solved by radicals; the approximate roots can be expressed in terms of polynomial coefficients. This methodology is a combination of two methods. The first part of the procedure pertains to the reduction of a general sextic equation H(x) to a depressed equation G(y), followed by the determination of solutions by radicals of G(y) which does not include a quintic term, provided that the fixed term of the equation depends on its other coefficients. The second method is a continuation of the first and pertains to the numerical correlation of the roots and the fixed term of a given sextic polynomial P(x) with the radicals and the fixed term of the sextic polynomial Q(x), where the two polynomials P(x) and Q(x) have the same coefficients except for the fixed term which might be different. From the application of the methodology presented above, the following formulation is derived; For any given general sextic polynomial equation P with coefficients within the interval [a, b], a defined polynomial equation Q corresponds which has equal coefficients to P except for its fixed term which might be different and dependent on the other coefficients so that Q has radical solutions. If we assume a pair of equations P, Q with coefficients within a predetermined interval [a, b], the numerical correlation through regression analysis of the radicals of Q, the roots of P and the fixed terms of P, Q, leads to the derivation of a mathematical model for the approximate estimation of the roots of sextic equations whose coefficients belong to the interval [a, b].

A combined methodology for the approximate estimation of the roots of the general sextic polynomial equation

In this article we present the methodology, according to which it is possible to derive approximate solutions for the roots of the general sextic polynomial equation as well as some other forms of sextic polynomial equations that normally cannot be solved by radicals; the approximate roots can be expressed in terms of polynomial coefficients. This methodology is a combination of two methods. The first part of the procedure pertains to the reduction of a general sextic equation H(x) to a depressed equation G(y), followed by the determination of solutions by radicals of G(y) which does not include a quintic term, provided that the fixed term of the equation depends on its other coefficients. The second method is a continuation of the first and pertains to the numerical correlation of the roots and the fixed term of a given sextic polynomial P(x) with the radicals and the fixed term of the sextic polynomial Q(x), where the two polynomials P(x) and Q(x) have the same coefficients except for the fixed term which might be different. From the application of the methodology presented above, the following formulation is derived; For any given general sextic polynomial equation P with coefficients within the interval [a, b], a defined polynomial equation Q corresponds which has equal coefficients to P except for its fixed term which might be different and dependent on the other coefficients so that Q has radical solutions. If we assume a pair of equations P, Q with coefficients within a predetermined interval [a, b], the numerical correlation through regression analysis of the radicals of Q, the roots of P and the fixed terms of P, Q, leads to the derivation of a mathematical model for the approximate estimation of the roots of sextic equations whose coefficients belong to the interval [a, b].